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Probability Abstracts 101
This document contains abstracts 6228-6510 from
November-1-2007 to December-31-2007.
They have been mailed on January 4th, 2008.
Author(s): Magda Peligrad and Sunder Sethuraman
Abstract: A well-known result with respect to the one dimensional nearest-
neighbor
symmetric simple exclusion process is the convergence to fractional
Brownian
motion with Hurst parameter 1/4, in the sense of finite-dimensional
distributions, of the subdiffusively rescaled current across the
origin, and
the subdiffusively rescaled tagged particle position.
The purpose of this note is to improve this convergence to a
functional
central limit theorem, with respect to the uniform topology, and so
complete
the solution to a conjecture in the literature with respect to simple
exclusion
processes.
http://arxiv.org/abs/0711.0017
Author(s): Erwin Bolthausen and Francesco Caravenna and B\'eatrice de Tili \`ere
Abstract: We consider a model for a polymer interacting with an attractive
wall through
a random sequence of charges. We focus on the so-called diluted
limit, when the
charges are very rare but have strong intensity. In this regime, we
determine
the quenched critical point of the model, showing that it is
different from the
annealed one. The proof is based on a rigorous renormalization
procedure.
Applications of our results to the problem of a copolymer near a
selective
interface are discussed.
http://arxiv.org/abs/0711.0141
Author(s): Peter Friz and Harald Oberhauser
Abstract: Optimal sample path properties of stochastic processes often involve
generalized H\"{o}lder- or variation norms. Following a classical
result of
Taylor, the exact variation of Brownian motion is measured in terms
of $\psi
(x) \equiv $ $x^{2}/\log \log (1/x) $ near $0+$. Such $\psi $-
variation results
extend to classes of processes with values in abstract metric spaces.
(No
Gaussian or Markovian properties are assumed.) To establish
integrability
properties of the $\psi $-variation we turn to a large class of
Gaussian rough
paths (e.g. Brownian motion and L\'{e}vy's area viewed as a process
in a Lie
group) and prove Gaussian integrability properties using Borell's
inequality on
abstract Wiener spaces. The interest in such results is that they are
compatible with rough path theory and yield certain sharp regularity and
integrability properties (for iterated Stratonovich integrals, for
example)
which would be difficult to obtain otherwise. At last, $\psi $-
variation is
identified as robust regularity property of solutions to (random) rough
differential equations beyond semimartingales.
http://arxiv.org/abs/0711.0163
Author(s): Christian L\'eonard (MODAL'x and Cmap)
Abstract: Generalized entropic projections and dominating points are
solutions to
convex minimization problems related to conditional laws of large
numbers. They
appear in many areas of applied mathematics such as statistical physics,
information theory, mathematical statistics, ill-posed inverse
problems or
large deviation theory. By means of convex conjugate duality and
functional
analysis, criteria are derived for their existence. Representations
of the
generalized entropic projections are obtained: they are the ``measure
component" of some extended entropy minimization problem.
http://arxiv.org/abs/0711.0206
Author(s): Claudio Albanese
Abstract: We are interested in the kernel of one-dimensional diffusion
equations with
continuous coefficients as evaluated by means of explicit discretization
schemes of uniform step $h>0$ in the limit as $h\to0$. We consider both
semidiscrete triangulations with continuous time and explicit Euler
schemes
with time step small enough for the method to be stable. We find
sharp uniform
bounds for the convergence rate as a function of the degree of
smoothness which
we conjecture. The bounds also apply to the time derivative of the
kernel and
its first two space derivatives. Our proof is constructive and is
based on a
new technique of path conditioning for Markov chains and a
renormalization
group argument. Convergence rates depend on the degree of smoothness and
H\"older differentiability of the coefficients. We find that the fastest
convergence rate is of order $O(h^2)$ and is achieved if the
coefficients have
a bounded second derivative. Otherwise, explicit schemes still
converge for any
degree of H\"older differentiability except that the convergence rate is
slower. H\"older continuity itself is not strictly necessary and can
be relaxed
by an hypothesis of uniform continuity.
http://arxiv.org/abs/0711.0132
Author(s): Tanja Gernhard and Dennis Wong
Abstract: We calculate the density and expectation for the number of
lineages in a
reconstructed tree with $n$ extant species. This is done with
conditioning on
the age of the tree as well as with assuming a uniform prior for the
age of the
tree.
http://arxiv.org/abs/0711.0269
Author(s): Johan Tykesson
Abstract: We consider the Poisson Boolean model of continuum percolation on a
homogeneous Riemannian manifold $M$. Let $lambda$ be intensity of the
Poisson
process in the model and let $lambda_u$ be the infimum of the set of
intensities that a.s. produce a unique unbounded component. We show
that above
$\lambda_u$ there is a.s. a unique unbounded component. We also study
what
happens at $\lambda_u$ for some spaces. In particular, if $M$ is the
product of
the hyperbolic disc and the real line, then at $\lambda_u$ there is
a.s. not a
unique unbounded component. The results are inspired by results for
Bernoulli
bond percolation on graphs due to Haggstrom, Peres and Schonmann.
http://arxiv.org/abs/0711.0307
Author(s): G. Morvai and B. Weiss
Abstract: Let $\{X_n\}_{n=0}^{\infty}$ be a stationary real-valued time
series with
unknown distribution. Our goal is to estimate the conditional
expectation of
$X_{n+1}$ based on the observations $X_i$, $0\le i\le n$ in a strongly
consistent way. Bailey and Ryabko proved that this is not possible
even for
ergodic binary time series if one estimates at all values of $n$. We
propose a
very simple algorithm which will make prediction infinitely often at
carefully
selected stopping times chosen by our rule. We show that under certain
conditions our procedure is strongly (pointwise) consistent, and $L_2$
consistent without any condition. An upper bound on the growth of the
stopping
times is also presented in this paper.
http://arxiv.org/abs/0711.0350
Author(s): G. Morvai and S. Yakowitz and and L. Gyorfi
Abstract: The setting is a stationary, ergodic time series. The challenge is to
construct a sequence of functions, each based on only finite segments
of the
past, which together provide a strongly consistent estimator for the
conditional probability of the next observation, given the infinite
past.
Ornstein gave such a construction for the case that the values are
from a
finite set, and recently Algoet extended the scheme to time series with
coordinates in a Polish space.
The present study relates a different solution to the challenge. The
algorithm is simple and its verification is fairly transparent. Some
extensions
to regression, pattern recognition, and on-line forecasting are
mentioned.
http://arxiv.org/abs/0711.0367
Author(s): Eric Schmutz
Abstract: For random maps, the expected value of the order (i.e. the period
of the
sequence of compositional iterates) is approximated asymptotically.
It is much
smaller than the expected value for the product of the cycle lengths.
http://arxiv.org/abs/0711.0312
Author(s): G. Morvai and B. Weiss
Abstract: Let $\{X_n\}$ be a stationary and ergodic time series taking
values from a
finite or countably infinite set ${\cal X}$. Assume that the
distribution of
the process is otherwise unknown. We propose a sequence of stopping
times
$\lambda_n$ along which we will be able to estimate the conditional
probability
$P(X_{\lambda_n+1}=x|X_0,...,X_{\lambda_n})$ from data segment
$(X_0,...,X_{\lambda_n})$ in a pointwise consistent way for a
restricted class
of stationary and ergodic finite or countably infinite alphabet time
series
which includes among others all stationary and ergodic finitarily
Markovian
processes. If the stationary and ergodic process turns out to be
finitarily
Markovian (among others, all stationary and ergodic Markov chains are
included
in this class) then $ \lim_{n\to \infty} {n\over \lambda_n}>0$ almost
surely.
If the stationary and ergodic process turns out to possess finite
entropy rate
then $\lambda_n$ is upperbounded by a polynomial, eventually almost
surely.
http://arxiv.org/abs/0711.0471
Author(s): G. Morvai and B. Weiss
Abstract: We describe estimators $\chi_n(X_0,X_1,...,X_n)$, which when
applied to an
unknown stationary process taking values from a countable alphabet $
{\cal X}$,
converge almost surely to $k$ in case the process is a $k$-th order
Markov
chain and to infinity otherwise.
http://arxiv.org/abs/0711.0472
Author(s): Krzysztof Burdzy and David White
Abstract: We study a class of Markov processes with finite state space and
continuous
time that have product form stationary distributions. We obtain a
number of
examples that can generate conjectures for diffusions with inert drift.
http://arxiv.org/abs/0711.0493
Author(s): Sourav Chatterjee
Abstract: We give a new proof of the Komlos-Major-Tusnady embedding theorem
for the
simple random walk. The only external tool that we use is the
Schauder-Tychonoff fixed point theorem for locally convex spaces.
Besides that,
the proof is almost entirely based on a series of soft arguments and
easy
inequalities, and no hard computations (implicit or explicit) are
involved.
This provides the first genuine alternative to the quantile transform
and the
Hungarian construction.
http://arxiv.org/abs/0711.0501
Author(s): Boris L.Granovsky and Michael M. Erlihson
Abstract: We establish a characterization of coagulation-fragmentation
processes, such
that the induced birth and death processes depicting the total number
of groups
at time $t\ge 0$ are Markov and time homogeneous. Based on this, we
provide a
characterization of Gibbs coagulation-fragmentation models, which
extends the
one derived by Hendriks et al. As a by- product of our results, the
class of
solvable models is widened and two questions posed by N. Berestycki
and Pitman
are answered.
http://arxiv.org/abs/0711.0503
Author(s): Lorenzo Zambotti
Abstract: We consider an effective interface model on a hard wall in (1+1)
dimensions,
with conservation of the area between the interface and the wall. We
prove that
the equilibrium fluctuations of the height variable converge in law
to the
solution of a SPDE with reflection and conservation of the space
average. The
proof is based on recent results obtained with L. Ambrosio and G.
Savare on
stability properties of Markov processes with log-concave invariant
measures.
http://arxiv.org/abs/0711.0583
Author(s): Andrew Neate and Aubrey Truman
Abstract: We summarise a selection of results on the inviscid limit of the
stochastic
Burgers equation emphasising geometric properties of the caustic,
Maxwell set
and Hamilton-Jacobi level surfaces and relating these results to a
discussion
of stochastic turbulence. We show that for small viscosities there
exists a
vortex filament structure near to the Maxwell set. We discuss how this
vorticity is directly related to the adhesion model for the evolution
of the
early universe and include new explicit formulas for the distribution
of mass
within the shock.
http://arxiv.org/abs/0711.0617
Author(s): Balint Toth and Balint Veto
Abstract: Let B(t), X(t) and Y(t) be independent standard 1d Borwnian
motions. Define
X^+(t) and Y^-(t) as the trajectories of the processes X(t) and Y(t)
pushed
upwards and, respectively, downwards by B(t), according to Skorohod-
reflection.
In a recent paper, Jon Warren proves inter alia that Z(t):= X^+(t)-Y^-
(t) is a
three-dimensional Bessel-process. In this note, we present an
alternative,
elementary proof of this fact.
http://arxiv.org/abs/0711.0631
Author(s): Rami Atar and Amarjit Budhiraja and Ruth J. Williams
Abstract: Given a closed, bounded convex set $\mathcal{W}\subset{\mathbb {R}}
^d$ with
nonempty interior, we consider a control problem in which the state
process $W$
and the control process $U$ satisfy \[W_t= w_0+\int_0^t\vartheta(W_s)
ds+\int_0^t\sigma(W_s) dZ_s+GU_t\in \mathcal{W},\qquad t\ge0,\] where
$Z$ is a
standard, multi-dimensional Brownian motion, $\vartheta,\sigma\in
C^{0,1}(\mathcal{W})$, $G$ is a fixed matrix, and $w_0\in\mathcal{W}
$. The
process $U$ is locally of bounded variation and has increments in a
given
closed convex cone $\mathcal{U}\subset{\mathbb{R}}^p$. Given $g\in
C(\mathcal{W})$, $\kappa\in{\mathbb{R}}^p$, and $\alpha>0$, consider the
objective that is to minimize the cost
\[J(w_0,U)\doteq\mathbb{E}\biggl[\int_0^{\infty}e^{-\alpha s}g(W_s)
ds+\int_{[0,\infty)}e^{-\alpha s} d(\kappa\cdot U_s)\biggr]\] over the
admissible controls $U$. Both $g$ and $\kappa\cdot u$ ($u\in\mathcal
{U}$) may
take positive and negative values. This paper studies the
corresponding dynamic
programming equation (DPE), a second-order degenerate elliptic partial
differential equation of HJB-type with a state constraint boundary
condition.
Under the controllability condition $G\mathcal{U}={\mathbb{R}}^d$ and
the
finiteness of $\mathcal{H}(q)=\sup_{u\in\mathcal{U}_1}\{-Gu\cdot q-
\kappa\cdot
u\}$, $q\in {\mathbb{R}}^d$, where $\mathcal{U}_1=\{u\in\mathcal{U}:|
Gu|=1\}$,
we show that the cost, that involves an improper integral, is well
defined. We
establish the following: (i) the value function for the control problem
satisfies the DPE (in the viscosity sense), and (ii) the condition
$\inf_{q\in{\mathbb{R}}^d}\mathcal{H}(q)<0$ is necessary and
sufficient for
uniqueness of solutions to the DPE. The existence and uniqueness of
solutions
are shown to be connected to an intuitive ``no arbitrage'' condition.
Our
results apply to Brownian control problems that represent formal
diffusion
approximations to control problems associated with stochastic processing
networks.
http://arxiv.org/abs/0711.0641
Author(s): Matthias Birkner and Andrej Depperschmidt
Abstract: We study a discrete time spatial branching system on $\mathbb{Z}^d
$ with
logistic-type local regulation at each deme depending on a weighted
average of
the population in neighboring demes. We show that the system survives
for all
time with positive probability if the competition term is small
enough. For a
restricted set of parameter values, we also obtain uniqueness of the
nontrivial
equilibrium and complete convergence, as well as long-term
coexistence in a
related two-type model. Along the way we classify the equilibria and
their
domain of attraction for the corresponding deterministic coupled map
lattice on
$\mathbb{Z}^d$.
http://arxiv.org/abs/0711.0649
Author(s): Antonio Mura and Francesco Mainardi
Abstract: In this paper we present a general mathematical construction that
allows us
to define a parametric class of $H$-sssi stochastic processes (self-
similar
with stationary increments), which have marginal probability density
function
that evolves in time according to a partial integro-differential
equation of
fractional type. This construction is based on the theory of finite
measures on
functional spaces. Since the variance evolves in time as a power
function,
these $H$-sssi processes naturally provide models for slow and fast
anomalous
diffusion. Such a class includes, as particular cases, fractional
Brownian
motion, grey Brownian motion and Brownian motion.
http://arxiv.org/abs/0711.0665
Author(s): Peter Friz and Nicolas Victoir
Abstract: Large classes of multi-dimensional Gaussian processes can be
enhanced with
stochastic Levy area(s). In a previous paper, we gave sufficient and
essentially necessary conditions, only involving variational
properties of the
covariance. Following T. Lyons, the resulting lift to a "Gaussian
rough path"
gives a robust theory of (stochastic) differential equations driven
by Gaussian
signals with sample path regularity worse than Brownian motion.
The purpose of this sequel paper is to establish convergence of
Karhunen-Loeve approximations in rough path metrics. Particular care is
necessary since martingale arguments are not enough to deal with
third iterated
integrals. An abstract support criterion for approximately continuous
Wiener
functionals then gives a description of the support of Gaussian rough
paths as
the closure of the (canonically lifted) Cameron-Martin space.
http://arxiv.org/abs/0711.0668
Author(s): Karine Bertin and Soledad Torres and Ciprian Tudor (CES and SAMOS)
Abstract: We consider statistical models driven by Gaussian and non-Gaussian
self-similar processes with long memory and we construct maximum
likelihood
estimators (MLE) for the drift parameter. Our approach is based on the
approximation by random walks of the driving noise. We study the
asymptotic
behavior of the estimators and we give some numerical simulations to
illustrate
our results.
http://arxiv.org/abs/0711.0513
Author(s): Gennadiy Averkov and Gabriele Bianchi
Abstract: The covariogram g_K of a convex body K in E^d is the function which
associates to each x in E^d the volume of the intersection of K with K
+x. In
1986 G. Matheron conjectured that for d=2 the covariogram g_K
determines K
within the class of all planar convex bodies, up to translations and
reflections in a point. This problem is equivalent to some problems in
stochastic geometry and probability as well as to a particular case
of the
phase retrieval problem in Fourier analysis. It is also relevant for the
inverse problem of determining the atomic structure of a quasicrystal
from its
X-ray diffraction image. In this paper, using some results previously
proved by
the second named author, we confirm Matheron's conjecture completely.
http://arxiv.org/abs/0711.0572
Author(s): J.-P. Lepeltier and M. Xu
Abstract: In this paper we prove the existence of a solution for reflected
BSDE's\
whose coefficient is of quadratic growth in $z$ and of linear growth
in $y$,
with an unbounded terminal value.
http://arxiv.org/abs/0711.0619
Author(s): Stephen B. Connor and Wilfrid S. Kendall
Abstract: Correction to Annals of Applied Probability 17 (2007) 781--808
[doi:10.1214/105051607000000032].
http://arxiv.org/abs/0711.0804
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (LSTA)
Abstract: We characterize the asymptotic behaviour of the weighted power
variation
processes associated with iterated Brownian motion. We prove weak
convergence
results in the sense of finite dimensional distributions, and show
that the
laws of the limiting objects can always be expressed in terms of three
independent Brownian motions X, Y and B, as well as of the local
times of Y. In
particular, our results involve "weighted'' versions of Kesten and
Spitzer's
Brownian motion in random scenery. Our findings extend the theory of
stochastic
integration developed theory initiated by Khoshnevisan and Lewis
(1999), and
should be compared with the recent results by Nourdin, Nualart and
Tudor (2007)
and Swanson (2007), concerning the weighted power variations of self-
similar
Gaussian processes.
http://arxiv.org/abs/0711.0858
Author(s): J. Lember and A. Koloydenko
Abstract: We show that, unlike in the general case, in the case of the two
state HMM,
the existence of infinite Viterbi alignments needs no special
assumptions and
can be proved considerably more easily.
http://arxiv.org/abs/0711.0928
Author(s): Peter Imkeller and Ilya Pavlyukevich and Torsten Wetzel
Abstract: We consider a dynamical system described by the differential equation
dY_t=-U'(Y_t)dt with a unique stable point at the origin. We perturb
the system
by L\'evy noise of intensity \e, to obtain the stochastic
differential equation
dX^\e_t=-U'(X^\e_{t-})dt+\e dL_t. The process L is a symmetric L\'evy
process
whose jump measure \nu has exponentially light tails,
\nu([u,\infty))\sim\exp(-u^\alpha), \alpha>0, u\to \infty. We study
the first
exit problem for the trajectories of the solutions of the stochastic
differential equation from the interval (-1,1). In the small noise
limit \e\to
0, the law of the first exit time \sigma_x, x\in(-1,1), is
exponential with the
mean value exhibiting an intriguing phase transition at the critical
index
\alpha=1, namely \log E \sigma\sim \e^{-\alpha} for 0<\alpha<1,
whereas \log \E
\sigma\sim \e^{-1}|\ln\e|^{1-\frac{1}{\alpha}} for \alpha>1.
http://arxiv.org/abs/0711.0982
Author(s): Leonid (Aryeh) Kontorovich
Abstract: The rate at which dependencies between future and past
observations decay in
a random process may be quantified in terms of mixing coefficients.
The latter
in turn appear in strong laws of large numbers and concentration of
measure
results for dependent random variables. Questions regarding what
rates are
possible for various notions of mixing have been posed since the
1960's, and
have important implications for some open problems in the theory of
strong
mixing conditions.
This paper deals with $\eta$-mixing, a notion defined in
[Kontorovich and
Ramanan], which is closely related to $\phi$-mixing. We show that
there exist
measures on finite sequences with essentially arbitrary $\eta$-mixing
coefficients, as well as processes with arbitrarily slow mixing rates.
http://arxiv.org/abs/0711.0986
Author(s): Leonid (Aryeh) Kontorovich
Abstract: Concentration bounds for non-product, non-Haar measures are fairly
recent:
the first such result was obtained for contracting Markov chains by
Marton in
1996. Since then, several other such results have been proved; with few
exceptions, these rely on coupling techniques. Though coupling is of
unquestionable utility as a theoretical tool, it appears to have some
limitations. Coupling has yet to be used to obtain bounds for more
general
Markov-type processes: hidden (or partially observed) Markov chains,
Markov
trees, etc. As an alternative to coupling, we apply the elementary
Markov
contraction lemma to obtain simple, useful, and apparently novel
concentration
results for the various Markov-type processes. Our technique consists of
expressing probabilities as matrix products and applying Markov
contraction to
these expressions; thus it is fairly general and holds the potential
to yield
numerous results in this vein.
http://arxiv.org/abs/0711.0987
Author(s): Lorenzo Zambotti
Abstract: We consider the problem of conditioning the Brownian excursion to
have a
fixed time average over the interval [0,1] and we study an associated
stochastic partial differential equation with reflection at 0 and
with the
constraint of conservation of the space average. The equation is
driven by the
derivative in space of a space-time white noise and contains a double
Laplacian
in the drift. Due to the lack of the maximum principle for the double
Laplacian, the standard techniques based on the penalization method
do not
yield existence of a solution.
http://arxiv.org/abs/0711.1068
Author(s): Gesine Reinert and Adrian R\"ollin
Abstract: We establish Stein's method of exchangeable pairs to assess
distributional
distances to potentially singular multivariate normal distributions,
in terms
of both smooth and non-smooth test functions. As examples we treat
runs on the
line, the joint count of edges, two-stars and triangles in Bernoulli
random
graphs, and complete $U$-statistics. Auxiliary random variables such as
Hoeffding projections arise naturally in the construction of
exchangeable
pairs.
http://arxiv.org/abs/0711.1082
Author(s): Nathana\"el Enriquez (MODAL'X and PMA) and Christophe Sabot (ICJ) and Olivier
Zindy (WIAS)
Abstract: We consider transient one-dimensional random walks in random
environment with
zero asymptotic speed. An aging phenomenon involving the generalized
Arcsine
law is proved using the localization of the walk at the foot of
"valleys" of
height $\log t$. In the quenched setting, we also sharply estimate the
distribution of the walk at time $t$.
http://arxiv.org/abs/0711.1095
Author(s): Margarete Knape and Ralph Neininger
Abstract: We propose and analyze an algorithm to approximate distribution
functions and
densities of perpetuities. Our algorithm refines an earlier approach
based on
iterating discretized versions of the fixed point equation that
defines the
perpetuity. We significantly reduce the complexity of the earlier
algorithm.
Also one particular perpetuity arising in the analysis of the selection
algorithm Quickselect is studied in more detail. Our approach works
well for
distribution functions. For densities we have weaker error bounds
although
computer experiments indicate that densities can also be approximated
well.
http://arxiv.org/abs/0711.1099
Author(s): Soumik Pal and Philip Protter
Abstract: We show pathological behavior of asset price processes modeled by
continuous
strict local martingales under a risk-neutral measure. The
inspiration comes
from recent results on financial bubbles. We analyze, in particular,
the effect
of the strict nature of the local martingale on the usual formula for
the price
of a European call option, especially a strong anomaly when call
prices decay
monotonically with maturity. A complete and detailed analysis for the
archetypical strict local martingale, the reciprocal of a three
dimensional
Bessel process, has been provided. Our main tool is based on a general
h-transform technique (due to Delbaen and Schachermayer) to generate
positive
strict local martingales. This gives the basis for a statistical test
to verify
whether a suspected bubble is indeed one (or not).
http://arxiv.org/abs/0711.1136
Author(s): Kei Fukuda and Akihiko Inoue and Yumiharu Nakano
Abstract: We present a general approach to the pricing of products in
finance and
insurance in the multi-period setting. It is a combination of the
utility
indifference pricing and optimal intertemporal risk allocation. We
give a
characterization of the optimal intertemporal risk allocation by a
first order
condition. Applying this result to the exponential utility function,
we obtain
an essentially new type of premium calculation method for a popular
type of
multi-period insurance contract. This method is simple and can be easily
implemented numerically. We see that the results of numerical
calculations are
well coincident with the risk loading level determined by traditional
practices. The results also suggest a possible implied utility
approach to
insurance pricing.
http://arxiv.org/abs/0711.1143
Author(s): Paul Balister and B\'ela Bollob\'as
Abstract: In this paper we have shall generalize Shearer's entropy
inequality and its
recent extensions by Madiman and Tetali, and shall apply projection
inequalities to deduce extensions of some of the inequalities
concerning sums
of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa.
We shall
also discuss projection and entropy inequalities and their connections.
http://arxiv.org/abs/0711.1151
Author(s): Nabin Kumar Jana
Abstract: In this thesis, we consider several Random Energy Models. This
includes
Derrida's Random Energy Model (REM) and Generalized Random Energy
Model (GREM)
and a nonhierarchical version (BK-GREM) by Bolthausen and Kistler.
The limiting
free energy in all these models along with Word GREM, a model
proposed by us,
turn out to be a cute consequence of large deviation principle (LDP).
This LDP
argument allows us to consider non-Gaussian driving distributions as
well as
external field. We could also consider random trees as the underlying
tree
structure in GREM. In all these models, as expected, limiting free
energy is
not 'universal' unlike the SK model. However it is 'rate specific'.
Consideration of non-Gaussian driving distribution as well as
different driving
distributions for the different levels of the underlying trees in
GREM leads to
interesting phenomena. For example in REM, if the Hamiltonian is
Binomial with
parameter $N$ and $p$ then the existence of phase transition depends
on the
parameter $p$. More precisely, phase transition takes place only when
$p>{1/2}$. For another example, consider a 2 level GREM with exponential
driving distribution at the first level and Gaussian in the second
with equal
weights at both the levels. Then even if the limiting ratio for the
second
level particles, $p_2$ is 0.00001 (very small), the system reduces to a
Gaussian REM. On the other hand, if we consider a 2 level GREM with
Gaussian
driving distribution at the first level and exponential in the
second, the
system will never reduce to a Gaussian REM. In either case, the
system will
never reduce to that of an exponential REM. etc.
http://arxiv.org/abs/0711.1249
Author(s): Jeffrey E. Steif and Marcus Warfheimer
Abstract: Bezuidenhout and Grimmett proved that the critical contact process
dies out.
Here, we generalize the result to the so called contact process in a
random
evolving environment (CPREE), introduced by Erik Broman. This process
is a
generalization of the contact process where the recovery rate can
vary between
two values. The rate which it chooses is determined by a background
process,
which evolves independently at different sites. As for the contact
process, we
can similarly define a critical value in terms of survival for this
process. In
this paper we prove that this definition is independent of how we
start the
background process, that finite and infinite survival (meaning
nontriviality of
the upper invariant measure) are equivalent and finally that the
process dies
out at criticality.
http://arxiv.org/abs/0711.1258
Author(s): Walter Schachermayer and Josef Teichmann
Abstract: We compare the option pricing formulas of Louis Bachelier and
Black-Merton-Scholes and observe -- theoretically as well as for
Bachelier's
original data -- that the prices coincide very well. We illustrate Louis
Bachelier's efforts to obtain applicable formulas for option pricing in
pre-computer time. Furthermore we explain -- by simple methods from
chaos
expansion -- why Bachelier's model yields good short-time
approximations of
prices and volatilities.
http://arxiv.org/abs/0711.1272
Author(s): Vladimir Vatutin and Vitali Wachtel
Abstract: Let S_0=0,{S_n, n>0} be a random walk generated by a sequence of
i.i.d.
random variables X_1,X_2,... and let \tau^{-} be the first descending
ladder
epoch. Assuming that the distribution of X_1 belongs to the domain of
attraction of an \alpha-stable law we study the asymptotic behavior
of the
local probabilities P(\tau ^{-}=n) and the conditional local
probabilities
P(S_n\in [x,x+y)|\tau^{-}>n) for fixed y and x=x(n)\in (0,\infty).
http://arxiv.org/abs/0711.1302
Author(s): Yaozhong Hu and David Nualart and Jian Song
Abstract: In this paper we introduce the notion of $\alpha$-martingale as the
fractional derivative of order $\alpha$ of a continuous local
martingale, where
$ \alpha\in (-\frac 12, \frac 12)$, and we show that it has a nonzero
finite
variation of order $\frac 2{1+2\alpha}$, under some integrability
assumptions
on the quadratic variation of the local martingale. As an application we
establish an extension of L\'evy's characterization theorem for the
fractional
Brownian motion.
http://arxiv.org/abs/0711.1313
Author(s): Manjunath Krishnapur
Abstract: We consider two families of random matrix-valued analytic
functions: (1)
G_1-zG_2 and (2) G_0 + zG_1 +z^2G_2+ ..., where G_i are n x n
independent
random matrices with independent standard complex Gaussian entries.
The set of
z where these matrix-valued analytic functions become singular, are
shown to be
determinantal point processes on the sphere and the hyperbolic plane,
respectively. The kernels of these determinantal processes are
reproducing
kernels of certain natural Hilbert spaces of analytic functions on the
corresponding surfaces. This gives a unified framework in which to
view a
result of Peres and Virag (n=1 in the second setting) and a well
known theorem
of Ginibre on Gaussian random matrices (which may be viewed as an
analogue of
our results in the whole plane).
http://arxiv.org/abs/0711.1378
Author(s): Mikl\'os Cs\"org\H{o} and Barbara Szyszkowicz and Qiying Wang
Abstract: Let $X, X_1, X_2,...$ be a sequence of non-degenerate i.i.d.
random variables
with mean zero. The best possible weighted approximations are
investigated in
$D[0, 1]$ for the partial sum processes $\{S_{[nt]}, 0\le t\le 1\}$,
where
$S_n=\sum_{j=1}^nX_j$, under the assumption that $X$ belongs to the
domain of
attraction of the normal law. The conclusions then are used to establish
similar results for the sequence of self-normalized partial sum
processes
$\{S_{[nt]}/V_n, 0\le t\le 1\}$, where $V_n^2=\sum_{j=1}^nX_j^2$. $L_p$
approximations of self-normalized partial sum processes are also
discussed.
http://arxiv.org/abs/0711.1384
Author(s): Mikl\'os Cs\"org\H{o} and Barbara Szyszkowicz and Qiying Wang
Abstract: This paper investigates weighted approximations for studentized
$U$-statistics type processes, both with symmetric and antisymmetric
kernels,
only under the assumption that the distribution of the projection
variate is in
the domain of attraction of the normal law. The classical second moment
condition $E|h(X_1,X_2)|^2 < \infty$ is also relaxed in both cases.
The results
can be used for testing the null assumption of having a random sample
versus
the alternative that there is a change in distribution in the sequence.
http://arxiv.org/abs/0711.1385
Author(s): Stefan Geiss and Anni Toivola
Abstract: We consider the weak convergence of the rescaled error processes
for Riemann
discretizations of certain stochastic integrals and relate the
integrability of
their weak limit to the fractional smoothness of the stochastic
integral.
http://arxiv.org/abs/0711.1439
Author(s): Rafa{\l} Lata{\l}a
Abstract: Motivated by a question of Krzysztof Oleszkiewicz we study a
notion of weak
tail domination of random vectors. We show that if the dominating random
variable is sufficiently regular weak tail domination implies strong
tail
domination. In particular positive answer to Oleszkiewicz question
would follow
from the so-called Bernoulli conjecture.
http://arxiv.org/abs/0711.1477
Author(s): Makoto Katori and Minami Izumi and Naoki Kobayashi
Abstract: It is known that the moments of the maximum value of a one-
dimensional
conditional Brownian motion, the three-dimensional Bessel bridge with
duration
1 started from the origin, are expressed using the Riemann zeta
function. We
consider a system of two Bessel bridges, in which noncolliding
condition is
imposed. We show that the moments of the maximum value is then
expressed using
the double Dirichlet series, or using the integrals of products of
the Jacobi
theta functions and its derivatives. Since the present system will be
provided
as a diffusion scaling limit of a version of vicious walker model,
the ensemble
of 2-watermelons with a wall, the dominant terms in long-time
asymptotics of
moments of height of 2-watermelons are completely determined, for
which only
the first moment, i.e. the average height, was recently studied by
Fulmek by a
method of enumerative combinatorics.
http://arxiv.org/abs/0711.1710
Author(s): Adam Timar
Abstract: We generalize theorems of Kesten and Deuschel-Pisztora about the
connectedness of the exterior boundary of a connected subset of $\Z^d
$, where
"connectedness" and "boundary" are understood with respect to various
graphs on
the vertices of $\Z^d$. We provide simple and elementary proofs of their
results. It turns out that the proper way of viewing these questions
is graph
theory, instead of topology.
http://arxiv.org/abs/0711.1713
Author(s): Alessandro De Gregorio and Stefano Iacus
Abstract: In this paper we derive explicit formulas of the R\'enyi
information, Shannon
entropy and Song measure for the invariant density of one dimensional
ergodic
diffusion processes. In particular, the diffusion models considered
include the
hyperbolic, the generalized inverse Gaussian, the Pearson, the
exponential
familiy and a new class of skew-$t$ diffusions.
http://arxiv.org/abs/0711.1789
Author(s): Maria Emilia Caballero and Victor Rivero
Abstract: We are interested by the rate of growth of increasing positive
self-similar
Markov processes (ipssMp) such that the subordinator associated to it
via
Lamperti's transformation has infinite mean. We prove that the
logarithm of an
ipssMp normalized by the logarithm of the time converges weakly, as
the time
tends to infinity, if and only if the Laplace exponent of the underlying
subordinator is regularly varying at zero. Moreover, we prove that
the regular
variation at zero of the Laplace exponent is essentially nasc for the
existence
of a function that normalizes the logarithm of an ipssMp. We obtain a
law of
iterated logarithm for the liminf of the logarithm of an ipssMp and
an integral
test to study the upper envelope of it. Furthermore, results
concerning the
rate of growth of the random clock appearing in Lamperti's
transformation are
obtained.
http://arxiv.org/abs/0711.1834
Author(s): Damien Gaboriau (UMPA-ENSL) and Russell Lyons
Abstract: We give a positive answer, in the measurable-group-theory context,
to von
Neumann's problem of knowing whether a non-amenable countable
discrete group
contains a non-cyclic free subgroup. We also get an embedding result
of the
free-group von Neumann factor into restricted wreath product factors.
http://arxiv.org/abs/0711.1643
Author(s): Adam Timar
Abstract: We answer three questions posed in a paper by Babson and
Benjamini. They
introduced a parameter $C_G$ for Cayley graphs $G$ that has significant
application to percolation. For a minimal cutset of $G$ and a
partition of this
cutset into two classes, take the minimal distance between the two
classes. The
supremum of this number over all minimal cutsets and all partitions
is $C_G$.
We show that if it is finite for some Cayley graph of the group then
it is
finite for any (finitely generated) Cayley graph. Having an
exponential bound
for the number of minimal cutsets of size $n$ separating $o$ from
infinity also
turns out to be independent of the Cayley graph chosen. We show a 1-
ended
example (the lamplighter group), where $C_G$ is infinite. Finally, we
give a
new proof for a question of de la Harpe, proving that the number of $n
$-element
cutsets separating $o$ from infinity is finite unless $G$ is a finite
extension
of $Z$.
http://arxiv.org/abs/0711.1711
Author(s): Bernard Shiffman and Steve Zelditch
Abstract: We consider the zero sets $Z_N$ of systems of $m$ random
polynomials of
degree $N$ in $m$ complex variables, and we give asymptotic formulas
for the
random variables given by summing a smooth test function over $Z_N$. Our
asymptotic formulas show that the variances for these smooth
statistics have
the growth $N^{m-2}$. We also prove analogues for the integrals of
smooth test
forms over the subvarieties defined by $k
http://arxiv.org/abs/0711.1840
Author(s): Julien Dubedat
Abstract: In this note, we prove a version of the conjectured duality of
Schramm-Loewner Evolutions, by establishing exact identities in
distribution
between some boundary arcs of chordal $\SLE_\kappa$, $\kappa>4$, and
appropriate versions of $\SLE_{\hat\kappa}$, $\hat\kappa=16/\kappa$.
http://arxiv.org/abs/0711.1884
Author(s): Shui Feng and Feng-Yu Wang
Abstract: Starting from a sequence of independent Wright-Fisher diffusion
processes on
$[0,1]$, we construct a class of reversible infinite dimensional
diffusion
processes on $\DD_\infty:= \{{\bf x}\in [0,1]^\N: \sum_{i\ge 1} x_i=1
\}$ with
GEM distribution as the reversible measure. Log-Sobolev inequalities are
established for these diffusions, which lead to the exponential
convergence to
the corresponding reversible measures in the entropy. Extensions are
made to a
class of measure-valued processes over an abstract space $S$. This
provides a
reasonable alternative to the Fleming-Viot process which does not
satisfy the
log-Sobolev inequality when $S$ is infinite as observed by W. Stannat
\cite{S}.
http://arxiv.org/abs/0711.1887
Author(s): Russell Lyons and Ron Peled and Oded Schramm
Abstract: The number of spanning trees in the giant component of the random
graph
$\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\}
$ as
$n\to\infty$, where $m$ is the number of vertices in the giant
component. The
function $f$ is not known explicitly, but we show that it is strictly
increasing and infinitely differentiable. Moreover, we give an
explicit lower
bound on $f'(c)$. A key lemma is the following. Let $\PGW(\lambda)$
denote a
Galton-Watson tree having Poisson offspring distribution with parameter
$\lambda$. Suppose that $\lambda^*>\lambda>1$. We show that $\PGW
(\lambda^*)$
conditioned to survive forever stochastically dominates $\PGW(\lambda)$
conditioned to survive forever.
http://arxiv.org/abs/0711.1893
Author(s): Mohammud Foondun and Davar Khoshnevisan and Eulalia Nualart
Abstract: It is frequently the case that a white-noise-driven parabolic and/or
hyperbolic stochastic partial differential equation (SPDE) can have
random-field solutions only in spatial dimension one. Here we show
that in many
cases, where the ``spatial operator'' is the L^2-generator of a L
\'evy process
X, a linear SPDE has a random-field solution if and only if the
symmetrization
of X possesses local times. This result gives a probabilistic reason
for the
lack of existence of random-field solutions in dimensions strictly
bigger than
one. In addition, we prove that the solution to the SPDE is [H\"older]
continuous in its spatial variable if and only if the said local time is
[H\"older] continuous in its spatial variable. We also produce
examples where
the random-field solution exists, but is almost surely unbounded in
every open
subset of space-time. Our results are based on first establishing a
quasi-isometry between the linear L^2-space of the weak solutions of
a family
of linear SPDEs, on one hand, and the Dirichlet space generated by the
symmetrization of X, on the other hand. We study mainly linear
equations in
order to present the local-time correspondence at a modest technical
level.
However, some of our work has consequences for nonlinear SPDEs as
well. We
demonstrate this assertion by studying a family of parabolic SPDEs
that have
additive nonlinearities. For those equations we prove that if the
linearized
problem has a random-field solution, then so does the nonlinear SPDE.
Moreover,
the solution to the linearized equation is [H\"older] continuous if
and only if
the solution to the nonlinear equation is. And the solutions are
bounded and
unbounded together as well. Finally, we prove that in the cases that the
solutions are unbounded, they almost surely blow up at exactly the
same points.
http://arxiv.org/abs/0711.1913
Author(s): Thomas Dean and Paul Dupuis
Abstract: Particle splitting methods are considered for the estimation of
rare events.
The probability of interest is that a Markov process first enters a
set $B$
before another set $A$, and it is assumed that this probability
satisfies a
large deviation scaling. A notion of subsolution is defined for the
related
calculus of variations problem, and two main results are proved under
mild
conditions. The first is that the number of particles generated by the
algorithm grows subexponentially if and only if a certain scalar
multiple of
the importance function is a subsolution. The second is that, under
the same
condition, the variance of the algorithm is characterized
(asymptotically) in
terms of the subsolution. The design of asymptotically optimal
schemes is
discussed, and numerical examples are presented.
http://arxiv.org/abs/0711.2037
Author(s): Masanori Hino
Abstract: We prove that the martingale dimensions for canonical diffusion
processes on
a class of self-similar sets including nested fractals are always
one. This
provides an affirmative answer to the conjecture of S. Kusuoka [Publ.
Res.
Inst. Math. Sci. 25 (1989) 659--680].
http://arxiv.org/abs/0711.2135
Author(s): Andrew J. F. Jack and Timothy C. Johnson and Mihail Zervos
Abstract: We consider a stochastic system whose uncontrolled state dynamics are
modelled by a general one-dimensional It\^{o} diffusion. The control
effort
that can be applied to this system takes the form that is associated
with the
so-called monotone follower problem of singular stochastic control.
The control
problem that we address aims at maximising a performance criterion
that rewards
high values of the utility derived from the system's controlled state
but
penalises any expenditure of control effort. This problem has been
motivated by
applications such as the so-called goodwill problem in which the
system's state
is used to represent the image that a product has in a market, while
control
expenditure is associated with raising the product's image, e.g.,
through
advertising. We obtain the solution to the optimisation problem that we
consider in a closed analytic form under rather general assumptions.
Also, our
analysis establishes a number of results that are concerned with
analytic as
well as probabilistic expressions for the first derivative of the
solution to a
second order linear non-homogeneous ordinary differential equation.
These
results have independent interest and can potentially be of use to
the solution
of other one-dimensional stochastic control problems.
http://arxiv.org/abs/0711.2143
Author(s): Masanori Hino and Hiroto Uchida
Abstract: Consider a set of continuous maps from the interval $[0,1]$ to a
domain in
${\mathbb R}^d$. Although the topological boundary of this set in the
path
space is not smooth in general, by using the theory of functions of
bounded
variation (BV functions) on the Wiener space and the theory of
Dirichlet forms,
we can discuss the existence of the surface measure and the Skorokhod
representation of the reflecting Ornstein-Uhlenbeck process
associated with the
canonical Dirichlet form on this set.
http://arxiv.org/abs/0711.2144
Author(s): Dmitry Korshunov
Abstract: We consider a time-homogeneous Markov chain $X_n$, $n\ge0$, valued
in ${\bf
R}$. Suppose that this chain is transient, that is, $X_n$ generates a
$\sigma$-finite renewal measure. We prove the key renewal theorem under
condition that this chain has asymptotically homogeneous at infinity
jumps and
asymptotically positive drift.
http://arxiv.org/abs/0711.2169
Author(s): Arie Leizarowitz and Adam Shwartz
Abstract: For a countable-state Markov decision process we introduce an
embedding which
produces a finite-state Markov decision process. The finite-state
embedded
process has the same optimal cost, and moreover, it has the same
dynamics as
the original process when restricting to the approximating set. The
embedded
process can be used as an approximation which, being finite, is more
convenient
for computation and implementation.
http://arxiv.org/abs/0711.2185
Author(s): Rami Atar and Adam Shwartz
Abstract: We consider a queue with renewal arrivals and n exponential
servers in the
Halfin-Whitt heavy traffic regime, where n and the arrival rate increase
without bound, so that a critical loading condition holds. Server k
serves at
rate $\mu_k $, and the empirical distribution of the $\mu_k $ is
assumed to
converge weakly. We show that very little information on the service
rates is
required for a routing mechanism to perform well. More precisely, we
construct
a routing mechanism that has access to a single sample from the
service time
distribution of each of $n$ to the power of $1/2 + \epsilon $
randomly selected
servers, but not to the actual values of the service rates, the
performance of
which is asymptotically as good as the best among mechanisms that
have the
complete information on $ \mu_k $.
http://arxiv.org/abs/0711.2188
Author(s): Adam Shwartz and Alan Weiss
Abstract: We show global uniqueness of the solution to a class of constrained
variational problems, using scaling properties. This is used to
establish the
essential uniqueness of solutions of a large deviations problem in
multiple
dimensions. The result is motivated by models of buffers, and in
particular the
probability of, and typical path to overflow in the limit of small
buffers,
which we analyze.
http://arxiv.org/abs/0711.2191
Author(s): Anton Bovier and Anton Klimovsky
Abstract: The non-hierarchical correlation structure of the Sherrington-
Kirkpatrick
(SK) model with multidimensional (e.g. Heisenberg) spins is studied
at the
level of the logarithmic asymptotic of the corresponding sum of the
correlated
exponentials -- the thermodynamic pressure. For this purpose an abstract
quenched large deviations principle (LDP) of Gaertner-Ellis type is
obtained
under an assumption of measure concentration. With the aid of this
principle
the framework of the Aizenman-Sims-Starr comparison scheme ($\text{AS}
^2$
scheme) is extended to the case of the SK model with multidimensional
spins.
This extension, based the quenched LDP, shows how the Hadamard matrix
products
arise rigorously in the context of the Parisi formula. This allows
one to
relate the pressure of the non-hierarchical SK model with the
pressure of the
hierarchical GREM by a saddle-point variational formula of the Parisi
type
including a negative remainder term.
http://arxiv.org/abs/0711.2286
Author(s): Cristina Toninelli and Giulio Biroli
Abstract: We introduce a new class of two-dimensional cellular automata with a
bootstrap percolation-like dynamics. Each site can be either empty or
occupied
by a single particle and the dynamics follows a deterministic
updating rule at
discrete times which allows only emptying sites. We prove that the
threshold
density $\rho_c$ for convergence to a completely empty configuration
is non
trivial, $0<\rho_c<1$, contrary to standard bootstrap percolation.
Furthermore
we prove that in the subcritical regime, $\rho<\rho_c$, emptying
always occurs
exponentially fast and that $\rho_c$ coincides with the critical
density for
two-dimensional oriented site percolation on $\bZ^2$. This is known
to occur
also for some cellular automata with oriented rules for which the
transition is
continuous in the value of the asymptotic density and the crossover
length
determining finite size effects diverges as a power law when the
critical
density is approached from below. Instead for our model we prove that
the
transition is {\it discontinuous} and at the same time the crossover
length
diverges {\it faster than any power law}. The proofs of the
discontinuity and
the lower bound on the crossover length use a conjecture on the critical
behaviour for oriented percolation. The latter is supported by several
numerical simulations and by analytical (though non rigorous) works
through
renormalization techniques. Finally, we will discuss why, due to the
peculiar
{\it mixed critical/first order character} of this transition, the
model is
particularly relevant to study glassy and jamming transitions.
Indeed, we will
show that it leads to a dynamical glass transition for a Kinetically
Constrained Spin Model. Most of the results that we present are the
rigorous
proofs of physical arguments developed in a joint work with D.S.Fisher.
http://arxiv.org/abs/0709.0378
Author(s): Arthur Cunha and Minh Do and and Martin Vetterli
Abstract: The {\it plenoptic function} (Adelson and Bergen, 91) describes
the visual
information available to an observer at any point in space and time.
Samples of
the plenoptic function (POF) are seen in video and in general visual
content,
and represent large amounts of information. In this paper we propose a
stochastic model to study the compression limits of the plenoptic
function. In
the proposed framework, we isolate the two fundamental sources of
information
in the POF: the one representing the camera motion and the other
representing
the information complexity of the ``reality'' being acquired and
transmitted.
The sources of information are combined, generating a stochastic
process that
we study in detail. We first propose a model for ensembles of
realities that do
not change over time. The proposed model is simple in that it enables
us to
derive precise coding bounds in the information-theoretic sense that
are sharp
in a number of cases of practical interest. For this simple case of
static
realities and camera motion, our results indicate that coding
practice is in
accordance with optimal coding from an information-theoretic
standpoint. The
model is further extended to account for visual realities that change
over
time. We derive bounds on the lossless and lossy information rates
for this
dynamic reality model, stating conditions under which the bounds are
tight.
Examples with synthetic sources suggest that in the presence of scene
dynamics,
simple hybrid coding using motion/displacement estimation with DPCM
performs
considerably suboptimally relative to the true rate-distortion bound.
http://arxiv.org/abs/0711.2104
Author(s): Rainer Buckdahn and Juan Li and Shige Peng
Abstract: Mathematical mean-field approaches play an important role in
different fields
of Physics and Chemistry, but have found in recent works also their
application
in Economics, Finance and Game Theory. The objective of our paper is
to study a
special mean-field problem in a purely stochastic approach. We
consider a
stochastic differential equation that describes the dynamics of a
particle
$X^{(N)}$ influenced by the dynamics of $N$ other particles, which
are supposed
to be independent identically distributed and of the same law as $X^
{(N)}$.
This equation (of rank $N$) is then associated with a backward
stochastic
differential equation (BSDE). After proving the existence and the
uniqueness of
a solution $(X^{(N)},Y^{(N)},Z^{(N)})$ for this couple of equations we
investigate its limit behavior. With an approach which uses the
tightness of
the laws of the above sequence of triplets in a suitable space, and
combines it
with BSDE methods and the Law of Large Numbers, it is shown that
$(X^{(N)},Y^{(N)},Z^{(N)})$ converges in $L^2$ to the unique solution
of a
limit equation, formed by a Mean-Field forward and a Mean-Field backward
equation.
http://arxiv.org/abs/0711.2162
Author(s): Mikhail Menshikov and Stanislav Volkov
Abstract: We study a one-dimensional random walk whose expected drift
depends both on
time and the position of a particle. We establish a non-trivial phase
transition for the recurrence vs. transience of the walk, and show some
interesting applications to Friedman's urn, as well as showing the
connection
with Lamperti's walk with asymptotically zero drift.
http://arxiv.org/abs/0711.2373
Author(s): Nicolas Vergne (1) and Miguel Abadi (2) ((1) Laboratoire Statistique
et G\'enome France, (2) Universidade de Campinas Brazil)
Abstract: Using recent results on the occurrence times of a string of
symbols in a
stochastic process with mixing properties, we present a new method
for the
search of rare words in biological sequences generally modelled by a
Markov
chain. We obtain a bound on the error between the distribution of the
number of
occurrences of a word in a sequence (under a Markov model) and its
Poisson
approximation. A global bound is already given by a Chen-Stein
method. Our
approach, the psi-mixing method, gives local bounds. Since we only
need the
error in the tails of distribution, the global uniform bound of Chen-
Stein is
too large and it is a better way to consider local bounds. We search
for two
thresholds on the number of occurrences from which we can regard the
studied
word as an over-represented or an under-represented one. A biological
role is
suggested for these over- or under-represented words. Our method
gives such
thresholds for a panel of words much broader than the Chen-Stein method.
Comparing the methods, we observe a better accuracy for the psi-
mixing method
for the bound of the tails of distribution. We also present the
software PANOW
(available at http://stat.genopole.cnrs.fr/software/panowdir/)
dedicated to the
computation of the error term and the thresholds for a studied word.
http://arxiv.org/abs/0711.2382
Author(s): Jinshan Zhang
Abstract: To constitute atoms of a sigma algebra is not a easy job due to
the large
number of its elements. Thus, determining them via the generator seems a
feasible and simple way since most sigma algebras are generated by their
smaller proper subsets. Precisely, Under some conditions each atom of
a sigma
algebra equals the intersection of the elememts containing any point
of the
atom in the generator. In this paper, a very weak sufficient
condition for
determining atoms by the generator will be presented. Besides, such a
condition, though not a necessary one, will be shown to be almost the
weakest
one, say, almost can not be improved.
http://arxiv.org/abs/0711.2400
Author(s): Fabrice Baudoin and Martin Hairer and Josef Teichmann
Abstract: We consider Ornstein-Uhlenbeck processes (OU-processes) related to
hypoelliptic diffusion on finite-dimensional Lie groups: let $
\mathcal{L} $ be
a hypoelliptic, left-invariant ``sum of the squares''-operator on a
Lie group $
G $ with associated Markov process $ X $, then we construct OU-type
processes
by adding horizontal gradient drifts of functions $ U $. In the
natural case $
U(x) = - \log p(1,x) $, where $ p(1,x) $ is the density of the law of
the
Markov process $ X $ starting at the identity $ e $ at time $ t =1 $
with
respect to the right-invariant Haar measure on $G$, we show the
Poincar\'e
inequality by applying the Driver-Melcher inequality for ``sum of the
squares''
operators on Lie groups.
The Markov process associated to $ - \log p(1,x) $ is called the
OU-process
related to the given hypoelliptic diffusion on $ G $. We prove the
global
strong existence of this OU-process on $ G $. The Poincar\'e
inequality for a
large class of potentials $U$ is then shown by perturbation methods
and used to
obtain a hypoelliptic equivalent of the standard result on cooling
schedules
for simulated annealing. The relation between local results on $
\mathcal{L} $
and global results for the constructed OU-process is widely used in
this study.
http://arxiv.org/abs/0711.2419
Author(s): Florin Avram and Zbigniew Palmowski and Martijn Pistorius
Abstract: In this paper we study the joint ruin problem for two insurance
companies
that divide between them both claims and premia in some specified
proportions
(modeling two branches of the same insurance company or an insurance and
re-insurance company). Modeling the risk processes of the insurance
companies
by Cram\'{e}r-Lundberg processes we obtain the Laplace transform in
space of
the probability that either of the insurance companies is ruined in
finite
time. Subsequently, for exponentially distributed claims, we derive
an explicit
analytical expression for this joint ruin probability by explicitly
inverting
this Laplace transform. We also provide a characterization of the
Laplace
transform of the joint ruin time.
http://arxiv.org/abs/0711.2465
Author(s): Fabrizio Durante and Erich Peter Klement and Jos\'e Juan Quesada- Molina
Abstract: We determine under which conditions three bivariate copulas are
compatible,
viz. they are the bivariate marginals of the same trivariate copula,
and, then,
construct the class of these copulas. In particular, the upper and
lower bounds
for this class of trivariate copulas are determined.
http://arxiv.org/abs/0711.2409
Author(s): Yaozhong Hu and David Nualart and Xiaoming Song
Abstract: In this paper we study a singular stochastic differential equation
driven by
an additive fractional Brownian motion with Hurst parameter $H>\frac
12$. Under
some assumptions on the drift, we show that there is a unique
solution, which
has moments of all orders. We also apply the techniques of Malliavin
calculus
to prove that the solution has an absolutely continuous law at any
time $t>0$.
http://arxiv.org/abs/0711.2507
Author(s): Frank Aurzada
Abstract: We obtain some new results concerning the small deviation problem for
$S=\sum_n q^n X_n$ and $M=\sup_n q^n X_n$, where $0
http://arxiv.org/abs/0711.2576
Author(s): Andreas Neuenkirch and Ivan Nourdin (PMA) and Samy Tindel (IECN)
Abstract: In this article, we illustrate the flexibility of the algebraic
integration
formalism introduced by M. Gubinelli (2004), by establishing an
existence and
uniqueness result for delay equations driven by rough paths. We then
apply our
results to the case where the driving path is a fractional Brownian
motion with
Hurst parameter H>1/3.
http://arxiv.org/abs/0711.2633
Author(s): Stanislav Volkov and Timothy Wong
Abstract: We study a simple random walk on an n-dimensional hypercube. For
any starting
position we find the probability of hitting vertex a before hitting
vertex b,
whenever a and b share the same edge. This generalizes the model in
Doyle, P.,
and Snell, J., "Random Walks and Electric Networks", Mathematical
Association
of America, 1984 (see Exercise 1.3.7 there).
http://arxiv.org/abs/0711.2675
Author(s): Kevin P. Costello and Van Vu
Abstract: We investigate the rank of random (symmetric) sparse matrices. Our
main
finding is that with high probability, any dependency that occurs in
such a
matrix is formed by a set of few rows that contains an overwhelming
number of
zeros. This allows us to obtain an exact estimate for the co-rank.
http://arxiv.org/abs/0711.2696
Author(s): Dong Wang
Abstract: We solve the largest sample eigenvalue distribution problem in the
rank 1
spiked model of the quaternionic Wishart ensemble, which is the first
case of a
statistical generalization of the Laguerre symplectic ensemble (LSE)
on the
soft edge. We observe a phase change phenomenon similar to that in
the complex
case, and prove that the new distribution at the phase change point
is the GOE
Tracy-Widom distribution.
http://arxiv.org/abs/0711.2722
Author(s): Nizar Demni (PMA)
Abstract: We generalize a previous result concerning free martingale
polynomials for
the stationary free Jacobi process of parameters $\lambda \in ]0.1],
\theta =
1/2$. Hopelessly, apart from the case $\lambda = 1$, the polynomials
we derive
are no longer orthogonal with respect to the spectral measure. As a
matter of
fact, we use the multiplicative renormalization to write down the
corresponding
orthogonality measure.
http://arxiv.org/abs/0711.2734
Author(s): Soeren Asmussen and Dilip Madan and Martijn Pistorius
Abstract: The Wiener-Hopf factorization is obtained in closed form for a
phase type
approximation to the CGMY L\'{e}vy process. This allows, for the
approximation,
exact computation of first passage times to barrier levels via Laplace
transform inversion. Calibration of the CGMY model to market option
prices
defines the risk neutral process for which we infer the first passage
times of
stock prices to 30% of the price level at contract initiation. These
distributions are then used in pricing 50% recovery rate equity
default swap
(EDS) contracts and the resulting prices are compared with the prices
of credit
default swaps (CDS). An illustrative analysis is presented for these
contracts
on Ford and GM.
http://arxiv.org/abs/0711.2807
Author(s): Shige Peng
Abstract: We introduce a new notion of G-normal distributions. This will
bring us to a
new framework of stochastic calculus of Ito's type (Ito's integral,
Ito's
formula, Ito's equation) through the corresponding G-Brownian motion.
We will
also present analytical calculations and some new statistical methods
with
application to risk analysis in finance under volatility uncertainty.
Our basic point of view is: sublinear expectation theory is very
like its
special situation of linear expectation in the classical probability
theory.
Under a sublinear expectation space we still can introduce the notion of
distributions, of random variables, as well as the notions of joint
distributions, marginal distributions, etc. A particularly interesting
phenomenon in sublinear situations is that a random variable Y is
independent
to X does not automatically implies that X is independent to Y.
Two important theorems have been proved: The law of large number
and the
central limit theorem.
http://arxiv.org/abs/0711.2834
Author(s): Josep Llu\'is Sol\'e and Frederic Utzet
Abstract: An explicit procedure to construct a family of martingales
generated by a
process with independent increments is presented. The main tools are the
polynomials that give the relationship between the moments and
cumulants, and a
set of martingales related to the jumps of the process called Teugels
martingales
http://arxiv.org/abs/0711.2879
Author(s): Claudio Albanese
Abstract: We introduce a classical diffusion process which provides a full
description
of non-relativistic quantum mechanics and has the form of a Z_4 gauge
theory.
We first define a stochastic process on a discretization of physical
space of
the form (aZ)^3, where a is an elementary length scale. We then lift
this
process to the principal bundle (aZ)^3 x Z_4. Non-relativistic quantum
mechanics is recovered in the limit as a tends to 0, as we show in
the case of
a scalar particle in an electromagnetic field. Many-body interactions
can
easily be accommodated. In the case of tight binding Hamiltonians no
limit
needs to be taken, the equivalence is straightforward and sheds new
light on
the dynamics of quantum phases.
http://arxiv.org/abs/0711.2978
Author(s): Claudio Albanese
Abstract: We study triangulation schemes for the joint kernel of a diffusion
process
with uniformly continuous coefficients and an adapted, non-resonant
Abelian
process. The prototypical example of Abelian process to which our
methods apply
is given by stochastic integrals with uniformly continuous
coeffcients. The
range of applicability includes also a broader class of processes of
practical
relevance, such as the sup process and certain discrete time
summations we
discuss. We discretize the space coordinate in uniform steps and
assume that
time is either continuous or finely discretized as in a fully
explicit Euler
method and the Courant condition is satisfied. We show that the Fourier
transform of the joint kernel of a diffusion and a stochastic integral
converges in a uniform graph norm associated to the Markov generator.
Convergence also implies smoothness properties for the Fourier
transform of the
joint kernel. Stochastic integrals are straightforward to define for
finite
triangulations and the convergence result gives a new and entirely
constructive
way of defining stochastic integrals in the continuum. The method
relies on a
reinterpretation and extension of the classic theorems by Feynman-Kac,
Girsanov, Ito and Cameron-Martin, which are also re-obtained. We make
use of a
path-wise analysis without relying on a probabilistic interpretation.
The
Fourier representation is needed to regularize the hypo-elliptic
character of
the joint process of a diffusion and an adapted stochastic integral. The
argument extends as long as the Fourier analysis framework can be
generalized.
This condition leads to the notion of non-resonant Abelian process.
http://arxiv.org/abs/0711.2980
Author(s): Xinjia Chen
Abstract: In this paper, we consider the nonasymptotic sequential estimation
of means
of random variables bounded in between zero and one. We have rigorously
demonstrated that, in order to guarantee prescribed relative
precision and
confidence level, it suffices to continue sampling until the sample
sum is no
less than a certain bound and then take the average of samples as an
estimate
for the mean of the bounded random variable. We have developed an
explicit
formula and a bisection search method for the determination of such
bound of
sample sum, without any knowledge of the bounded variable. Moreover,
we have
derived bounds for the distribution of sample size. In the special
case of
Bernoulli random variables, we have established analytical and numerical
methods to further reduce the bound of sample sum and thus improve the
efficiency of sampling.
http://arxiv.org/abs/0711.2801
Author(s): Grigorii Melnichenko
Abstract: It is shown that the quantum logic of linear subspaces can be used
for
recognition of random signals by a Bayesian energy discriminant
classifier. The
energy distribution on linear subspaces is described by a correlation
matrix of
probability distribution. We show that the correlation matrix
corresponds to
von Neumann density matrix in the quantum theory. We offered the
interpretation
of quantum logic as a fuzzy logic of fuzzy sets. The use of quantum
logic for
recognition is based on the fact that the probability distribution of
each
class lies approximately on a lower-dimensional subspace of feature
space. It
is offered interpretation of discriminant functions as membership
functions of
fuzzy sets. Also we offer the quality functional for optimal choose of
discriminant functions for recognition from some class of discriminant
functions.
http://arxiv.org/abs/0711.1437
Author(s): Geir T. Helleloid
Abstract: This thesis has three goals related to the automorphism groups of
finite
$p$-groups. The primary goal is to provide a complete proof of a theorem
showing that, in some asymptotic sense, the automorphism group of
almost every
finite $p$-group is itself a $p$-group. We originally proved this
theorem in a
paper with Martin; the presentation of the proof here contains
omitted proof
details and revised exposition. We also give a survey of the extant
results on
automorphism groups of finite $p$-groups, focusing on the order of the
automorphism groups and on known examples. Finally, we explore a
connection
between automorphisms of finite $p$-groups and Markov chains.
Specifically, we
define a family of Markov chains on an elementary abelian $p$-group
and bound
the convergence rate of some of those chains.
http://arxiv.org/abs/0711.2816
Author(s): Jeff Kahn and Nicholas Weininger
Abstract: A fractional fuzzy Potts measure is a probability distribution on
spin
configurations of a finite graph $G$ obtained in two steps: first a
subgraph of
$G$ is chosen according to a random cluster measure $\phi_{p,q}$, and
then a
spin ($\pm1$) is chosen independently for each component of the
subgraph and
assigned to all vertices of that component. We show that whenever $q
\geq1$,
such a measure is positively associated, meaning that any two
increasing events
are positively correlated. This generalizes earlier results of
H\"{a}ggstr\"{o}m [Ann. Appl. Probab. 9 (1999) 1149--1159] and
H\"{a}ggstr\"{o}m and Schramm [Stochastic Process. Appl. 96 (2001)
213--242].
http://arxiv.org/abs/0711.3136
Author(s): Oded Schramm and Wang Zhou
Abstract: This paper examines how close the chordal $\SLE_\kappa$ curve gets
to the
real line asymptotically far away from its starting point. In
particular, when
$\kappa\in(0,4)$, it is shown that if $\beta>\beta_\kappa:=1/(8/
\kappa-2)$,
then the intersection of the $\SLE_\kappa$ curve with the graph of
the function
$y=x/(\log x)^{\beta}$, $x>e$, is a.s. bounded, while it is a.s.
unbounded if
$\beta=\beta_\kappa$. The critical $\SLE_4$ curve a.s. intersects the
graph of
$y=x^{-(\log\log x)^\alpha}$, $x>e^e$, in an unbounded set if $\alpha
\le 1$,
but not if $\alpha>1$. Under a very mild regularity assumption on the
function
$y(x)$, we give a necessary and sufficient integrability condition
for the
intersection of the $\SLE_\kappa$ path with the graph of $y$ to be
unbounded.
We also prove that the Hausdorff dimension of the intersection set
of the
$\SLE_{\kappa}$ curve and real axis is $2-8/\kappa$ when $4<\kappa<8$.
http://arxiv.org/abs/0711.3350
Author(s): J. van den Berg and B.N.B. de Lima
Abstract: The self-destructive percolation model is defined as follows:
Consider
percolation with parameter $p > p_c$. Remove the infinite occupied
cluster.
Finally, give each vertex (or, for bond percolation, each edge) that
at this
stage is vacant, an extra chance $\delta$ to become occupied. Let $
\delta_c(p)$
be the minimal value of $\delta$, needed to obtain an infinite
occupied cluster
in the final configuration. This model was introduced some years ago
by van den
Berg and Brouwer. They showed that, for the site model on the square
lattice
(and a few other 2D lattices satisfying a special technical
condition) that
$\delta_c(p)\geq\frac{(p-p_c)}{p}$. In particular, $\delta_c(p)$ is
at least
linear in $p-p_c$.
Although the arguments used by van den Berg and Brouwer look quite
rigid, we
show that they can be suitably modified to obtain similar linear
lower bounds
for $\delta_c(p)$ (with $p$ near $p_c$) for a much larger class of 2D
lattices,
including bond percolation on the square and triangular lattices, and
site
percolation on the star lattice (or matching lattice) of the square
lattice.
http://arxiv.org/abs/0711.3563
Author(s): Erik I. Broman
Abstract: The ordinary contact process is used to model the spread of a
disease in a
population. In this model, each infected individual waits an
exponentially
distributed time with parameter 1 before becoming healthy. In this
paper, we
introduce and study the contact process in a randomly evolving
environment.
Here we associate to every individual an independent two-state, $\{0,1
\},$
background process. Given $\delta_0<\delta_1,$ if the background
process is in
state $0,$ the individual (if infected) becomes healthy at rate $
\delta_0,$
while if the background process is in state $1,$ it becomes healthy
at rate
$\delta_1.$ By stochastically comparing the contact process in a
randomly
evolving environment to the ordinary contact process, we will
investigate
matters of extinction and that of weak and strong survival. A key
step in our
analysis is to obtain stochastic domination results between certain
point
processes. We do this by starting out in a discrete setting and then
taking
continuous time limits.
http://arxiv.org/abs/0711.3597
Author(s): Nayantara Bhatnagar and Juan Vera and and Eric Vigoda
Abstract: Consider $k$-colorings of the complete tree of depth $\ell$ and
branching
factor $\Delta$. If we fix the coloring of the leaves, for what range
of $k$ is
the root uniformly distributed over all $k$ colors (in the limit
$\ell\to\infty$)? This corresponds to the threshold for uniqueness of
the
infinite-volume Gibbs measure. It is straightforward to show the
existence of
colorings of the leaves which ``freeze'' the entire tree when $k\le
\Delta+1$.
For $k\geq\Delta+2$, Jonasson proved the root is ``unbiased'' for any
fixed
coloring of the leaves and thus the Gibbs measure is unique. What
happens for a
{\em typical} coloring of the leaves? When the leaves have a non-
vanishing
influence on the root in expectation, over random colorings of the
leaves,
reconstruction is said to hold. Non-reconstruction is equivalent to
extremality
of the Gibbs measure. When $k<\Delta/\ln{\Delta}$, it is
straightforward to
show that reconstruction is possible (and hence the measure is not
extremal).
We prove that for $C>2$ and $k =C\Delta/\ln{\Delta}$, non-
reconstruction
holds, i.e., the Gibbs measure is extremal. We prove a strong form of
extremality: with high probability over the colorings of the leaves the
influence at the root decays exponentially fast with the depth of the
tree.
These are the first results coming close to the threshold for
extremality for
colorings. Extremality on trees and random graphs has received
considerable
attention recently since it may have connections to the efficiency of
local
algorithms.
http://arxiv.org/abs/0711.3664
Author(s): Wei Biao Wu
Abstract: We establish strong invariance principles for sums of stationary
and ergodic
processes with nearly optimal bounds. Applications to linear and some
nonlinear
processes are discussed. Strong laws of large numbers and laws of the
iterated
logarithm are also obtained under easily verifiable conditions.
http://arxiv.org/abs/0711.3674
Author(s): Alexander Fribergh (ICJ) and Nina Gantert
Abstract: We consider an outwardly $\beta$-biased random walk $X_n$ on a
Galton-Watson
tree with leaves in the sub-ballistic regime. We prove that $X_n/n^
{\gamma}$
convergences in law and we characterize the limit law. The exponent $
\gamma\in
(0,1)$ is explicit and is a decreasing function of $\beta$. Key tools
for the
proof are classical decomposition results for Galton-Watson trees, a new
variant of regeneration times and the careful analysis of the time
the walker
spends in leaves.
http://arxiv.org/abs/0711.3686
Author(s): C. Kuelske and A. A. Opoku
Abstract: We present a general method to derive continuity estimates for
conditional
probabilities of general (possibly continuous) spin models sub jected
to local
transformations. Such systems arise in the study of a stochastic time-
evolution
of Gibbs measures or as noisy observations. We exhibit the minimal
necessary
structure for such double-layer systems. Assuming no a priori metric
on the
local state spaces, we define the posterior metric on the local image
space. We
show that it allows in a natural way to divide the local part of the
continuity
estimates from the spatial part (which is treated by Dobrushin
uniqueness
here). We show in the concrete example of the time evolution of
rotators on the
q-1 dimensional sphere how this method can be used to obtain
estimates in terms
of the familiar Euclidean metric.
http://arxiv.org/abs/0711.3764
Author(s): Masumi Nakajima
Abstract: By Probability theory, that is, by a kind of quasi-law of the
iterated
logarithm, we prove the title claim.
http://arxiv.org/abs/0711.3784
Author(s): Serban T. Belinschi and Alexandru Nica
Abstract: Let D be the space of non-commutative distributions of k-tuples of
selfadjoints in a C*-probability space (for a fixed k). We introduce a
semigroup of transformations B_t of D, such that every distribution in D
evolves under the B_t towards infinite divisibility with respect to free
additive convolution. The very good properties of B_t come from some
special
connections that we put into evidence between free additive
convolution and the
operation of Boolean convolution.
On the other hand we put into evidence a relation between the
transformations
B_t and free Brownian motion. More precisely, we introduce a
transformation Phi
of D which converts the free Brownian motion started at an arbitrary
distribution m in D into the process B_t (Phi(m)), t>0.
http://arxiv.org/abs/0711.3787
Author(s): Dong Wang
Abstract: This paper gives a PDE for multi-time joint probability of the
Airy process,
which generalizes Adler and van Moerbeke's result on the 2-time case.
As an
intermediate step, the PDE for the multi-time joint probability of
the Dyson
Brownian motion is also given.
http://arxiv.org/abs/0711.3797
Author(s): Oded Schramm
Abstract: G\'abor Elek introduced the notion of a hyperfinite graph family: a
collection of graphs is hypefinite if for every $\epsilon>0$ there is
some
finite $k$ such that each graph $G$ in the collection can be broken into
connected components of size at most $k$ by removing a set of edges
of size at
most $\epsilon|V(G)|$. We presently extend this notion to a certain
compactification of finite bounded-degree graphs, and show that if a
sequence
of finite graphs converges to a hyperfinite limit, then the sequence
itself is
hyperfinite.
http://arxiv.org/abs/0711.3808
Author(s): Jean Bertoin (DMA and Pma)
Abstract: We consider a (sub) critical Galton-Watson process with neutral
mutations
(infinite alleles model), and decompose the entire population into
clusters of
individuals carrying the same allele. We specify the law of this allelic
partition in terms of the distribution of the number of clone-
children and the
number of mutant-children of a typical individual. The approach
combines an
extension of Harris representation of Galton-Watson processes and a
version of
the ballot theorem. Some limit theorems related to the distribution
of the
allelic partition are also given.
http://arxiv.org/abs/0711.3852
Author(s): Gusztav Morvai and Benjamin Weiss
Abstract: The forward estimation problem for stationary and ergodic time series
$\{X_n\}_{n=0}^{\infty}$ taking values from a finite alphabet ${\cal
X}$ is to
estimate the probability that $X_{n+1}=x$ based on the observations
$X_i$,
$0\le i\le n$ without prior knowledge of the distribution of the process
$\{X_n\}$. We present a simple procedure $g_n$ which is evaluated on
the data
segment $(X_0,...,X_n)$ and for which, ${\rm error}(n) = |g_{n}(x)-P
(X_{n+1}=x
|X_0,...,X_n)|\to 0$ almost surely for a subclass of all stationary
and ergodic
time series, while for the full class the Cesaro average of the error
tends to
zero almost surely and moreover, the error tends to zero in probability.
http://arxiv.org/abs/0711.3856
Author(s): Nicholas Crawford
Abstract: We explore the joint behavior of a finite number of multi-
overlaps in the
high temperature phase of the SK model. Extending work by M. Tala-
grand, we
show that, when these objects are scaled to have non-trivial limiting
distributions, the joint behavior is described by a Gaussian process
with an
explicit covariance structure.
http://arxiv.org/abs/0711.3873
Author(s): J\'er\^ome Dedecker (LSTA) and Florence Merlev\`ede (PMA) and Magda Peligrad, Sergey Utev
Abstract: In this paper we derive the moderate deviation principle for
stationary
sequences of bounded random variables under martingale-type conditions.
Applications to functions of $\phi$-mixing sequences, contracting Markov
chains, expanding maps of the interval, and symmetric random walks on
the
circle are given.
http://arxiv.org/abs/0711.3924
Author(s): H. Dehling and S. R. Fleurke and C. Kuelske
Abstract: Consider an infinite tree with random degrees, i.i.d. over the
sites, with a
prescribed probability distribution with generating function G(s). We
consider
the following variation of Renyi's parking problem, alternatively called
blocking RSA: at every vertex of the tree a particle (or car) arrives
with rate
one. The particle sticks to the vertex whenever the vertex and all of
its
nearest neighbors are not occupied yet. We provide an explicit
expression for
the so-called parking constant in terms of the generating function.
http://arxiv.org/abs/0711.4061
Author(s): Tom Alberts and Scott Sheffield
Abstract: We establish an upper bound on the asymptotic probability of an SLE
(kappa)
curve hitting two small intervals on the real line as the interval
width goes
to zero, for the range 4 < kappa < 8. As a consequence we are able to
prove
that the SLE curve intersected with the real line has Hausdorff
dimension
2-8/kappa, almost surely.
http://arxiv.org/abs/0711.4070
Author(s): A.C.D. van Enter and W.M.Ruszel
Abstract: We study the Gibbsian character of time-evolved planar rotor
systems on Z^d,
d at least 2, in the transient regime, evolving with stochastic
dynamics and
starting with an initial Gibbs measure. We model the system by
interacting
Brownian diffusions, moving on circles. We prove that for small times
and
arbitrary initial Gibbs measures \nu, or for long times and both
high- or
infinite-temperature measure and dynamics, the evolved measure \nu^t
stays
Gibbsian. Furthermore we show that for a low-temperature initial
measure \nu,
evolving under infinite-temperature dynamics thee is a time interval
(t_0, t_1)
such that \nu^t fails to be Gibbsian in d=2.
http://arxiv.org/abs/0711.3621
Author(s): Arvind Ayyer and Carlangelo Liverani and Mikko Stenlund
Abstract: We establish a quenched Central Limit Theorem (CLT) for a smooth
observable
of random sequences of iterated linear hyperbolic maps on the torus.
To this
end we also obtain an annealed CLT for the same system. We show that,
almost
surely, the variance of the quenched system is the same as for the
annealed
system. Our technique is the study of the transfer operator on an
anisotropic
Banach space specifically tailored to use the cone condition
satisfied by the
maps.
http://arxiv.org/abs/0711.3818
Author(s): Janko Gravner and David Griffeath
Abstract: We introduce a three-dimensional, computationally feasible,
mesoscopic model
for snow crystal growth, based on diffusion of vapor, anisotropic
attachment,
and a semi-liquid boundary layer. Several case studies are presented
that
faithfully emulate a wide variety of physical snowflakes.
http://arxiv.org/abs/0711.4020
Author(s): Elena Villa
Abstract: The mean density of a random closed set $\Theta$ in $R^d$ with
Hausdorff
dimension $n$ is the Radon-Nikodym derivative of the expected measure
$E[H^n(\Theta\cap\cdot)]$ induced by $\Theta$ with respect to the usual
$d$-dimensional Lebesgue measure. We consider here inhomogeneous
Boolean models
with lower dimensional typical grain. Under general regularity
assumptions on
the typical grain, related to the existence of its Minkowski content,
and on
the intensity measure of the underlying Poisson point process, we
provide an
explicit formula for the mean density. Particular cases and examples
are also
discussed. Moreover, an estimator of the mean density naturally
arises in terms
of the empirical capacity functional, which turns to be closely
related to the
well known random variable density estimation by histograms in the
extreme case
$n=0$.
http://arxiv.org/abs/0711.4202
Author(s): Fabio Zucca
Abstract: A classical inequality, which is known for families of monotone
functions, is
generalized to a larger class of families of measurable functions.
Moreover we
characterize all the families of functions for which the equality
holds. We
apply this result to a problem arising from probability theory.
http://arxiv.org/abs/0711.4127
Author(s): Miklos Bona
Abstract: We review how the monotone pattern compares to other patterns in
terms of
enumerative results on pattern avoiding permutations. We consider
three natural
definitions of pattern avoidance, give an overview of classic and recent
formulas, and provide some new results related to limiting
distributions.
http://arxiv.org/abs/0711.4325
Author(s): V\'ictor Rivero
Abstract: We prove that a positive self-similar Markov process $(X,\mathbb
{P})$ that
hits 0 in a finite time admits a self-similar recurrent extension
that leaves 0
continuously if and only if the underlying L\'{e}vy process satisfies
Cram\'{e}r's condition.
http://arxiv.org/abs/0711.4442
Author(s): J\"urgen G\"artner and Rongfeng Sun
Abstract: Let $X$ and $Y$ be two independent random walks on $\Z^2$ with
zero mean and
finite variances, and let $L_t(X,Y)$ be the local time of $X-Y$ at
the origin
at time $t$. We show that almost surely with respect to $Y$, $L_t
(X,Y)/\log t$
conditioned on $Y$ converges in distribution to an exponential random
variable
with the same mean as the distributional limit of $L_t(X,Y)/\log t$
without
conditioning. This question arises naturally from the study of the
parabolic
Anderson model with a single moving catalyst, which is closely
related to a
pinning model.
http://arxiv.org/abs/0711.4488
Author(s): Denis Denisov and Serguei Foss and Dmitry Korshunov
Abstract: We study lower limits for the ratio $\frac{\bar{F^{*\tau}}(x)}
{\bar F(x)}$ of
tail distributions where $ F^{*\tau}$ is a distribution of a sum of a
random
size $\tau$ of i.i.d. random variables having a common distribution $F
$, and a
random variable $\tau$ does not depend on summands.
http://arxiv.org/abs/0711.4491
Author(s): Bruce K. Driver and Maria Gordina
Abstract: We prove an integrated Harnack inequality for heat kernels on uni-
modular Lie
groups. A key feature of these inequalities is that they only involve a
constant depending on a lower bound for the Ricci curvature tensor. In
particular, they are independent of dimension and hence are
applicable in
infinite--dimensional settings.
http://arxiv.org/abs/0711.4392
Author(s): Giambattista Giacomin and Hubert Lacoin and Fabio Lucio Toninelli
Abstract: We consider a hierarchical model of polymer pinning in presence of
quenched
disorder, introduced by B. Derrida, V. Hakim and J. Vannimenius in
1992, which
can be re-interpreted as an infinite dimensional dynamical system
with random
initial condition (the disorder). It is defined through a recurrence
relation
for the law of a random variable {R_n}_{n=1,2,...}, which in absence of
disorder (i.e., when the initial condition is degenerate) reduces to a
particular case of the well-known Logistic Map. The large-n limit of the
sequence of random variables 2^{-n} log R_n, a non-random quantity
which is
naturally interpreted as a free energy, plays a central role in our
analysis.
The model depends on a parameter alpha>0, related to the geometry of the
hierarchical lattice, and has a phase transition in the sense that
the free
energy is positive if the expectation of R_0 is larger than a certain
threshold
value, and it is zero otherwise. It was conjectured by Derrida et al.
(1992)
that disorder is relevant (respectively, irrelevant or marginally
relevant) if
1/21/2 we find the correct scaling form (for
weak
disorder) of the critical point shift.
http://arxiv.org/abs/0711.4649
Author(s): Ilkka Norros and Eero Saksman
Abstract: Let S(t,t') be the sigma-algebra generated by the differences X(s)-
X(s) with
s,s' in the interval(t,t'), where (X_t) is the fractional Brownian
motion
process with Hurst index H between 0 and 1. We prove that for any two
distinct
t and t' the sigma-algebras S(t-a,t+a) and S(t'-a,t'+a) are
asymptotically
independent as a tends to 0. We show this in the strong sense that
Shannon's
mutual information between these two sigma-algebras tends to zero as
a tends to
0. Some generalizations and quantitative estimates are provided also.
http://arxiv.org/abs/0711.4809
Author(s): Marc Kesseb\"ohmer and Bernd O. Stratmann
Abstract: In this paper we give non-trivial applications of the
thermodynamic formalism
to the theory of distribution functions of Gibbs measures (devil's
staircases)
supported on limit sets of finitely generated conformal iterated
function
systems in $\R$. For a large class of these Gibbs states we determine
the
Hausdorff dimension of the set of points at which the distribution
function of
these measures is not $\alpha$-H\"older-differentiable. The obtained
results
give significant extensions of recent work by Darst, Dekking,
Falconer, Li,
Morris, and Xiao. In particular, our results clearly show that the
results of
these authors have their natural home within thermodynamic formalism.
http://arxiv.org/abs/0711.4698
Author(s): Iddo Ben-Ari and Mathieu Merle and Alexander Roitershtein
Abstract: We consider a nearest neighbor random walk on the one-dimensional
integer
lattice with drift towards the origin determined by an asymptotically
vanishing
function of the number of visits to zero. We show the existence of
distinct
regimes according to the rate of decay of the drift. In particular,
when the
rate is sufficiently slow, the position of the random walk, properly
normalized, converges to a symmetric exponential law. In this regime, in
contrast to the classical case, the range of the walk scales
differently from
its position.
http://arxiv.org/abs/0711.4871
Author(s): Atilla Yilmaz
Abstract: We consider random walk $(X_n)_{n\geq0}$ on $\mathbb{Z}^d$ in a
space-time
product environment $\omega\in\Omega$. We take the point of view of the
particle and focus on the environment Markov chain $(T_{n,X_n}\omega)_
{n\geq0}$
where $T$ denotes the shift on $\Omega$. Conditioned on the particle
having
asymptotic speed equal to any given $\xi$, we show that the
environment Markov
chain converges to a stationary process $\mu_\xi$ under the annealed
measure.
When $d\geq3$ and $\xi$ is sufficiently close to the typical speed,
we prove
that annealed and quenched large deviations are equivalent and when
conditioned
on the particle having asymptotic speed $\xi$, the environment Markov
chain
converges to $\mu_\xi$ under the quenched measure as well. In this
case, we
show that $\mu_\xi$ is a stationary Markov process whose kernel is
obtained
from the original kernel by a Doob transform.
http://arxiv.org/abs/0711.4872
Author(s): Pierre Nolin (LM-Orsay and DMA)
Abstract: We give a self-contained and detailed presentation of Kesten's
results that
allow to relate critical and near-critical percolation on the triangular
lattice. They constitute an important step in the derivation of the
exponents
describing the near-critical behavior of this model. For future use and
reference, we also show how these results can be obtained in more
general
situations, and we state some new consequences.
http://arxiv.org/abs/0711.4948
Author(s): Jean B\'erard and Didier Piau
Abstract: We define a notion of coupling time with ambiguities for
interacting particle
systems, and show how this can be used to prove ergodicity and to
bound the
convergence time to equilibrium and the decay of correlations at
equilibrium. A
motivation is to provide simple conditions which ensure that
perturbed particle
systems share some properties of the underlying unperturbed system.
We apply
these results to context-dependent substitution models recently
introduced by
molecular biologists as descriptions of DNA evolution processes.
These models
take into account the influence of the neighboring bases on the
substitution
probabilities at a site of the DNA sequence, as opposed to most usual
substitution models which assume that sites evolve independently of
each other.
http://arxiv.org/abs/0712.0072
Author(s): Gusztav Morvai and Benjamin Weiss
Abstract: Finitarily Markovian processes are those processes
$\{X_n\}_{n=-\infty}^{\infty}$ for which there is a finite $K$ ($K =
K(\{X_n\}_{n=-\infty}^0$) such that the conditional distribution of
$X_1$ given
the entire past is equal to the conditional distribution of $X_1$
given only
$\{X_n\}_{n=1-K}^0$. The least such value of $K$ is called the memory
length.
We give a rather complete analysis of the problems of universally
estimating
the least such value of $K$, both in the backward sense that we have
just
described and in the forward sense, where one observes successive
values of
$\{X_n\}$ for $n \geq 0$ and asks for the least value $K$ such that the
conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=n-K+1}^n$ is
the same
as the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=-
\infty}^n$. We
allow for finite or countably infinite alphabet size.
http://arxiv.org/abs/0712.0105
Author(s): Marek Bozejko and Nizar Demni
Abstract: We characterize by the use of free probability the family of
measures for
which the mulitiplicative renormalization method applies with $h(x) =
(1-x)^_{-1}$. This provides a representation formula for their
Voiculescu
Transforms.
http://arxiv.org/abs/0712.0156
Author(s): Makoto Maejima and Ken-iti Sato
Abstract: It is shown that the limits of the nested subclasses of five
classes of
infinitely divisible distributions on $R^d$, which are the Jurek
class, the
Goldie-Steutel-Bondesson class, the class of selfdecomposable
distributions,
the Thorin class and the class of generalized type $G$ distributions,
are
identical with the closure of the class of stable distributions. More
general
results are also given.
http://arxiv.org/abs/0712.0206
Author(s): Jeong Han Kim and Ravi Montenegro and Yuval Peres and and Prasad Tetali
Abstract: We show a Birthday Paradox for self-intersections of Markov chains
with
uniform stationary distribution. As an application, we analyze
Pollard's Rho
algorithm for finding the discrete logarithm in a cyclic group G and
find that,
if the partition in the algorithm is given by a random oracle, then
with high
probability a collision occurs in order |G|^0.5 steps. This is the
first proof
of the correct order bound which does not assume that every step of the
algorithm produces an i.i.d. sample from G.
http://arxiv.org/abs/0712.0220
Author(s): Patrick Cattiaux (CMAP and MODAL'X) and Arnaud Guillin (LATP) and Feng-Yu Wang,
Liming Wu
Abstract: We show how to use Lyapunov functions to obtain functional
inequalities which
are stronger than Poincar\'e inequality (for instance logarithmic
Sobolev or
$F$-Sobolev). The case of Poincar\'e and weak Poincar\'e inequalities
was
studied in Bakry and al. This approach allows us to recover and
extend in an
unified way some known criteria in the euclidean case (Bakry-Emery,
Wang,
Kusuoka-Stroock ...).
http://arxiv.org/abs/0712.0235
Author(s): Geoffrey Grimmett and Svante Janson
Abstract: We study the random graph G_{n,\lambda/n} conditioned on the event
that all
vertex degrees lie in some given subset S of the non-negative
integers. Subject
to a certain hypothesis on S, the empirical distribution of the
vertex degrees
is asymptotically Poisson with some parameter \mux given as the root
of a
certain `characteristic equation' of S that maximises a certain function
\psis(\mu). Subject to a hypothesis on S, we obtain a partial
description of
the structure of such a random graph, including a condition for the
existence
(or not) of a giant component. The requisite hypothesis is in many cases
benign, and applications are presented to a number of choices for the
set S
including the sets of (respectively) even and odd numbers. The random
\emph{even} graph is related to the random-cluster model on the
complete graph
K_n.
http://arxiv.org/abs/0712.0270
Author(s): Markus Heydenreich and Remco van der Hofstad and Akira Sakai
Abstract: We consider self-avoiding walk, percolation and the Ising model
with long and
finite range. By means of the lace expansion we prove mean-field
behavior for
these models if $d>2(\alpha\wedge2)$ for self-avoiding walk and the
Ising
model, and $d>3(\alpha\wedge2)$ for percolation, where $d$ denotes the
dimension and $\alpha$ the power-law decay exponent of the coupling
function.
We provide a simplified analysis of the lace expansion based on the
trigonometric approach in Borgs et al. (2007)
http://arxiv.org/abs/0712.0312
Author(s): Dapeng Zhan
Abstract: We prove that the outer boundary of the final hull of some chordal
SLE$(\kappa;\vec{\rho})$ process has the same distribution as the
image of some
chordal SLE$(\kappa';\vec{\rho'})$ trace, where $\kappa>4$ and
$\kappa'=16/\kappa$; and the reversal of some SLE$(4;\vec{\rho})$
trace has the
same distribution as the time-change of some SLE$(4;\vec{\rho'})$
trace. And we
also study some geometric properties of some chordal SLE$(\kappa;\vec
{\rho})$
traces.
http://arxiv.org/abs/0712.0332
Author(s): J\'er\^ome Dedecker (LSTA) and Florence Merlev\`ede (PMA) and Emmanuel Rio
(LM-Versailles)
Abstract: In this paper, we give estimates of ideal or minimal distances
between the
distribution of the normalized partial sum and the limiting Gaussian
distribution for stationary martingale difference sequences or
stationary
sequences satisfying projective criteria. Applications to functions
of linear
processes and to functions of expanding maps of the interval are given.
http://arxiv.org/abs/0712.0179
Author(s): Shige Peng and Mingyu Xu
Abstract: In this paper, we study the relation between the smallest $g$-
supersolution
of constraint backward stochastic differential equation and viscosity
solution
of constraint semilineare parabolic PDE, i.e. variation inequalities.
And we
get an existence result of variation inequalities via constraint
BSDE, and
prove a uniqueness result under certain condition.
http://arxiv.org/abs/0712.0306
Author(s): Boualem Djehiche and Jens Svensson
Abstract: We study large deviation probabilities for a sum of dependent random
variables from a heavy-tailed factor model, assuming that the
components are
regularly varying. We identify conditions where both the factor and the
idiosyncratic terms contribute to the behaviour of the tail-
probability of the
sum. A simple conditional Monte Carlo algorithm is also provided
together with
a comparison between the simulations and the large deviation
approximation. We
also study large deviation probabilities for stochastic processes
with factor
structure. The processes involved are assumed to be Levy processes with
regularly varying jump measures. Based on the results of the first
part of the
paper, we show that large deviations on a finite time interval are
due to one
large jump that can come from either the factor or the idiosyncratic
part of
the process.
http://arxiv.org/abs/0712.0459
Author(s): Marianne Akian and Stephane Gaubert and Laure Ninove
Abstract: PageRank is a ranking of the web pages that measures how often a
given web
page is visited by a random surfer on the web graph, for a simple
model of web
surfing. It seems realistic that PageRank may also have an influence
on the
behavior of web surfers. We propose here a simple model taking into
account the
mutual influence between web ranking and web surfing. Our ranking, the
T-PageRank, is a nonlinear generalization of the PageRank. It is
defined as the
limit, if it exists, of some nonlinear iterates. A positive parameter
T, the
temperature, measures the confidence of the web surfer in the web
ranking. We
prove that, when the temperature is large enough, the T-PageRank is
unique and
the iterates converge globally on the domain. But when the
temperature is
small, there may be several T-PageRanks, that may strongly depend on the
initial ranking. Our analysis uses results of nonlinear Perron-Frobenius
theory, Hilbert projective metric and Birkhoff's coefficient of
ergodicity.
http://arxiv.org/abs/0712.0469
Author(s): Alessandra Bianchi
Abstract: We study a continuous time Glauber dynamics reversible with
respect to the
Ising model on hyperbolic graphs and analyze the effect of boundary
conditions
on the mixing time. Specifically, we consider the dynamics on an n-
vertex ball
of the hyperbolic graph $\H(v,s)$, where v is the number of neighbors
of each
vertex and s is the number of sides of each face, conditioned on having
(+)-boundary. If v>4, s>3 and for all low enough temperatures (phase
coexistence region) we prove that the spectral gap of this dynamics
is bounded
below by a constant independent of n. This implies that the mixing
time grows
at most linearly in n, in contrast to the free boundary case where it is
polynomial with exponent growing with the inverse temperature $\b$.
Such a
result extends to hyperbolic graphs the work done by Martinelli,
Sinclair and
Weitz for the analogous system on regular tree graphs, and provides a
further
example of influence of the boundary condition on the mixing time.
http://arxiv.org/abs/0712.0489
Author(s): Christina Goldschmidt and James B. Martin and Dario Span\`o
Abstract: Problem 1.5.7 from Pitman's Saint-Flour lecture notes: Does there
exist for
each n a fragmentation process (\Pi_{n,k}, 1 \leq k \leq n) taking
values in
the space of partitions of {1,2,...,n} such that \Pi_{n,k} is
distributed like
the partition generated by cycles of a uniform random permutation of
{1,2,...,n} conditioned to have k cycles? We show that the answer is
yes. We
also give a partial extension to general exchangeable Gibbs partitions.
http://arxiv.org/abs/0712.0556
Author(s): Mark M. Meerschaert and Erkan Nane and Yimin Xiao
Abstract: Let $W^H=\{W^H(t), t \in \rr\}$ be a fractional Brownian motion of
Hurst
index $H \in (0, 1)$ with values in $\rr$, and let $L = \{L_t, t \ge 0
\}$ be
the local time process at zero of a strictly stable L\'evy process $X=
\{X_t, t
\ge 0\}$ of index $1<\alpha\leq 2$ independent of $W^H$. The $\a$-
stable local
time fractional Brownian motion $Z^H=\{Z^H(t), t \ge 0\}$ is defined
by $Z^H(t)
= W^H(L_t)$. The process $Z^H$ is self-similar with self-similarity
index $H(1
- \frac 1 \alpha)$ and is related to the scaling limit of a
continuous time
random walk with heavy-tailed waiting times between jumps
(\cite{coupleCTRW,limitCTRW}). However, $Z^H$ does not have stationary
increments and is non-Gaussian.
In this paper we establish large deviation results for the process
$Z^H$. As
applications we derive upper bounds for the uniform modulus of
continuity and
the laws of the iterated logarithm for $Z^H$.
http://arxiv.org/abs/0712.0574
Author(s): Martin Jacobsen and Thomas Mikosch and Jan Rosinski and Gennady Samorodnitsky
Abstract: We study a group of related problems: the extent to which the
presence of
regular variation in the tail of certain $\sigma$-finite measures at
the output
of a linear filter determines the corresponding regular variation of
a measure
at the input to the filter. This turns out to be related to the
presence of a
particular cancellation property in $\sigma$-finite measures, which,
in turn,
is related to the uniqueness of the solution of certain functional
equations.
The techniques we develop are applied to weighted sums of iid random
variables,
to products of independent random variables, and to stochastic
integrals with
respect to L\'evy motions.
http://arxiv.org/abs/0712.0576
Author(s): Marie Albenque (LIAFA) and Jean-Fran\c{c}ois Marckert (LaBRI)
Abstract: Stack-triangulations appear as natural objects when one wants to
define some
increasing families of triangulations by successive additions of
faces. We
investigate the asymptotic behavior of rooted stack-triangulations
with $2n$
faces under two different distributions. We show that the uniform
distribution
on this set of maps converges, for a topology of local convergence, to a
distribution on the set of infinite maps. In the other hand, we show
that
rescaled by $n^{1/2}$, they converge for the Gromov-Hausdorff
topology on
metric spaces to the continuum random tree introduced by Aldous. Under a
distribution induced by a natural random construction, the distance
between
random points rescaled by $(6/11)\log n$ converge to 1 in probability.
We obtain similar asymptotic results for a family of increasing
quadrangulations.
http://arxiv.org/abs/0712.0593
Author(s): Erhan Bayraktar and Mike Ludkovski
Abstract: We study finite horizon optimal switching problems for hidden
Markov chain
models under partially observable Poisson processes. The controller
possesses a
finite range of strategies and attempts to track the state of the
unobserved
state variable using Bayesian updates over the discrete observations.
Such a
model has applications in economic policy making, staffing under
variable
demand levels and generalized Poisson disorder problems. We show
regularity of
the value function and explicitly characterize an optimal strategy.
We also
provide an efficient numerical scheme and illustrate our results with
several
computational examples.
http://arxiv.org/abs/0712.0413
Author(s): Stefano Bonaccorsi and Carlo Marinelli and Giacomo Ziglio
Abstract: We prove global well-posedness in the mild sense for a stochastic
partial
differential equation with a power-type nonlinearity and L\'evy noise.
Equations of this type arise in models of neurophysiology.
http://arxiv.org/abs/0712.0580
Author(s): Delia Coculescu and Ashkan Nikeghbali
Abstract: In this article, we define the notion of a filtration and then
give the basic
theorems on initial and progressive enlargements of filtrations.
http://arxiv.org/abs/0712.0622
Author(s): Nobuo Yoshida
Abstract: We consider branching random walks in $d$-dimensional integer
lattice with
time-space i.i.d. offspring distributions. When $d \ge 3$ and the
fluctuation
of the environment is well moderated by the random walk, we prove a
central
limit theorem for the density of the population, together with upper
bounds for
the density of the most populated site and the replica overlap. We
also discuss
the phase transition of this model in connection with directed
polymers in
random environment.
http://arxiv.org/abs/0712.0648
Author(s): Yueyun Hu and Nobuo Yoshida
Abstract: We consider branching random walks in $d$-dimensional integer
lattice with
time-space i.i.d. offspring distributions. This model is known to
exhibit a
phase transition: If $d \ge 3$ and the environment is "not too
random", then,
the total population grows as fast as its expectation with strictly
positive
probability. If,on the other hand, $d \le 2$, or the environment is
``random
enough", then the total population grows strictly slower than its
expectation
almost surely. We show the equivalence between the slow population
growth and a
natural localization property in terms of "replica overlap". We also
prove a
certain stronger localization property, whenever the total population
grows
strictly slower than its expectation almost surely.
http://arxiv.org/abs/0712.0649
Author(s): Parthanil Roy
Abstract: We consider a point process sequence induced by a stationary
symmetric
alpha-stable (0 < alpha < 2) discrete parameter random field. It is
easy to
prove, following the arguments in the one-dimensional case in Resnick
and
Samorodnitsky (2004), that if the random field is generated by a
dissipative
group action then the point process sequence converges weakly to a
cluster
Poisson process. For the conservative case, no general result is
known even in
the one-dimensional case. We look at a specific class of stable
random fields
generated by conservative actions whose effective dimensions can be
computed
using the structure theorem of finitely generated abelian groups. The
corresponding point processes sequence is not tight and hence needs
to be
properly normalized in order to ensure weak convergence. This weak
limit is
computed using extreme value theory and some counting techniques.
http://arxiv.org/abs/0712.0688
Author(s): Alexander Dukhovny and Jean-Luc Marichal
Abstract: The lifetime of a system of connected units under some natural
assumptions
can be represented as a random variable Y defined as a weighted lattice
polynomial of random lifetimes of its components. As such, the
concept of a
random variable Y defined by a weighted lattice polynomial of
(lattice-valued)
random variables is considered in general and in some special cases. The
central object of interest is the cumulative distribution function of
Y. In
particular, numerous results are obtained for lattice polynomials and
weighted
lattice polynomials in case of independent arguments and in general.
For the
general case, the technique consists in considering the joint
probability
generating function of "indicator" variables. A connection is studied
between Y
and order statistics of the set of arguments.
http://arxiv.org/abs/0712.0707
Author(s): Denis Denisov and Vsevolod Shneer
Abstract: We study the exact asymptotics for the distribution of the first time
$\tau_x$ a L\'evy process $X_t$ crosses a negative level $-x$. We
prove that
$\mathbf P(\tau_x>t)\sim V(x)\mathbf P(X_t\ge 0)/t$ as $t\to\infty$
for a
certain function $V(x)$. Using known results for the large deviations
of random
walks we obtain asymptotics for $\mathbf P(\tau_x>t)$ explicitly in
both light
and heavy tailed cases. We also apply our results to find asymptotics
for the
distribution of the busy period in an M/G/1 queue.
http://arxiv.org/abs/0712.0728
Author(s): David A. Levin and Malwina J. Luczak and Yuval Peres
Abstract: We study the Glauber dynamics for the Ising model on the complete
graph, also
known as the Curie-Weiss Model. For beta < 1, we prove that the dynamics
exhibits a cut-off: the distance to stationarity drops from near 1 to
near 0 in
a window of order n centered at [2(1-beta)]^{-1} n log n. For beta =
1, we
prove that the mixing time is of order n^{3/2}. For beta > 1, we study
metastability. In particular, we show that the Glauber dynamics
restricted to
states of non-negative magnetization has mixing time O(n log n).
http://arxiv.org/abs/0712.0790
Author(s): A. B. Dieker and J. Moriarty
Abstract: We give necessary and sufficient conditions for the stationary
density of
reflected Brownian motion (RBM) in a wedge to be written as a finite
sum of
terms of exponential product form. Relying on geometric ideas
reminiscent of
the reflection principle, we give an explicit formula for the density
in such
cases. We also show that the density can be written as a determinant.
http://arxiv.org/abs/0712.0844
Author(s): Paul Bourgade and Ashkan Nikeghbali and Alain Rouault
Abstract: Take a generic subgroup $\mathcal{G}$, endowed with its Haar
measure, from
$U(n,K)$, the unitary group of dimension $n$ over the field $K$ of real,
complex or quaternion numbers. We give some equalities in law for
$Z:=\det(\Id-G)$, $G\in\mathcal{G}$ : under some general conditions,
$Z$ can be
decomposed as a product of independent random variables, whose laws are
explicitly known (Section 2). Consequently $\mathcal{G}$, endowed with a
generalization of its Haar measure (the Hua-Pickrell measure), can be
generated
as a product of independent reflections. This constitutes a
generalization of
the well known Ewens sampling formula, corresponding to
$\mathcal{G}=\mathcal{S}_n$, the $n$-dimensional symmetric group.
Eventually, explicit determinantal point processes can be
associated to the
spectrum induced by the Hua-Pickrell measures, implying asymptotics on
correlation functions.
http://arxiv.org/abs/0712.0848
Author(s): V. Limic and P. Tarres
Abstract: We offer a reader-friendly introduction to the attracting edge
problem (also
known as the "triangle conjecture") and its most general current
solution of
Limic and Tarr\`es (2007). Little original research is reported;
rather this
article ``zooms in'' to describe the essential characteristics of two
different
techniques/approaches verifying the almost sure existence of the
attracting
edge for the strongly edge reinforced random walk (SERRW) on a
square. Both
arguments extend straightforwardly to the SERRW on even cycles.
Finally, we
show that the case where the underlying graph is a triangle cannot be
studied
by a simple modification of either of the two techniques.
http://arxiv.org/abs/0712.0958
Author(s): Frank Aurzada and Steffen Dereich and Michael Scheutzow and Christian Vormoor
Abstract: We study the quantization problem for certain types of jump
processes. The
probabilities for the number of jumps are assumed to be bounded by
Poisson
weights. Otherwise, jump positions and increments can be rather
generally
distributed and correlated. We show in particular that in many cases
entropy
coding error and quantization error have distinct rates. Finally, we
investigate the quantization problem for the special case of
$\mathbb{R}^d$-valued compound Poisson processes.
http://arxiv.org/abs/0712.0964
Author(s): A.E. Kyprianou and J.C. Pardo
Abstract: In this paper, for $\alpha\in (1, 2}$ we show that the $\alpha$-
stable
continuous-state branching process and the associated process
conditioned never
to become extinct are positive self-similar Markov processes.
Understanding the
interaction of the Lamperti transformation for continuous-state
branching
processes and the Lamperti transformation for positive self-similar
Markov
processes permits accessto a number of explicit results concerning
the paths of
stable-continuous branching processes and its conditioned version.
http://arxiv.org/abs/0712.0987
Author(s): Jungmin Choi and Kyounghee Kim
Abstract: The distribution of a time integral of geometric Brownian motion
is not well
understood. To price an Asian option and to obtain measures of its
dependence
on the parameters of time, strike price, and underlying market price,
it is
essential to have the distribution of time integral of geometric
Brownian
motion and it is also required to have a way to manipulate its
distribution. We
present integral forms for key quantities in the price of Asian
option and its
derivatives ({\it{delta, gamma,theta, and vega}}). For example for
any $a>0$
$\mathbb{E} [ (A_t -a)^+] = t -a + a^{2} \mathbb{E} [ (a+A_t)^{-1} \exp
(\frac{2M_t}{a+ A_t} - \frac{2}{a}) ]$, where $A_t = \int^t_0 \exp
(B_s -s/2)
ds$ and $M_t =\exp (B_t -t/2).$
http://arxiv.org/abs/0712.1093
Author(s): P. Patie
Abstract: We first introduce and derive some basic properties of a two-
parameters
family of one-sided Levy processes. Their Laplace exponents are given
in terms
of the Pochhammer symbol. This family includes, in a limit case, the
family of
Brownian motion with drifts. Then, we proceed by computing the
density of the
law of the exponential functional associated to some elements of this
family
(and their dual) and some transformations of these elements. These
densities
are expressed in terms of the Wright hypergeometric functions. By
means of
probabilistic arguments, we derive some interesting properties
enjoyed by these
functions. On the way we also characterize explicitly the density of the
semi-groups of the family of self-similar continuous state branching
processes
with immigration.
http://arxiv.org/abs/0712.1115
Author(s): Arnaud Le Ny
Abstract: These notes have been written to complete a mini-course
"Introduction to
(generalized) Gibbs measures" given at the universities UFMG
(Universidade
Federal de Minas Gerais, Belo Horizonte, Brasil) and UFRGS (Universidade
Federal do Rio Grande do Sul, Porto Alegre, Brasil) during the first
semester
2007. The main goal of the lectures was to describe Gibbs and
generalized Gibbs
measures on lattices at a rigorous mathematical level, as equilibirum
states of
systems of a huge number of particles in interaction. In particular,
our main
message is that although the historical approach based on potentials
has been
rather successful from a physical point of view, one has to insist on
(almost
sure) continuity properties of conditional probabilities to get a proper
mathematical framework.
http://arxiv.org/abs/0712.1171
Author(s): A. Borodin; S. Peche
Abstract: We introduce a generalization of the extended Airy kernel with two
sets of
real parameters. We show that this kernel arises in the edge scaling
limit of
correlation kernels of determinantal processes related to a directed
percolation model and to an ensemble of random matrices.
http://arxiv.org/abs/0712.1086
Author(s): Vladimir Vovk
Abstract: A new definition of events of game-theoretic probability zero in
continuous
time is proposed and used to prove results suggesting that trading in
financial
markets results in the emergence of properties usually associated with
randomness. This paper concentrates on "qualitative" results, stated
in terms
of order (or order topology) rather than in terms of the precise
values taken
by the price processes.
http://arxiv.org/abs/0712.1275
Author(s): G. Fabbri and B. Goldys
Abstract: We present an Hilbert space formulation for a set of implied
volatility
models introduced in \cite{BraceGoldys01} in which the authors studied
conditions for a family of European call options, varying the
maturing time and
the strike price $T$ an $K$, to be arbitrage free. The arbitrage free
conditions give a system of stochastic PDEs for the evolution of the
implied
volatility surface ${\hat\sigma}_t(T,K)$. We will focus on the family
obtained
fixing a strike $K$ and varying $T$. In order to give conditions to
prove an
existence-and-uniqueness result for the solution of the system it is
here
expressed in terms of the square root of the forward implied
volatility and
rewritten in an Hilbert space setting. The existence and the
uniqueness for the
(arbitrage free) evolution of the forward implied volatility, and
then of the
the implied volatility, among a class of models, are proved. Specific
examples
are also given.
http://arxiv.org/abs/0712.1343
Author(s): Jason Fulman
Abstract: It is shown that the combinatorics of commutation relations is
well suited
for analyzing the convergence rate of certain Markov chains. Examples
studied
include random walk on irreducible representations, a local random
walk on
partitions whose stationary distribution is the Ewens distribution,
and some
birth-death chains.
http://arxiv.org/abs/0712.1375
Author(s): Vladislav Kargin
Abstract: Lyapunov exponents of a dynamical system are a useful tool to
gauge the
stability and complexity of the system. This paper offers a
definition of
Lyapunov exponents for a sequence of free linear operators. The
definition is
based on the concept of the extended Fuglede-Kadison determinant. We
establish
the existence of Lyapunov exponents, derive formulas for their
calculation, and
show that Lyapunov exponents are additive with respect to operator
product. We
illustrate these results using an example of free operators whose
singular
values are distributed by the Marchenko-Pastur law, and relate this
example to
C. M. Newman's "triangle" law for the distribution of Lyapunov
exponents of
large random matrices with independent Gaussian entries. As an
interesting
by-product of our results, we derive a relation between the extended
Fuglede-Kadison determinant and Voiculescu's S-transform, which sheds
some
light on the multiplicativity of the S-transform.
http://arxiv.org/abs/0712.1378
Author(s): Vladimir Vovk
Abstract: This paper continues investigation of randomness-type properties
emerging in
idealized financial markets with continuous price processes. It is
shown that
the strong variation exponent of non-constant price processes has to
be 2, as
in the case of Brownian motion.
http://arxiv.org/abs/0712.1483
Author(s): Alain-Sol Sznitman
Abstract: We investigate the disconnection time of a simple random walk in a
discrete
cylinder with a large finite connected base. In a recent article of
A. Dembo
and the author it was found that for large $N$ the disconnection time of
$G_N\times\mathbb{Z}$ has rough order $|G_N|^2$, when
$G_N=(\mathbb{Z}/N\mathbb{Z})^d$. In agreement with a conjecture by I.
Benjamini, we show here that this behavior has broad generality when
the bases
of the discrete cylinders are large connected graphs of uniformly
bounded
degree.
http://arxiv.org/abs/0712.1497
Author(s): John H. Hubbard and Robyn L. Miller
Abstract: We provide a self-contained, accessible introduction to Ratner's
Equidistribution Theorem in the special case of horocyclic flow on a
complete
hyperbolic surface of finite area. We also prove a result due to
Breuillard: on
the modular surface an arbitrary uncentered random walk on the horocycle
through almost any point will fail to equidistribute, even though the
horocycles are themselves equidistributed.
http://arxiv.org/abs/0712.1300
Author(s): Bernard Bercu and Jean-Francois Bony and Vincent Bruneau
Abstract: We study the spectrum of the product of two Toeplitz operators.
Assume that
the symbols of these operators are continuous and real-valued and
that one of
them is non-negative. We prove that the spectrum of the product of
finite
section Toeplitz matrices converges to the spectrum of the product of
the
semi-infinite Toeplitz operators. We give an example showing that the
supremum
of this set is not always the supremum of the product of the two
symbols.
Finally, we provide an application in probability which is the first
motivation
of this study. More precisely, we obtain a large deviation principle for
Gaussian quadratic forms.
http://arxiv.org/abs/0712.1302
Author(s): A.B.J. Kuijlaars and A. Martinez-Finkelshtein and and F. Wielonsky
Abstract: We study a model of $n$ non-intersecting squared Bessel processes
in the
confluent case: all paths start at time $t = 0$ at the same positive
value $x =
a$, remain positive, and are conditioned to end at time $t = T$ at $x
= 0$. In
the limit $n \to \infty$, after appropriate rescaling, the paths fill
out a
region in the $tx$-plane that we describe explicitly. In particular,
the paths
initially stay away from the hard edge at $x = 0$, but at a certain
critical
time $t^*$ the smallest paths hit the hard edge and from then on are
stuck to
it. For $t \neq t^*$ we obtain the usual scaling limits from random
matrix
theory, namely the sine, Airy, and Bessel kernels. A key fact is that
the
positions of the paths at any time $t$ constitute a multiple orthogonal
polynomial ensemble, corresponding to a system of two modified Bessel-
type
weights. As a consequence, there is a $3 \times 3$ matrix valued
Riemann-Hilbert problem characterizing this model, that we analyze in
the large
$n$ limit using the Deift-Zhou steepest descent method. There are
some novel
ingredients in the Riemann-Hilbert analysis that are of independent
interest.
http://arxiv.org/abs/0712.1333
Author(s): D.L. Finkelshtein
Abstract: We study measures on the configuration spaces of two type
particles. Gibbs
measures on the such spaces are described. Main properties of
corresponding
relative energies densities and correlation functions are considered. In
particular, we show that a support set for the such Gibbs measure is
the set of
pairs of non-intersected configurations.
http://arxiv.org/abs/0712.1401
Author(s): Kshitij Khare
Abstract: For a skip-free Markov process on non-negative integers with
generator matrix
Q, we evaluate the joint Laplace transform of the occupation times
before
hitting the state n (starting at 0). This Laplace transform has a very
straightforward and familiar expression. We investigate the
properties of this
Laplace transform, especially the conditions under which the
occupation times
form a Markov chain.
http://arxiv.org/abs/0712.1646
Author(s): Luc Devroye and G\'abor Lugosi
Abstract: It is shown that functions defined on $\{0,1,...,r-1\}^n$
satisfying certain
conditions of bounded differences that guarantee sub-Gaussian tail
behavior
also satisfy a much stronger ``local'' sub-Gaussian property. For
self-bounding
and configuration functions we derive analogous locally subexponential
behavior. The key tool is Talagrand's [Ann. Probab. 22 (1994)
1576--1587]
variance inequality for functions defined on the binary hypercube
which we
extend to functions of uniformly distributed random variables defined on
$\{0,1,...,r-1\}^n$ for $r\ge2$.
http://arxiv.org/abs/0712.1686
Author(s): Hannu Reittu and Ilkka Norros
Abstract: We consider a variant of so called power-law random graph. A
sequence of
expected degrees corresponds to a power-law degree distribution with
finite
mean and infinite variance. In previous works the asymptotic picture
with
number of nodes limiting to infinity has been considered. It was
found that an
interesting structure appears. It has resemblance with such graphs
like the
Internet graph. Some simulations have shown that a finite sized
variant has
similar properties as well. Here we investigate this case in more
analytical
fashion, and, with help of some simple lower bounds for large valued
expectations of relevant random variables, we can shed some light
into this
issue. A new term, 'communication range random graph' is introduced to
emphasize that some further restrictions are needed to have a
relevant random
graph model for a reasonable sized communication network, like the
Internet. In
this case a pleasant model is obtained, giving the opportunity to
understand
such networks on an intuitive level. This would be beneficial in
order to
understand, say, how a particular routing works in such networks.
http://arxiv.org/abs/0712.1690
Author(s): Kai Matzutt
Abstract: This article describes, in elementary terms, a generic approach to
produce
discrete random tilings and similar random structures by using point
process
theory. The standard Voronoi and Delone tilings can be constructed in
this way.
For this purpose, convex polytopes are replaced by their vertex sets.
Three
explicit constructions are given to illustrate the concept.
http://arxiv.org/abs/0712.1684
Author(s): Alexander E. Holroyd and Robin Pemantle and Yuval Peres and Oded Schramm
Abstract: Suppose that red and blue points occur as independent homogeneous
Poisson
processes in R^d. We investigate translation-invariant schemes for
perfectly
matching the red points to the blue points. For any such scheme in
dimensions
d=1,2, the matching distance X from a typical point to its partner
must have
infinite d/2-th moment, while in dimensions d>=3 there exist schemes
where X
has finite exponential moments. The Gale-Shapley stable marriage is
one natural
matching scheme, obtained by iteratively matching mutually closest
pairs. A
principal result of this paper is a power law upper bound on the
matching
distance $X$ for this scheme. A power law lower bound holds also. In
particular, stable marriage is close to optimal (in tail behavior) in
d=1, but
far from optimal in d>=3. For the problem of matching Poisson points
of a
single color to each other, in d=1 there exist schemes where X has
finite
exponential moments, but if we insist that the matching is a
deterministic
factor of the point process then X must have infinite mean.
http://arxiv.org/abs/0712.1867
Author(s): Peter Jagers and Andreas Nordvall Lager{\aa}s
Abstract: It is well known that a simple, supercritical Bienaym\'e-Galton-
Watson
process turns into a subcritical such process, if conditioned to die
out. We
prove that the corresponding holds true for general, multi-type
branching,
where child-bearing may occur at different ages, life span may depend
upon
reproduction, and the whole course of events is thus affected by
conditioning
upon extinction.
http://arxiv.org/abs/0712.1872
Author(s): Florent Benaych-Georges (PMA)
Abstract: We prove some general results about the asymptotics of the
distribution of
the number of cycles of given length of a random permutation which
distribution
is invariant under conjugation. These results were first established
to be
applied in a forthcoming paper (Cycles of free words in several random
permutations with restricted cycles lengths) were we prove results
about cycles
of random permutations which can be written as free words in several
independent random permutations. However, we also apply them here to
prove
asymptotic results about random permutations with restricted cycle
lengths.
More specifically, for $A$ set of positive integers, we consider a
random
permutation chosen uniformly among permutations of $\{1,..., n\}$
which have
all their cycle lengths in $A$, and then let $n$ tend to infinity. We
prove
that if $A$ is infinite and large enough, then the number of cycles of
different given cycle lengths of this random permutation are
asymptotically
independent and distributed according to Poisson distributions. In
the case
where $A$ is finite, we prove that the behavior of these random
variables is
completely different: cycles with length $\max A$ are predominant.
http://arxiv.org/abs/0712.1903
Author(s): Ward Whitt
Abstract: This is an expository review paper elaborating on the proof of the
martingale
functional central limit theorem (FCLT). This paper also reviews
tightness and
stochastic boundedness, highlighting one-dimensional criteria for
tightness
used in the proof of the martingale FCLT. This paper supplements the
expository
review paper Pang, Talreja and Whitt (2007) illustrating the
``martingale
method'' for proving many-server heavy-traffic stochastic-process
limits for
queueing models, supporting diffusion-process approximations.
http://arxiv.org/abs/0712.1929
Author(s): Nicoletta Cancrini and Fabio Martinelli and Cyril Roberto and Cristina Toninelli
Abstract: Facilitated or kinetically constrained spin models (KCSM) are a
class of
interacting particle systems reversible w.r.t. to a simple product
measure.
Each dynamical variable (spin) is re-sampled from its equilibrium
distribution
only if the surrounding configuration fulfills a simple local
constraint which
\emph{does not involve} the chosen variable itself. Such simple
models are
quite popular in the glass community since they display some of the
peculiar
features of glassy dynamics, in particular they can undergo a
dynamical arrest
reminiscent of the liquid/glass transitiom. Due to the fact that the
jumps
rates of the Markov process can be zero, the whole analysis of the
long time
behavior becomes quite delicate and, until recently, KCSM have escaped a
rigorous analysis with the notable exception of the East model. In
these notes
we will mainly review several recent mathematical results which,
besides being
applicable to a wide class of KCSM, have contributed to settle some
debated
questions arising in numerical simulations made by physicists. We
will also
provide some interesting new extensions. In particular we will show
how to deal
with interacting models reversible w.r.t. to a high temperature Gibbs
measure
and we will provide a detailed analysis of the so called one spin
facilitated
model on a general connected graph.
http://arxiv.org/abs/0712.1934
Author(s): Olivier Zindy (WIAS)
Abstract: We consider one-dimensional directed trap models and suppose that the
trapping times are heavy-tailed. We obtain the inverse of a stable
subordinator
as scaling limit and prove an aging phenomenon expressed in terms of the
generalized arcsine law. These results confirm the status of
universality
described by Ben Arous and \v{C}ern\'y for a large class of graphs.
http://arxiv.org/abs/0712.1951
Author(s): Benedek Valko and Balint Virag
Abstract: We show that at any location away from the spectral edge, the
eigenvalues of
the Gaussian unitary ensemble and its general beta siblings converge to
Sine_beta, a translation invariant point process. This process has a
geometric
description in term of the Brownian carousel, a deterministic
function of
Brownian motion in the hyperbolic plane.
The Brownian carousel, a description of the a continuum limit of
random
matrices, provides a convenient way to analyze the limiting point
processes. We
show that the gap probability of Sine_beta is continuous in the gap
size and
$\beta$, and compute its asymptotics for large gaps. Moreover, the
stochastic
differential equation version of the Brownian carousel exhibits a phase
transition at beta=2.
http://arxiv.org/abs/0712.2000
Author(s): Michel Fliess (INRIA Futurs)
Abstract: The signal to noise ratio, which plays such an important role in
information
theory, is shown to become pointless in digital communications where
- symbols
are modulating carriers, which are solutions of linear differential
equations
with polynomial coefficients, - demodulations is achieved thanks to new
algebraic estimation techniques. Operational calculus, differential
algebra and
nonstandard analysis are the main mathematical tools.
http://arxiv.org/abs/0712.1875
Author(s): Malwina J. Luczak and Colin McDiarmid
Abstract: In the supermarket model there are n queues, each with a unit rate
server.
Customers arrive in a Poisson process at rate \lambda n, where 0<
\lambda <1.
Each customer chooses d > 2 queues uniformly at random, and joins a
shortest
one. It is known that the equilibrium distribution of a typical queue
length
converges to a certain explicit limiting distribution as n -> oo. We
quantify
the rate of convergence by showing that the total variation distance
between
the equilibrium distribution and the limiting distribution is
essentially of
order n^{-1}; and we give a corresponding result for systems starting
from
quite general initial conditions (not in equilibrium). Further, we
quantify the
result that the systems exhibit chaotic behaviour: we show that the
total
variation distance between the joint law of a fixed set of queue
lengths and
the corresponding product law is essentially of order at most n^{-1}.
http://arxiv.org/abs/0712.2091
Author(s): Auguste Aman (LMAI)
Abstract: In this paper we study the class of backward doubly stochastic
differential
equation (BDSDE, for short) whose terminal value depends on the
history of
forward diffusion. We first establish a probabilistic representation
for the
spatial gradient of the stochastic viscosity solution to a quasilinear
parabolic SPDE in the spirit of the Feynman-Kac formula, without
using the
derivatives of the coefficients of the corresponding BDSDE. Then such a
representation leads to a closed-form representation of the martingale
integrand of BDSDE, under only standard Lipschitz condition on the
coefficients.
http://arxiv.org/abs/0712.2219
Author(s): Yuri Bakhtin and Christine Heitsch
Abstract: We consider large random trees under Gibbs distributions and prove
a Large
Deviation Principle (LDP) for the distribution of degrees of vertices
of the
tree. The LDP rate function is given explicitly. An immediate
consequence is a
Law of Large Numbers for the distribution of vertex degrees in a
large random
tree. Our motivation for this study comes from the analysis of RNA
secondary
structures.
http://arxiv.org/abs/0712.2253
Author(s): Louis-Pierre Arguin
Abstract: We study point processes on the real line whose configurations X
can be
ordered decreasingly and evolve by increments which are functions of
correlated
gaussian variables. The correlations are intrinsic to the points and
quantified
by a matrix Q={q_ij}. Quasi-stationary systems are those for which
the law of
(X,Q) is invariant under the evolution up to translation of X. It was
conjectured by Aizenman and co-authors that the matrix Q of robustly
quasi-stationary systems must exhibit a hierarchal structure. This was
established recently, up to a natural decomposition of the system,
whenever the
set S_Q of values assumed by q_ij is finite. In this paper, we study the
general case where S_Q may be infinite. Using the past increments of the
evolution, we show that the law of robustly quasi-stationary systems
must obey
the Ghirlanda-Guerra identities, which first appear in the study of
spin glass
models. This provides strong evidence that the above conjecture also
holds in
the general case.
http://arxiv.org/abs/0712.2338
Author(s): Sebastian Andres and Max-K. von Renesse
Abstract: We construct a system of interacting two-sided Bessel processes on
the unit
interval and show that the associated empirical measure process
converges to
the Wasserstein Diffusion, assuming that Markov uniqueness holds for the
generating Wasserstein Dirichlet form. The proof is based on the
variational
convergence of an associated sequence of Dirichlet forms in the
generalized
Mosco sense of Kuwae and Shioya.
http://arxiv.org/abs/0712.2387
Author(s): R. Bass and X. Chen and J. Rosen
Abstract: We study functionals of the form \[\zeta_{t}=\int_0^{t}...\int_0^
{t} |
X_1(s_1)+...+ X_p(s_p)|^{-\sigma}ds_1... ds_p\] where $X_1(t),..., X_p
(t)$ are
i.i.d. $d$-dimensional symmetric stable processes of index $0<\bb\le 2
$. We
obtain results about the large deviations and laws of the iterated
logarithm
for $\zeta_{t}$.
http://arxiv.org/abs/0712.2401
Author(s): George Lowther
Abstract: In this paper we look at the properties of limits of a sequence of
real
valued time inhomogeneous diffusions. When convergence is only in the
sense of
finite-dimensional distributions then the limit does not have to be a
diffusion. However, we show that as long as the drift terms satisfy a
Lipschitz
condition and the limit is continuous in probability, then it will
lie in a
class of processes that we refer to as almost-continuous diffusions.
These
processes are strong Markov and satisfy an `almost-continuity'
condition. We
also give a simple condition for the limit to be a continuous
diffusion. These
results contrast with the multidimensional case where, as we show
with an
example, a sequence of two dimensional martingale diffusions can
converge to a
process that is both discontinuous and non-Markov.
http://arxiv.org/abs/0712.2428
Author(s): L. Gyorfi and G. Morvai and and S. Yakowitz
Abstract: This study concerns problems of time-series forecasting under the
weakest of
assumptions. Related results are surveyed and are points of departure
for the
developments here, some of which are new and others are new
derivations of
previous findings. The contributions in this study are all negative,
showing
that various plausible prediction problems are unsolvable, or in
other cases,
are not solvable by predictors which are known to be consistent when
mixing
conditions hold.
http://arxiv.org/abs/0712.2430
Author(s): Eric Chojnacki (IRSN) and Jean Baccou (IRSN) and S\'ebastien Destercke
(IRSN, IRIT)
Abstract: The treatment of both aleatory and epistemic uncertainty by recent
methods
often requires an high computational effort. In this abstract, we
propose a
numerical sampling method allowing to lighten the computational
burden of
treating the information by means of so-called fuzzy random variables.
http://arxiv.org/abs/0712.2141
Author(s): Vyacheslav M. Abramov
Abstract: The book of Lajos Tak\'acs \emph{Combinatorial Methods in the
Theory of
Stochastic Processes} has been published in 1967. It discusses
various problems
associated with $$ P_{k,i}=\mathrm{P}\left\{\sup_{1\leq
n\leq\rho(i)}(N_n-n)0$, and $\rho(i)$ is the smallest $n$ such that $N_n=n-i$, $i
\geq1$. (If
there is no such $n$, then $\rho(i)=\infty$.)
(*) is a discrete generalization of the classic ruin probability,
and its
value is represented as $P_{k,i}={Q_{k-i}}/{Q_k}$, where the sequence
$\{Q_k\}_{k\geq0}$ satisfies the recurrence relation of convolution
type:
$Q_0\neq0$ and $Q_k=\sum_{j=0}^k\pi_jQ_{k-j+1}$.
Since 1967 there have been many papers related to applications of the
generalized classic ruin probability. The present survey concerns
only with one
of the areas of application associated with asymptotic behavior of
$Q_k$ as
$k\to\infty$. The theorem on asymptotic behaviour of $Q_k$ as $k\to
\infty$ and
further properties of that limiting sequence are given on pages 22-23
of the
aforementioned book by Tak\'acs. In the present survey we discuss
applications
of Tak\'acs' asymptotic theorem and other related results in queueing
theory,
telecommunication systems and dams. Most of the results of this
survey are
based on the work of the author and have appeared during the last years.
http://arxiv.org/abs/0712.2480
Author(s): B. Derrida and G. Giacomin and H. Lacoin and F. L. Toninelli
Abstract: We study the critical point of directed pinning/wetting models
with quenched
disorder. The distribution K(.) of the location of the first contact
of the
(free) polymer with the defect line is assumed to be of the form
K(n)=n^{-\alpha-1}L(n), with L(.) slowly varying. The model undergoes a
(de)-localization phase transition: the free energy (per unit length)
is zero
in the delocalized phase and positive in the localized phase. For
\alpha<1/2 it
is known that disorder is irrelevant: quenched and annealed critical
points
coincide for small disorder, as well as quenched and annealed critical
exponents. The same has been proven also for \alpha=1/2, but under the
assumption that L(.) diverges sufficiently fast at infinity, an
hypothesis that
is not satisfied in the (1+1)-dimensional wetting model considered by
Forgacs
et al. (1986) and Derrida et al. (1992), where L(.) is asymptotically
constant.
Here we prove that, if 1/2<\alpha<1 or \alpha >1, then quenched and
annealed
critical points differ whenever disorder is present, and we give the
scaling
form of their difference for small disorder. In agreement with the so-
called
Harris criterion, disorder is therefore relevant in this case. In the
marginal
case \alpha=1/2, under the assumption that L(.) vanishes sufficiently
fast at
infinity, we prove that the difference between quenched and annealed
critical
points, which is known to be smaller than any power of the disorder
strength,
is positive: disorder is marginally relevant. Again, the case
considered by
Forgacs et al. (1986) and Derrida et al. (1992) is out of our
analysis and
remains open.
http://arxiv.org/abs/0712.2515
Author(s): Glenn Merlet (LIAFA)
Abstract: We analyze the asymptotic behavior of the sequence of random
variables (x(n,
x0))n \in N defined by x(0, x0) = x0 and x(n+1, x0) = A(n)x(n, x0),
where
(A(n))n \in N is a stationary and ergodic sequence of random matrices
with
entries in the semiring (R \cup {-\infinity}, max, +). Such sequences
model a
large class of discrete event systems, among which timed event graphs,
1-bounded Petri nets, some queuing networks, train or computer
networks. We
give a necessary condition for 1/n x(n, x0) n \in N to converge
almost-surely,
which proves to be sufficient when the A(n) are i.i.d. Moreover, we
construct a
new example, in which (A(n))n \in N is strongly mixing, that
condition is
satisfied, but 1/n x(n, x0) n \in N do not converge almost-surely.
http://arxiv.org/abs/0712.2559
Author(s): Pietro Caputo and Paolo Dai Pra and Gustavo Posta
Abstract: We develop a method, based on a Bochner-type identity, to obtain
estimates on
the exponential rate of decay of the relative entropy from
equilibrium of
Markov processes in discrete settings. When this method applies the
relative
entropy decays in a convex way. The method is shown to be rather
powerful when
applied to a class of birth and death processes. We then consider other
examples, including inhomogeneous zero-range processes and Bernoulli-
Laplace
models. For these two models, known results were limited to the
homogeneous
case, and obtained via the martingale approach, whose applicability to
inhomogeneous models is still unclear.
http://arxiv.org/abs/0712.2578
Author(s): L. Addario-Berry and B.A. Reed
Abstract: Given a branching random walk, let M_n be the minimum position of
any member
of the n'th generation. We calculate the expected value of M_n to
within O(1)
and prove exponential tail bounds for M_n around its expected value,
under
quite general conditions on the branching random walk. In particular,
together
with work of Bramson (1978), our results fully characterize the possible
behavior of M_n when the branching random walk has bounded branching
and step
size.
http://arxiv.org/abs/0712.2582
Author(s): S. Yakowitz and L. Gyorfi and J. Kieffer and G. Morvai
Abstract: Let $\{(X_i,Y_i)\}$ be a stationary ergodic time series with $(X,Y)
$ values
in the product space $\R^d\bigotimes \R .$ This study offers what is
believed
to be the first strongly consistent (with respect to pointwise, least-
squares,
and uniform distance) algorithm for inferring $m(x)=E[Y_0|X_0=x]$
under the
presumption that $m(x)$ is uniformly Lipschitz continuous. Auto-
regression, or
forecasting, is an important special case, and as such our work
extends the
literature of nonparametric, nonlinear forecasting by circumventing
customary
mixing assumptions. The work is motivated by a time series model in
stochastic
finance and by perspectives of its contribution to the issues of
universal time
series estimation.
http://arxiv.org/abs/0712.2592
Author(s): B.T. Graham
Abstract: We introduce a mean-field zero-range process. It is a Markov chain
model for
a microcanonical ensemble. We prove that the process converges to a
fluid
limit. The fluid limit rapidly relaxes to the appropriate Gibbs
distribution.
http://arxiv.org/abs/0712.2599
Author(s): Sergey G. Foss and Anatolii A. Puhalskii
Abstract: Let $\xi_1,\xi_2,...$ be i.i.d. random variables with negative
mean. Suppose
that $\mathbf{E}\exp(\lambda\xi_1)<\infty$ for some $\lambda>0$ and
that there
exists $\gamma>0$ with $\mathbf{E}\exp(\gamma\xi_1)=1$ . It is known
that if,
in addition, $\mathbf{E} \xi_1\exp(\gamma\xi_1)<\infty$, then the
most likely
way for the random walk $S_k=\sum_{i=1}^k\xi_i$ to reach a high level
is to
follow a straight line with a positive slope. We study the case where
$\mathbf{E} \xi_1\exp(\gamma\xi_1)=\infty$. Assuming that the
distribution
$\exp(\gamma x) \mathbf{P}(\xi_1\in dx) $ belongs to the domain of
attraction
of a spectrally positive stable law, we obtain a weak convergence
limit theorem
as $r\to\infty$ for the conditional distribution of the process
$\bl(r^{-1}\sum_{i=1}^{\lfloor t/(1- F (r))\rfloor}\xi_i, t\ge0\br)$
stopped at
the time when it reaches level 1 given that the latter event occurs.
The limit
is an increasing jump process. It is shown to be distributed as an
increasing
stable L\'evy process stopped at the time when it reaches level 1
conditioned
on the event this level is not overshot. Some properties of this
process are
studied.
http://arxiv.org/abs/0712.2637
Author(s): Francis Comets (PMA) and Francois Delarue (PMA) and Ren\'e Schott (IECN,
LORIA)
Abstract: We provide a large deviations analysis of deadlock phenomena
occurring in
distributed systems sharing common resources. In our model transition
probabilities of resource allocation and deallocation are time and space
dependent. The process is driven by an ergodic Markov chain and is
reflected on
the boundary of the d-dimensional cube. In the large resource limit,
we prove
Freidlin-Wentzell estimates, we study the asymptotic of the deadlock
time and
we show that the quasi-potential is a viscosity solution of a
Hamilton-Jacobi
equation with a Neumann boundary condition. We give a complete
analysis of the
colliding 2-stacks problem and show an example where the system has a
stable
attractor which is a limit cycle.
http://arxiv.org/abs/0712.2676
Author(s): J. Dedecker and C. Prieur
Abstract: We compute some dependence coefficients for the stationary Markov
chain whose
transition kernel is the Perron-Frobenius operator of an expanding
map $T$ of
$[0, 1]$ with a neutral fixed point. We use these coefficients to
prove a
central limit theorem for the partial sums of $f\circ T^i$, when $f$
belongs to
a large class of unbounded functions from $[0, 1]$ to ${\mathbb R}$.
We also
prove other limit theorems and moment inequalities.
http://arxiv.org/abs/0712.2726
Author(s): Persi Diaconis and Svante Janson
Abstract: We develop a clear connection between deFinetti's theorem for
exchangeable
arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of
graph
limits (work of Lovasz and many coauthors). Along the way, we
translate the
graph theory into more classical probability.
http://arxiv.org/abs/0712.2749
Author(s): Peter Buergisser and Felipe Cucker and Martin Lotz
Abstract: This paper has two agendas. Firstly, we exhibit new results for
coverage
processes. Let $p(n,m,\a)$ be the probability that $n$ spherical caps of
angular radius~$\a$ in $S^m$ do not cover the whole sphere $S^m$. We
give an
exact formula for $p(n,m,\a)$ in the case $\a\in [\pi/2,\pi]$ and an
upper
bound for $p(n,m,\a)$ in the case $\a\in [0,\pi/2]$, which tends to
$p(n,m,\pi/2)$ when $\a\to\pi/2$. In the case $\a\in [0,\pi/2]$ this
yields
upper bounds for the expected number of spherical caps of radius~$\a$
that are
needed to cover $S^m$.
Secondly, we study the condition number $\CC(A)$ of the linear
programming
feasibility problem $\exists x\in\R^{m+1}\, Ax\le 0,\, x\ne 0$ where
$A\in\R^{n\times (m+1)}$ is randomly chosen according to the standard
normal
distribution. We exactly determine the distribution of $\CC(A)$
conditioned
to~$A$ being feasible and provide an upper bound on the distribution
function
in the infeasible case. Using these results, we show that $\bE(\ln\CC
(A))\le
2\ln(m+1) + 3.31$ for all $n>m$, the sharpest bound for this
expectancy as of
today. Both agendas are related through a result which translates
between
coverage and condition.
http://arxiv.org/abs/0712.2816
Author(s): Zhenning Kong and Edmund M. Yeh
Abstract: Previous studies of connectivity in wireless networks have focused on
undirected geometric graphs. More sophisticated models such as
Signal-to-Interference-and-Noise-Ratio (SINR) model, however, usually
leads to
directed graphs. In this paper, we study percolation processes in
wireless
networks modelled by directed SINR graphs. We first investigate
interference-free networks, where we define four types of phase
transitions and
show that they take place at the same time. By coupling the directed
SINR graph
with two other undirected SINR graphs, we further obtain analytical
upper and
lower bounds on the critical density. Then, we show that with
interference,
percolation in directed SINR graphs depends not only on the density
but also on
the inverse system processing gain. We also provide bounds on the
critical
value of the inverse system processing gain.
http://arxiv.org/abs/0712.2469
Author(s): Miklos Bona
Abstract: We prove that the number of copies of any given permutation
pattern $q$ has
an asymptotically normal distribution in random permutations.
http://arxiv.org/abs/0712.2792
Author(s): Krzysztof Burdzy and Weining Kang and Kavita Ramanan
Abstract: We consider the Skorokhod problem in a time-varying interval. We
prove
existence and uniqueness for the solution. We also express the
solution in
terms of an explicit formula. Moving boundaries may generate
singularities when
they touch. We establish two sets of sufficient conditions on the moving
boundaries that guarantee that the variation of the local time of the
associated reflected Brownian motion is, respectively, finite and
infinite. We
also apply these results to study the semimartingale property of a
class of
two-dimensional reflected Brownian motions.
http://arxiv.org/abs/0712.2863
Author(s): Marcus Warfheimer
Abstract: We consider spin systems on the integers (i.e. interacting
particle systems
on the integers in which each coordinate only has two possible values
and only
one coordinate changes in each transition) whose rates are determined by
another process, called a background process. A canonical example is
the so
called contact process in randomly evolving environment (CPREE),
introduced and
analysed by E. Broman and furthermore studied by J. Steif and the
author, where
the marginals of the background process independently evolve as 2-
state Markov
chains and determine the recovery rates for a contact process. We
prove a
generalization of a result by Liggett, that under certain conditions
on the
rates there are only two extremal stationary distributions.
http://arxiv.org/abs/0712.2929
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (LSTA)
Abstract: We combine Malliavin calculus with Stein's method, in order to derive
explicit bounds in the Gaussian and Gamma approximations of random
variables in
a fixed Wiener chaos of a general Gaussian process. We also prove
results
concerning random variables admitting a possibly infinite Wiener chaotic
decomposition. Our approach generalizes, refines and unifies the
central and
non-central limit theorems for multiple Wiener-It\^o integrals
recently proved
(in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-
Latorre,
Peccati and Tudor. We apply our techniques to prove Berry-Esseen
bounds in the
Breuer-Major CLT for subordinated functionals of fractional Brownian
motion. By
using the well-known Mehler's formula for Ornstein-Uhlenbeck
semigroups, we
also recover a result recently proved by Chatterjee, in the context
of limit
theorems for linear statistics of eigenvalues of random matrices.
http://arxiv.org/abs/0712.2940
Author(s): Auguste Aman (LMAI) and Modeste Nzi
Abstract: We study the homogenization problem of one valued semi linear
refected
partial dif- ferential equation (reflected PDE for short) with nonlinear
Neumann condition. The non- linear term is a function of the solution
but not
of its gradient. The proof are fully probabilistic and use weak
convergence of
an associated reflected generalized backward differential stochastic
equation
(reflected GBSDE in short). We also give an homogeniza- tion property
for
solution of semi linear reflected PDE with Neumann boundary condition in
Sobolev space.
http://arxiv.org/abs/0712.2986
Author(s): Julien Dubedat
Abstract: Schramm-Loewner Evolutions ($\SLE$) are random curves in planar
simply
connected domains; the massless (Euclidean) free field in such a
domain is a
random distribution. Both have conformal invariance properties in
law. In the
present article, some relations between the two objects are studied. We
establish identities of partition functions between different
versions of
$\SLE$ and the free field with appropriate boundary conditions; this
involves
$\zeta$-regularization and the Polyakov-Alvarez conformal anomaly
formula. We
proceed with a construction of couplings of $\SLE$ with the free
field, showing
that, in a precise sense, chordal $\SLE$ is the solution of a stochastic
"differential" equation driven by the free field. Existence and
uniqueness in
law for these SDEs are proved for general $\kappa>0$; pathwise
uniqueness is
proved for chordal $\SLE_4$.
http://arxiv.org/abs/0712.3018
Author(s): Ariel Yadin
Abstract: Let A,B be two random subsets of a finite group G. We consider the
event that
the products of elements from A and B span the whole group; i.e. (AB
union BA)
= G. The study of this event gives rise to a group invariant we call
\Theta(G).
\Theta(G) is between 1/2 and 1, and is 1 if and only if the group is
abelian.
We show that a phase transition occurs as the size of A and B passes
\sqrt{\Theta(G)|G|\log|G|}; i.e. for any c>0, if the size of A and B
is less
than (1-c)\sqrt{\Theta(G)|G|\log|G|}, then with high probability (AB
union BA)
does not equal G. If A and B are larger than (1+c)\sqrt{\Theta(G)|G|
\log|G|}
then (AB union BA) equals G with high probability.
http://arxiv.org/abs/0712.3019
Author(s): Russell Lyons
Abstract: The notion of tree entropy was introduced by the author as a
normalized limit
of the number of spanning trees in finite graphs, but is defined on
random
infinite rooted graphs. We give some new expressions for tree entropy
and use
one of them to prove that tree entropy respects stochastic
domination. We also
prove that tree entropy is non-negative in the unweighted case.
http://arxiv.org/abs/0712.3035
Author(s): Wei Liu and Feng-Yu Wang
Abstract: This paper presents analogous results for stochastic fast-diffusion
equations. Since the fast-diffusion equation possesses weaker
dissipativity
than the porous medium one does, some technical difficulties appear
in the
study. As a compensation to the weaker dissipativity condition, a
Sobolev-Nash
inequality is assumed for the underlying self-adjoint operator in
applications.
Some concrete examples are constructed to illustrate the main results.
http://arxiv.org/abs/0712.3136
Author(s): Shizan Fang and Feng-Yu Wang and Bo Wu
Abstract: Starting from a sequence of independent Wright-Fisher diffusion
processes on
$[0,1]$, we construct a class of reversible infinite dimensional
diffusion
processes on $\DD_\infty:= \{{\bf x}\in
Let $M$ be a complete Riemnnian manifold and $\mu$ the
distribution of the
diffusion process generated by $\ff 1 2\DD+Z$ where $Z$ is a $C^1$-
vector
field. When $\Ric-\nn Z$ is bounded below and $Z$ has, for instance,
linear
growth, the transportation-cost inequality with respect to the
uniform distance
is established for $\mu$ on the path space over $M$. A simple example
is given
to show the optimality of the condition.
http://arxiv.org/abs/0712.3139
Author(s): Feng-Yu Wang
Abstract: We derive weighted log-Sobolev inequalities from a class of super
Poincar\'e
inequalities. As an application, the Talagrand inequality with larger
distances
are obtained. In particular, on a complete connected Riemannian
manifold, we
prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$
implies the
$L^{2/(2-\dd)}$-transportation cost inequality
$$W^\rr_{2/(2-\dd)}(f\mu,\mu)^{2/(2-\dd)}\le C\mu(f\log f), \mu(f)
=1, f\ge
0$$ for some constant $C>0$, and they are equivalent if the curvature
of the
corresponding generator is bounded below. Weighted log-Sobolev and
entropy-cost
inequalities are also derived for a large class of probability
measures on
$\R^d$.
http://arxiv.org/abs/0712.3142
Author(s): Feng-Yu Wang
Abstract: Let $P_t$ be the diffusion semigroup generated by $L:= \DD+\nn V$
on a
complete connected Riemannian manifold with
$\Ric\ge -(\si^2 \rr_o^2 +c)$ for some constants $\si, c>0$ and $
\rr_o$ the
Riemannian distance to a fixed point. It is shown that $P_t$ is
hypercontractive, or the log-Sobolev inequality holds for the associated
Dirichlet form, provided $-\Hess_V\ge \dd$ holds outside of a compact
set for
some constant $\dd>(1+\ss 2)\si\ss{d-1}.$ This indicates, at least in
finite
dimensions, that $\Ric$ and $-\Hess_V$ play quite different roles for
the
log-Sobolev inequality to hold. The supercontractivity and the
ultracontractivity are also studied.
http://arxiv.org/abs/0712.3143
Author(s): Feng-Yu Wang
Abstract: By establishing the intrinsic super-Poincar\'e inequality, some
explicit
conditions are presented for diffusion semigroups on a non-compact
complete
Riemannian manifold to be intrinsically ultracontractive. These
conditions, as
well as the resulting uniform upper bounds on the intrinsic heat
kernels, are
sharp for some concrete examples.
http://arxiv.org/abs/0712.3144
Author(s): Matthieu Marouby
Abstract: In this paper, we simulate sample paths of a class of symmetric
$\alpha$-stable processes using their series expression. We will
develop a
result in the approximation of shot-noise series. And finally, we
will get a
convergence rate for the approximation.
http://arxiv.org/abs/0712.3210
Author(s): Paul Doukhan (CREST and CES) and Olivier Wintenberger (CES and SAMOS)
Abstract: We prove the existence of a weakly dependent strictly stationary
solution of
the equation $ X_t=F(X_{t-1},X_{t-2},X_{t-3},...;\xi_t)$ called {\em
chain with
infinite memory}. Here the {\em innovations} $\xi_t$ constitute an
independent
and identically distributed sequence of random variables. The
function $F$
takes values in some Banach space and satisfies a Lipschitz-type
condition. We
also study the interplay between the existence of moments and the
rate of decay
of the Lipschitz coefficients of the function $F$. With the help of
the weak
dependence properties, we derive Strong Laws of Large Number, a
Central Limit
Theorem and a Strong Invariance Principle.
http://arxiv.org/abs/0712.3231
Author(s): Gregory F. Lawler
Abstract: This is the first expository set of notes on SLE I have written since
publishing a book two years ago [45]. That book covers material from a
year-long class, so I cannot cover everything there. However, these
notes are
not just a subset of those notes, because there is a slight change of
perspective. The main differences are:
o I have defined SLE as a finite measure on paths that is not
necessarily a
probability measure. This seems more natural from the perspective of
limits of
lattice systems and seems to be more useful when extending SLE to non-
simply
connected domains. (However, I do not discuss non-simply connected
domains in
these notes.)
o I have made more use of the Girsanov theorem in studying
corresponding
martingales and local martingales.
As in [45], I will focus these notes on the continuous process SLE
and will
not prove any results about convergence of discrete processes to SLE.
However,
my first lecture will be about discrete processes -- it is very hard to
appreciate SLE if one does not understand what it is trying to model.
http://arxiv.org/abs/0712.3256
Author(s): Gregory F. Lawler
Abstract: Some possible definitions for the natural parametrization of SLE
(Schramm-Loewner evolution) paths are proposed in terms of various
limits. One
of the definitions is used to give a new proof of the Hausdorff
dimension of
SLE paths.
http://arxiv.org/abs/0712.3263
Author(s): Franz Lehner and Markus Neuhauser and Wolfgang Woess
Abstract: Let $G$ be a finitely generated group and $X$ its Cayley graph
with respect
to a finite, symmetric generating set $S$. Furthermore, let $H$ be a
finite
group and $H \wr G$ the lamplighter group (wreath product) over $G$
with group
of "lamps" $H$. We show that the spectral measure (Plancherel
measure) of any
symmetric "switch--walk--switch" random walk on $H \wr G$ coincides
with the
expected spectral measure (integrated density of states) of the
random walk
with absorbing boundary on the cluster of the group identity for
Bernoulli site
percolation on $X$ with parameter $p = 1/|H|$. The return
probabilities of the
lamplighter random walk coincide with the expected (annealed) return
probabilites on the percolation cluster. In particular, if the
clusters of
percolation with parameter $p$ are almost surely finite then the
spectrum of
the lamplighter group is pure point. This generalizes results of
Grigorchuk and
Zuk, resp. Dicks and Schick regarding the case when $G$ is infinite
cyclic.
Analogous results relate bond percolation with another lamplighter
random walk.
In general, the integrated density of states of site (or bond)
percolation with
arbitrary parameter $p$ is always related with the Plancherel measure
of a
convolution operator by a signed measure on $H \wr G$, where $H = Z$
or another
suitable group.
http://arxiv.org/abs/0712.3135
Author(s): Francois Delarue (PMA) and Fr\'ed\'eric Lagouti\`ere (LJLL)
Abstract: We provide a probabilistic analysis of the upwind scheme for
multi-dimensional transport equations. We associate a Markov chain
with the
numerical scheme and then obtain a backward representation formula of
Kolmogorov type for the numerical solution. We then understand that
the error
induced by the scheme is governed by the fluctuations of the Markov
chain
around the characteristics of the flow. We show, in various
situations, that
the fluctuations are of diffusive type. As a by-product, we prove
that the
scheme is of order 1/2 for an initial datum in BV and of order 1/2-a,
for all
a>0, for a Lipschitz continuous initial datum. Our analysis provides
a new
interpretation of the numerical diffusion phenomenon.
http://arxiv.org/abs/0712.3217
Author(s): Erhan Bayraktar and Hao Xing
Abstract: In this paper we show that the optimal exercise boundary/free
boundary of the
American put option pricing problem for jump diffusions is continuously
differentiable (except at the maturity). We also discuss its higher
regularity.
http://arxiv.org/abs/0712.3323
Author(s): Wael Bahsoun and Igor V. Evstigneev and Michael I. Taksar
Abstract: The paper examines random dynamical systems related to the
classical von
Neumann and Gale models of economic growth. Such systems are defined
in terms
of multivalued operators in spaces of random vectors, possessing certain
properties of convexity and homogeneity. A central role in the theory
of von
Neumann-Gale dynamics is played by a special class of paths called
rapid (they
maximize properly defined growth rates). Up to now the theory lacked
quite
satisfactory results on the existence of such paths. This work
provides a
general existence theorem holding under assumptions analogous to the
standard
deterministic ones. The result solves a problem that remained open
for more
than three decades.
http://arxiv.org/abs/0712.3353
Author(s): Dirk Tasche
Abstract: Intuitively, the default risk of a single borrower is higher when
her or his
assets and debt are denominated in different currencies.
Additionally, the
default dependence of borrowers with assets and debt in different
currencies
should be stronger than in the one-currency case. By combining well-
known
models by Merton (1974), Garman and Kohlhagen (1983), and Vasicek
(2002) we
develop simple representations of PDs and asset correlations that
take into
account exchange rate risk. From these results, consistency
conditions can be
derived that link the changes in PD and asset correlation and do not
require
knowledge of hard-to-estimate parameters like asset value volatility.
http://arxiv.org/abs/0712.3363
Author(s): Lionel Levine and Yuval Peres
Abstract: We study the scaling limits of three different aggregation models
on Z^d:
internal DLA, in which particles perform random walks until reaching an
unoccupied site; the rotor-router model, in which particles perform
deterministic analogues of random walks; and the divisible sandpile,
in which
each site distributes its excess mass equally among its neighbors. As
the
lattice spacing tends to zero, all three models are found to have the
same
scaling limit, which we describe as the solution to a certain PDE
free boundary
problem in R^d. In particular, internal DLA has a deterministic
scaling limit.
We find that the scaling limits are quadrature domains, which have
arisen
independently in many fields such as potential theory and fluid
dynamics. Our
results apply both to the case of multiple point sources and to the
Diaconis-Fulton smash sum of domains.
http://arxiv.org/abs/0712.3378
Author(s): N. H. Bingham
Abstract: The link between Tauberian theorems and large deviations is
surveyed, with
particular reference to regular variation.
http://arxiv.org/abs/0712.3410
Author(s): R. A. Doney
Abstract: If $X$ is a spectrally positive stable process of index $\alpha\in
(1,2)$
whose L\'{e}vy measure has density $cx^{-\alpha-1}$ on $(0,\infty),$ and
$S_1=\sup_{0x)\backsim
c\alpha^{-1}x^{-\alpha}$ as $x\to\infty.$ It is also known that $S_1
$has a
continuous density, $s$ say. The point of this note is to show that
$s(x)\backsim cx^{-(\alpha+1)}$ as $x\to\infty.$
http://arxiv.org/abs/0712.3414
Author(s): J. du Toit and G. Peskir and A. N. Shiryaev
Abstract: Given a standard Brownian motion $B^{\mu}=(B_t^{\mu})_{0\le t\le T}
$ with
drift $\mu \in IR$ and letting $g$ denote the last zero of $B^{\mu}$
before
$T$, we consider the optimal prediction problem
V_*=\inf_{0\le \tau \le T}\mathsf {E}\:|\:g-\tau |
where the infimum is taken over all stopping times $\tau$ of $B^
{\mu}$.
Reducing the optimal prediction problem to a parabolic free-boundary
problem
and making use of local time-space calculus techniques, we show that the
following stopping time is optimal:
\tau_*=\inf {t\in [0,T] | B_t^{\mu} \le b_-(t) or B_t^{\mu} \ge b_+
(t)}
where the function $t\mapsto b_-(t)$ is continuous and increasing
on $[0,T]$
with $b_-(T)=0$, the function $t\mapsto b_+(t)$ is continuous and
decreasing on
$[0,T]$ with $b_+(T)=0$, and the pair $b_-$ and $b_+$ can be
characterised as
the unique solution to a coupled system of nonlinear Volterra integral
equations. This also yields an explicit formula for $V_*$ in terms of
$b_-$ and
$b_+$. If $\mu=0$ then $b_-=-b_+$ and there is a closed form
expression for
$b_{\pm}$ as shown in [10] using the method of time change from [4].
The latter
method cannot be extended to the case when $\mu \ne 0$ and the
present paper
settles the remaining cases using a different approach.
http://arxiv.org/abs/0712.3415
Author(s): Remi Rhodes
Abstract: We investigate stochastic homogenization for Reflected Stochastic
differential Equations on a half-plane. Our method relies on solving
the "third
boundary value problem" stated on a random medium and on a sector
condition for
the natural random Dirichlet form associated to the reflection term.
http://arxiv.org/abs/0712.3416
Author(s): Charles M. Goldie and Rudolf Gr\"ubel
Abstract: We obtain a strong law of large numbers and a functional central
limit
theorem, as $t\to\infty$, for the number of records up to time $t$
and the
Lebesgue measure (length) of the subset of the time interval $[0,t]$
during
which the Poisson process is in a record lifetime.
http://arxiv.org/abs/0712.3420
Author(s): Jean-Paul Ibrahim
Abstract: Following the recent investigations of J. Baik and T. Suidan in
\cite{baik2005gcl} and J. Martin and T. Bodineau in \cite
{bodineau2005upl}, we
prove large deviations properties for a last-passage percolation
model in
$\Z^{2}_{+}$ whose paths are close to the axis. The results are
obtained for
Gaussian as well as bounded weights and rely, as in \cite
{baik2005gcl} and
\cite{bodineau2005upl}, on a Skorokhod embedding in Brownian paths.
http://arxiv.org/abs/0712.3421
Author(s): Erik I. Broman and Federico Camia
Abstract: We study Mandelbrot's percolation process in dimension $d \geq 2$.
The
process generates random fractal sets by an iterative procedure which
starts by
dividing the unit cube $[0,1]^d$ in $N^d$ subcubes, and independently
retaining
or discarding each subcube with probability $p$ or $1-p$
respectively. This
step is then repeated within the retained subcubes at all scales. As
$p$ is
varied, there is a percolation phase transition in terms of paths for
all $d
\geq 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d \geq
3$.
For any $d \geq 2$, we consider the random fractal set produced at
the
path-percolation critical value $p_c(N,d)$, and show that the
probability that
it contains a path connecting two opposite faces of the cube $[0,1]^d
$ tends to
one as $N \to \infty$. As an immediate consequence, we obtain that
the above
probability has a discontinuity, as a function of $p$, at $p_c(N,d)$
for all
$N$ sufficiently large. This had previously been proved only for $d=2
$ (for any
$N \geq 2$). For $d \geq 3$, we prove analogous results for sheet-
percolation.
In dimension two, Chayes and Chayes proved that $p_c(N,2)$
converges, as $N
\to \infty$, to the critical density $p_c$ of site percolation on the
square
lattice. Assuming the existence of the correlation length exponent $
\nu$ for
site percolation on the square lattice, we establish the speed of
convergence
up to a logarithmic factor. In particular, our results imply that
$p_c(N,2)-p_c=(\frac{1}{N})^{1/\nu+o(1)}$ as $N \to \infty$.
http://arxiv.org/abs/0712.3422
Author(s): Anders Martin-L\"of
Abstract: Two modifications of the Petersburg game are considered: 1.
Truncation, so
that the player has a finite capital at his disposal. 2. A cost of
borrowing
capital, so that the player has to pay interest on the capital
needed. In both
cases limit theorems for the total net gain are derived, so that it
is easy to
judge if the game is favourable or not.
http://arxiv.org/abs/0712.3424
Author(s): Francesco Caravenna and Nicolas P\'etr\'elis
Abstract: We consider a model for a polymer chain interacting with a
sequence of
equi-spaced flat interfaces through a pinning potential. The
intensity \delta
\in R of the pinning interaction is constant, while the interface
spacing T =
T_N is allowed to vary with the size N of the polymer. Our main
result is the
explicit determination of the scaling behavior of the model in the
large N
limit, as a function of (T_N)_N and for fixed \delta > 0. In
particular, we
show that a transition occurs at T_N = O(\log N). Our approach is
based on
renewal theory.
http://arxiv.org/abs/0712.3426
Author(s): Nikita Ratanov and Alexander Melnikov
Abstract: The paper develops a new class of financial market models. These
models are
based on generalized telegraph processes: Markov random flows with
alternating
velocities and jumps occurring when the velocities are switching.
While such
markets may admit an arbitrage opportunity, the model under
consideration is
arbitrage-free and complete if directions of jumps in stock prices
are in a
certain correspondence with their velocity and interest rate
behaviour. An
analog of the Black-Scholes fundamental differential equation is
derived, but,
in contrast with the Black-Scholes model, this equation is
hyperbolic. Explicit
formulas for prices of European options are obtained using perfect
and quantile
hedging.
http://arxiv.org/abs/0712.3428
Author(s): Edward Omey
Abstract: Many statistics are based on functions of sample moments.
Important examples
are the sample variance $s_{n-1}^2$, the sample coefficient of
variation SV(n),
the sample dispersion SD(n) and the non-central $t$-statistic $t(n)$.
The
definition of these quantities makes clear that the vector defined by
(\sum_{i=1}^nX_i,\sum_{i=1}^nX_i^2)
plays an important role. In studying the asymptotic behaviour of
this vector
we start by formulating best possible conditions under which the vector
$(X,X^2)$ belongs to a bivariate domain of attraction of a stable
law. This
approach is new, uniform and simple. Our main results include a full
discussion
of the asymptotic behaviour of SV(n), SD(n) and $t^2(n)$. For
simplicity, in
restrict ourselves to positive random variables $X$.
http://arxiv.org/abs/0712.3440
Author(s): Sidney I. Resnick
Abstract: We attempt to bring some modest unity to three subareas of heavy tail
analysis and extreme value theory: limit laws for componentwise
maxima of iid
random variables;hidden regular variation and asymptotic
independence;conditioned limit laws when one component of a random
vector is
extreme. The common theme is multivariate regular variation on a cone
and the
three cases cited come from specifying the cones $[0,\infty]^d\setminus
\{\boldsymbol 0\};(0,\infty]^d;$ and $[0,\infty]\times (0,\infty]$.
http://arxiv.org/abs/0712.3442
Author(s): I.V. Evstigneev and N.H. Bingham
Abstract: This article contains the introduction to the special volume of
Stochastics
dedicated to Priscilla Greenwood, her CV and her list of publications.
http://arxiv.org/abs/0712.3459
Author(s): Alexander Novikov and Nino Kordzakhia
Abstract: Using the martingale approach we find sufficient conditions for
exponential
boundedness of first passage times over a level for ergodic first order
autoregressive sequences (AR(1)). Further, we prove a martingale
identity to be
used in obtaining explicit bounds for the expectation of first
passage times.
http://arxiv.org/abs/0712.3468
Author(s): Eric Benhamou (LJK) and Emmanuel Gobet (LJK) and Mohammed Miri (LJK)
Abstract: Using Malliavin calculus techniques, we derive an analytical
formula for the
price of European options, for any model including local volatility
and jump
Poisson process. We show that the accuracy of the formula depends on the
smoothness of the payoff. Our approach relies on an asymptotic expansion
related to small diffusion and small jump frequency. As a
consequence, the
calibration of such model becomes very fast.
http://arxiv.org/abs/0712.3485
Author(s): Michael Sch\"urmann and Michael Skeide and Silvia Volkwardt
Abstract: A L\'evy process on a *-bialgebra is given by its generator, a
conditionally
positive hermitian linear functional vanishing at the unit element. A
*-algebra
homomorphism k from a *-bialgebra C to a *-bialgebra B with the
property that k
respects the counits maps generators on B to generators on C. A
tranformation
between the corrresponding two L\'evy processes is given by forming
infinitesimal convolution products.
This general result is applied to various situations, e.g. to a *-
bialgebra
and its associated primitive tensor *-bialgebra (called "generator
process") as
well as its associated group-like *-bialgebra (called Weyl-*-
bialgebra). It
follows that a L\'evy process on a *-bialgebra can be realized on
Bose Fock
space as the infinitesimal convolution product of its generator
process such
that the vacuum vector is cyclic for the L\e'vy process. Moreover, we
obtain
convolution approximations of the Az\'ema martingale by the Wiener
process and
vice versa.
http://arxiv.org/abs/0712.3504
Author(s): Jeff Kahn and Michael Neiman
Abstract: We settle, mostly in the negative, a number of conjectures of R.
Pemantle and
D. Wagner concerning negative correlation and log-concavity
properties for
probability measures and relations between them. Most of the negative
results
have also been obtained, independently but somewhat earlier, by
Borcea et al.
We also give short proofs of a pair of results due to Pemantle and
Borcea at
al.; prove that "almost exchangeable" measures satisfy the "Feder-
Mihail"
property, thus providing the first "non-obvious" example of a class
of measures
for which this important property can be shown to hold; and mention some
further questions.
http://arxiv.org/abs/0712.3507
Author(s): Gr\'egory Benmenzer (LJK) and Emmanuel Gobet (LJK) and C\'eline J \'erusalem
(LJK)
Abstract: In this article we present a continuous time model for natural gas
and crude
oil future prices. Its main feature is the possibility to link both
energies in
the long term and in the short term. For each energy, the future
returns are
represented as the sum of volatility functions driven by motions.
Under the
risk neutral probability, the motions of both energies are correlated
Brownian
motions while under the historical probability, they are cointegrated
by a
Vectorial Error Correction Model. Our approach is equivalent to
defining the
market price of risk. This model is free of arbitrage: thus, it can
be used for
risk management as well for option pricing issues. Calibration on
European
market data and numerical simulations illustrate well its behavior.
http://arxiv.org/abs/0712.3537
Author(s): Federico Camia
Abstract: We consider a type of dependent percolation introduced by Aizenman
and
Grimmett, who showed that certain "enhancements" of independent
(Bernoulli)
percolation, called essential, make the percolation critical probability
strictly smaller. In this paper we first prove that, for two-dimensional
enhancements with a natural monotonicity property, being essential is
also a
necessary condition to shift the critical point. We then show that
(some)
critical exponents and the scaling limit of crossing probabilities of a
two-dimensional percolation process are unchanged if the process is
subjected
to a monotonic enhancement that is not essential. This proves a form of
universality for all dependent percolation models obtained via a
monotonic
enhancement (of Bernoulli percolation) that does not shift the
critical point.
For the case of site percolation on the triangular lattice, we also
prove a
stronger form of universality by showing that the full scaling limit
is not
affected by any monotonic enhancement that does not shift the
critical point.
http://arxiv.org/abs/0712.3412
Author(s): St\'ephane Attal (ICJ) and Ameur Dhahri (CEREMADE)
Abstract: Among the discrete evolution equations describing a quantum system
$\rH_S$
undergoing repeated quantum interactions with a chain of exterior
systems, we
study and characterize those which are directed by classical random
variables
in $\RR^N$. The characterization we obtain is entirely algebraical in
terms of
the unitary operator driving the elementary interaction. We show that
the
solutions of these equations are then random walks on the group $U
(\rH_0)$ of
unitary operators on $\rH_0$.
http://arxiv.org/abs/0712.3417
Author(s): St\'ephane Attal (ICJ) and Nadine Guillotin-Plantard (ICJ)
Abstract: We study the statistical properties of the triplet
$(\sigma_x,\sigma_y,\sigma_z)$ of Pauli matrices going through a
sequence of
noisy channels, modeled by the repetition of a general, trace-
preserving,
completely positive map. We show a non-commutative central limit
theorem for
the distribution of this triplet, which shows up a 3-dimensional
Brownian
motion in the limit with a non-trivial covariance matrix. We also
prove a large
deviation principle associated to this convergence, with an explicit
rate
function depending on the stationary state of the noisy channel.
http://arxiv.org/abs/0712.3418
Author(s): Andreas E. Kyprianou and V\'i ctor Rivero
Abstract: Following from recent developments by Hubalek and Kyprianou, the
objective of
this paper is to provide further methods for constructing new
families of scale
functions for spectrally negative L\'evy processes which are completely
explicit. This is the result of an observation in the aforementioned
paper
which permits feeding the theory of Bernstein functions directly into
the
Wiener-Hopf factorization for spectrally negative L\'evy processes.
Many new,
concrete examples of scale functions are offered although the
methodology in
principle delivers still more explicit examples than those listed.
http://arxiv.org/abs/0712.3588
Author(s): Rainer Avikainen
Abstract: We consider upper bounds for the approximation error E|g(X)-g(\hat
X)|^p,
where X and \hat X are random variables such that \hat X is an
approximation of
X in the L_p-norm, and the function g belongs to certain function
classes,
which contain e.g. functions of bounded variation. We apply the
results to the
approximations of a solution of a stochastic differential equation at
time T by
the Euler and Milstein schemes. For the Euler scheme we provide also
a lower
bound.
http://arxiv.org/abs/0712.3635
Author(s): Jean-Baptiste Gou\'er\'e (MAPMO)
Abstract: We consider some continuum percolation models. We are mainly
interested in
giving some sufficient conditions for absence of percolation. We give
some
general conditions and then focuse on two examples. The first one is a
multiscale percolation model based on the Boolean model. It was
introduced by
Meester and Roy and subsequently studied by Menshikov, Popov and
Vachkovskaia.
The second one is based on the stable marriage of Poisson and Lebesgue
introduced by Hoffman, Holroyd and Peres and whose percolation
properties have
been studied by Freire, Popov and Vachkovskaia. This is a preliminary
version:
in particular, some parts of the introduction need to be developped.
http://arxiv.org/abs/0712.3638
Author(s): Anne Fey and Remco van der Hofstad and Marten Klok
Abstract: We study sample covariance matrices of the form $W=\frac 1n C C^T
$, where $C$
is a $k\times n$ matrix with i.i.d. mean zero entries. This is a
generalization
of so-called Wishart matrices, where the entries of $C$ are
independent and
identically distributed standard normal random variables. Such
matrices arise
in statistics as sample covariance matrices, and the high-dimensional
case,
when $k$ is large, arises in the analysis of DNA experiments.
We investigate the large deviation properties of the largest and
smallest
eigenvalues of $W$ when either $k$ is fixed and $n\to \infty$, or $k_n
\to
\infty$ with $k_n=o(n/\log\log{n})$, in the case where the squares of
the
i.i.d. entries have finite exponential moments. Previous results,
proving a.s.
limits of the eigenvalues, only require finite fourth moments.
Our most explicit results for $k$ large are for the case where the
entries of
$C$ are $\pm1$ with equal probability. We relate the large deviation
rate
functions of the smallest and largest eigenvalue to the rate
functions for
independent and identically distributed standard normal entries of $C
$. This
case is of particular interest, since it is related to the problem of
the
decoding of a signal in a code division multiple access system
arising in
telecommunications. In this example, $k$ plays the role of the number
of users
in the system, and $n$ is the length of the coding sequence of each
of the
users. Each user transmits at the same time and uses the same
frequency, and
the codes are used to distinguish the signals of the separate users. The
results imply large deviation bounds for the probability of a bit
error due to
the interference of the various users.
http://arxiv.org/abs/0712.3650
Author(s): Zbigniew S. Szewczak
Abstract: It is shown that for sums of functionals of digits in continued
fraction
expansion the Kolmogorov-Feller weak laws of large numbers and the
Khinchine-L\'evy-Feller-Raikov characterization of the domain of
attraction of
the normal law hold.
http://arxiv.org/abs/0712.3681
Author(s): Gr\'egory Marc Miermont (LM-Orsay and PMA)
Abstract: We give a new proof of a theorem by Le Gall & Paulin, showing that
scaling
limits of random planar quadrangulations are homeomorphic to the 2-
sphere. The
main geometric tool is a reinforcement of the notion of Gromov-Hausdorff
convergence, called 1-regular convergence, that preserves topological
properties of metric surfaces.
http://arxiv.org/abs/0712.3687
Author(s): Gr\'egory Marc Miermont (PMA and LM-Orsay)
Abstract: We investigate Voronoi-like tessellations of bipartite
quadrangulations on
surfaces of arbitrary genus, by using a natural generalization of a
bijection
of Marcus and Schaeffer allowing to encode such structures into
labeled maps
with a fixed number of faces. We investigate the scaling limits of
the latter.
Applications include asymptotic enumeration results for
quadrangulations, and
typical metric properties of randomly sampled quadrangulations. In
particular,
we show that scaling limits of these random quadrangulations are such
that
almost every pair of points are linked by a unique geodesic.
http://arxiv.org/abs/0712.3688
Author(s): Nadine Guillotin-Plantard (ICJ) and Cl\'ementine Prieur (LSProba)
Abstract: We prove a central limit theorem for linear triangular arrays
under weak
dependence conditions. Our result is then applied to the study of
dependent
random variables sampled by a $\bbZ$-valued transient random walk.
This extends
the results obtained by Guillotin-Plantard & Schneider (2003). An
application
to parametric estimation by random sampling is also provided.
http://arxiv.org/abs/0712.3696
Author(s): F. G\"otze and A. Tikhomirov
Abstract: It is shown that the Kolmogorov distance between the spectral
distribution
function of a random covariance matrix $\frac1p XX^T$, where $X$ is a
$n\times
p$ matrix with independent entries and the distribution function of the
Marchenko-Pastur law is of order $O(n^{-1/2})$. The bounds hold {\it
uniformly}
for any $p$, including $\frac pn$ equal or close to 1.
http://arxiv.org/abs/0712.3725
Author(s): Stefan Ankirchner and Peter Imkeller and Goncalo dos Reis
Abstract: This paper is concerned with the study of insurance related
derivatives on
financial markets that are based on non-tradable underlyings, but are
correlated with tradable assets. We calculate exponential utility-based
indifference prices, and corresponding derivative hedges. We use the
fact that
they can be represented in terms of solutions of forward-backward
stochastic
differential equations (FBSDE) with quadratic growth generators. We
derive the
Markov property of such FBSDE and generalize results on the
differentiability
relative to the initial value of their forward components. In this
case the
optimal hedge can be represented by the price gradient multiplied
with the
correlation coefficient. This way we obtain a generalization of the
classical
'delta hedge' in complete markets.
http://arxiv.org/abs/0712.3746
Author(s): Christian Bayer and Josef Teichmann
Abstract: We prove a stochastic Taylor expansion for SPDEs and apply this
result to
obtain cubature methods, i. e. high order weak approximation schemes
for SPDEs,
in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak
convergence for well-defined classes of test functions if the process
starts at
sufficiently regular points. We can also derive analogous results in the
presence of L\'evy processes of finite type, here the results seem to
be new
even in finite dimension. Several numerical examples are added.
http://arxiv.org/abs/0712.3763
Author(s): Anders Szepessy
Abstract: Stochastic Langevin molecular dynamics for nuclei is derived from
quantum
classical molecular dynamics, also called Ehrenfest dynamics, at
positive
temperature, assuming that the molecular bulk system is in
equilibrium and that
the initial data for the electrons is stochastically perturbed from
the ground
state. The initial electron probability distribution is derived from the
Liouville equilibrium solution generated by the nuclei acting as a
heat bath
for the electrons. The diffusion and friction coefficients in the
Langevin
equation satisfy Einstein's fluctuation-dissipation relation. The
fluctuating
initial data yields, in addition to the fluctuating diffusion terms,
also a
contribution to the drift, modifying the standard ab initio Born-
Oppenheimer
solution at zero temperature, where the electrons are in their ground
state for
the current nuclear configuration. The dissipative friction mechanism
comes
from the evolution of the electron ground state, due to slow dynamics
of the
nuclei, while the modified drift can be understood as the mean field
Born-Oppenheimer solution, for the proposed initial electron
distribution at
positive temperature.
http://arxiv.org/abs/0712.3656
Author(s): Xavier Bardina and Maria Jolis and Ciprian Tudor (CES and SAMOS)
Abstract: We study the convergence to the multiple Wiener-It\^{o} integral from
processes with absolutely continuous paths. More precisely, consider
a family
of processes, with paths in the Cameron-Martin space, that converges
weakly to
a standard Brownian motion in $\mathcal C_0([0,T])$. Using these
processes, we
construct a family that converges weakly, in the sense of the finite
dimensional distributions, to the multiple Wiener-It\^{o} integral
process of a
function $f\in L^2([0,T]^n)$. We prove also the weak convergence in
the space
$\mathcal C_0([0,T])$ to the second order integral for two important
families
of processes that converge to a standard Brownian motion.
http://arxiv.org/abs/0712.3837
Author(s): Pierre van Moerbeke
Abstract: This set of Montreal lectures is an elementary and sketchy
introduction to
the general field of random matrices. The first half is devoted to
combinatorial models, whereas the second half deals with random matrix
questions(GUE, etc...).
http://arxiv.org/abs/0712.3847
Author(s): Tam\'as Szabados (Budapest University of Technology and Economics),
Bal\'azs Sz\'ekely (Budapest University of Technology and Economics)
Abstract: A new approach to stochastic integration is described, which is
based on an
a.s. pathwise approximation of the integrator by simple, symmetric
random
walks. Hopefully, this method is didactically more advantageous, more
transparent, and technically less demanding than other existing ones.
In a
large part of the theory one has a.s. uniform convergence on
compacts. In
particular, it gives a.s. convergence for the stochastic integral of
a finite
variation function of the integrator, which is not c\`adl\`ag in
general.
http://arxiv.org/abs/0712.3908
Author(s): Yuri Bakhtin
Abstract: We consider a white noise perturbation of dynamics in the
neighborhood of a
heteroclinic network. We show that under the logarithmic time
rescaling the
diffusion converges in distributon in a special topology to a piecewise
constant process that jumps between saddle points along the
heteroclinic orbits
of the network. We also obtain precise asymptotics for the exit
measure for a
domain containing the starting point of the diffusion.
http://arxiv.org/abs/0712.3952
Author(s): Huili Tang
Abstract: Let $L$ be the operator defined on $C^2$ functions by $$L
f(x)=\int[f(x+h)-f(x)-1_{(|h|\leq 1)}\nabla f(x)\cdot
h]\frac{n(x,h)}{|h|^{d+\alpha(x)}}dh.$$ This is an operator of
variable order
and the corresponding process is of pure jump type. We consider the
martingale
problem associated with $L$. Sufficient conditions for existence and
uniqueness
are given. Transition density estimates for $\alpha$-stable processes
are also
obtained.
http://arxiv.org/abs/0712.4137
Author(s): Zbigniew S. Szewczak
Abstract: An operator form of asymptotic expansions for Markov chains is
established.
Coefficients are given explicitly. Such expansions require a certain
modification of the classical spectral method. They prove to be
extremely
useful within the context of large deviations.
http://arxiv.org/abs/0712.4199
Author(s): Guodong Pang and Rishi Talreja and Ward Whitt
Abstract: This is an expository review paper illustrating the ``martingale
method'' for
proving many-server heavy-traffic stochastic-process limits for queueing
models, supporting diffusion-process approximations. Careful
treatment is given
to an elementary model -- the classical infinite-server model $M/M/
\infty$, but
models with finitely many servers and customer abandonment are also
treated.
The Markovian stochastic process representing the number of customers
in the
system is constructed in terms of rate-1 Poisson processes in two
ways: (i)
through random time changes and (ii) through random thinnings.
Associated
martingale representations are obtained for these constructions by
applying,
respectively: (i) optional stopping theorems where the random time
changes are
the stopping times and (ii) the integration theorem associated with
random
thinning of a counting process. Convergence to the diffusion process
limit for
the appropriate sequence of scaled queueing processes is obtained by
applying
the continuous mapping theorem. A key FCLT and a key FWLLN in this
framework
are established both with and without applying martingales.
http://arxiv.org/abs/0712.4211
Author(s): Robert J. Adler and Gennady Samorodnitsky and Jonathan E. Taylor
Abstract: Studying the geometry generated by Gaussian and Gaussian- related
random
fields via their excursion sets is now a well developed and well
understood
subject. The purely non-Gaussian scenario has, however, not been
studied at
all. In this paper we look at three classes of stable random fields,
and obtain
asymptotic formulae for the mean values of various geometric
characteristics of
their excursion sets over high levels.
While the formulae are asymptotic, they contain enough information
to show
that not only do stable random fields exhibit geometric behaviour very
different from that of Gaussian fields, but they also differ
significantly
among themselves.
http://arxiv.org/abs/0712.4276
Author(s): Anna Karczewska
Abstract: The aim of this work is to present, in self-contained form, results
concerning fundamental and the most important questions related to
linear
stochastic Volterra equations of convolution type. The paper is
devoted to
study the existence and some kind of regularity of solutions to
stochastic
Volterra equations in Hilbert space and the space of tempered
distributions, as
well.
In recent years the theory of Volterra equations, particularly
fractional
ones, has undergone a big development. This is an emerging area of
research
with interesting mathematical questions and various important
applications. The
increasing interest in these equations comes from their applications to
problems from physics and engeenering, particularly from
viscoelasticity, heat
conduction in materials with memory or electrodynamics with memory.
http://arxiv.org/abs/0712.4357
Author(s): Lionel Levine
Abstract: We study the scaling limits of three different aggregation models
on the
integer lattice Z^d: internal DLA, in which particles perform random
walks
until reaching an unoccupied site; the rotor-router model, in which
particles
perform deterministic analogues of random walks; and the divisible
sandpile, in
which each site distributes its excess mass equally among its
neighbors. As the
lattice spacing tends to zero, all three models are found to have the
same
scaling limit, which we describe as the solution to a certain PDE
free boundary
problem in R^d. In particular, internal DLA has a deterministic
scaling limit.
We find that the scaling limits are quadrature domains, which have
arisen
independently in many fields such as potential theory and fluid
dynamics. Our
results apply both to the case of multiple point sources and to the
Diaconis-Fulton smash sum of domains. In the special case when all
particles
start at a single site, we show that the scaling limit is a Euclidean
ball in
R^d, and give quantitative bounds on the rate of convergence to a
ball. We also
improve on the previously best known bounds of Le Borgne and Rossin
in Z^2 and
Fey and Redig in higher dimensions for the shape of the classical
abelian
sandpile model. Lastly, we study the sandpile group of a regular tree
whose
leaves are collapsed to a single sink vertex, and determine the
decomposition
of the full sandpile group as a product of cyclic groups. For the
regular
ternary tree of height n, for example, the sandpile group is
isomorphic to
(Z_3)^{2^{n-3}} x (Z_7)^{2^{n-4}} x ... x Z_{2^{n-1}-1} x Z_{2^n-1}.
We use
this result to prove that rotor-router aggregation on the regular
tree yields a
perfect ball.
http://arxiv.org/abs/0712.4358
Author(s): Ruadhan O'Flanagan
Abstract: The concept of a judgment as a logical action which introduces new
information into a deductive system is examined. This leads to a way of
mathematically representing implication which is distinct from the
familiar
material implication, according to which "If A then B" is considered
to be
equivalent to "B or not-A". This leads, in turn, to a resolution of
the paradox
of the raven.
http://arxiv.org/abs/0712.4402
Author(s): Ron Peled and Ariel Yadin and Amir Yehudayoff
Abstract: A k-wise independent distribution on n bits is a joint
distribution of the
bits such that each k of them are independent. In this paper we
consider k-wise
independent distributions with identical marginals, each bit has
probability p
to be 1. We address the following question: how high can the
probability that
all the bits are 1 be, for such a distribution? For a wide range of the
parameters n,k and p we find an explicit lower bound for this
probability which
matches an upper bound given by Benjamini et al., up to
multiplicative factors
of lower order. The question we investigate can be seen as a
relaxation of a
major open problem in error-correcting codes theory, namely, how
large can a
linear error correcting code with given parameters be?
The question is a type of discrete moment problem, and our
approach is based
on showing that bounds obtained from the theory of the classical
moment problem
provide good approximations for it. The main tool we use is a bound
controlling
the change in the expectation of a polynomial after small
perturbation of its
zeros.
http://arxiv.org/abs/0801.0059
Author(s): Rudolf Gorenflo and Entsar A.A. Abdel-Rehim
Abstract: Starting from the model of continuous time random walk, we focus
our interest
on random walks in which the probability distributions of the waiting
times and
jumps have fat tails characterized by power laws with exponent
between 0 and 1
for the waiting times, between 0 and 2 for the jumps. By stating the
relevant
lemmata (of Tauber type) for the distribution functions we need not
distinguish
between continuous and discrete space and time. We will see that, by a
well-scaled passage to the diffusion limit, generalized diffusion
processes,
fractional in time as well as in space, are obtained. The corresponding
equation of evolution is a linear partial pseudo-differential
equation with
fractional derivatives in time and in space, the orders being equal
to the
above exponents.
Such processes are well approximated and visualized by simulation
via various
types of random walks. For their explicit solutions there are available
integral representations that allow to investigate their detailed
structure.
http://arxiv.org/abs/0801.0142
Author(s): Rudolf Gorenflo and Francesco Mainardi
Abstract: To offer a view into the rapidly developing theory of fractional
diffusion
processes we describe in some detail three topics of present
interest: (i) the
well-scaled passage to the limit from continuous time random walk
under power
law assumptions to space-time fractional diffusion, (ii) the asymptotic
universality of the Mittag-Leffler waiting time law in time-fractional
processes, (iii) our method of parametric subordination for
generating particle
trajectories.
http://arxiv.org/abs/0801.0146
Author(s): Charles Bordenave and Marc Lelarge
Abstract: We analyze the convergence of the spectrum of large random graphs
to the
spectrum of a limit infinite graph. We apply these results to graphs
converging
locally to trees and derive a new formula for the Stieljes transform
of the
spectral measure of such graphs. We illustrate our results on the
uniform
regular graphs, Erdos-Renyi graphs and preferential attachment
graphs. We
sketch examples of application for weighted graphs, bipartite graphs
and the
uniform spanning tree of n vertices.
http://arxiv.org/abs/0801.0155
Author(s): Svante Janson
Abstract: It is well-known that a random variable, i.e., a function defined
on a
probability space, with values in a Borel space, can be represented
on the
special probability space consisting of the unit interval with Lebesgue
measure. We show an extension of this to multivariate functions. This is
motivated by some recent constructions of random graphs.
http://arxiv.org/abs/0801.0196
Author(s): George Lowther
Abstract: Given a real valued and time-inhomogeneous martingale diffusion X, we
investigate the properties of functions defined by the conditional
expectation
f(t,X_t)=E[g(X_T)|F_t]. We show that whenever g is monotonic or
Lipschitz
continuous then f(t,x) will also be monotonic or Lipschitz continuous
in x. If
g is convex then f(t,x) will be convex in x and decreasing in t. We
also define
the marginal support of a process and show that it almost surely
contains the
paths of the process. Although f need not be jointly continuous, we
show that
it will be continuous on the marginal support of X. We prove these
results for
a generalization of diffusion processes that we call `almost-continuous
diffusions', and includes all continuous and strong Markov processes.
http://arxiv.org/abs/0801.0330
Author(s): F. Hubalek and A.E. Kyprianou
Abstract: We give a review of the state of the art with regard to the theory
of scale
functions for spectrally negative Levy processes. From this we
introduce a
general method for generating new families of scale functions. Using
this
method we introduce a new family of scale functions belonging to the
Gaussian
Tempered Stable Convolution (GTSC) class. We give particular emphasis to
special cases as well as cross-referencing their analytical behaviour
against
known general considerations.
http://arxiv.org/abs/0801.0393
Author(s): Stefano Galatolo and Mathieu Hoyrup and Cristobal Rojas
Abstract: In the general context of computable metric spaces and computable
measures we
prove a kind of constructive Borel-Cantelli lemma: given a sequence
(recursive
in some way) of sets $A_{i}$ with recursively summable measures,
there are
computable points which are not contained in infinitely many $ A_{i}
$. As a
consequence of this we obtain the existence of computable points
which follow
the typical statistical behavior of a dynamical system (they satisfy the
Birkhoff theorem) for a large class of systems, having computable
invariant
measure and polynomial decay of correlation. This is applied to
uniformly
hyperbolic systems, piecewise expanding maps, systems on the interval
with an
indifferent fixed point and it directly implies the existence of
computable
numbers which are normal with respect to any base.
http://arxiv.org/abs/0711.1478
Author(s): J. Harnad and A. Yu. Orlov
Abstract: We give a simple algebraic derivation of a useful determinantal
identity for
multilevel systems such as random matrix chains and finite
determinantal point
processes, with applications to the calculation of point correlators,
gap
probabililties and Janossy densities.
http://arxiv.org/abs/0712.3892
Author(s): Stefano Galatolo and Mathieu Hoyrup and Cristobal Rojas
Abstract: We consider the dynamical behavior of Martin-L\"of random points
in dynamical
systems over metric spaces with a computable dynamics and a computable
invariant measure. We use computable partitions to define a sort of
effective
symbolic model for the dynamics. Trough this construction we prove
that such
points have typical statistical behavior (the behavior which is
typical in the
Birkhoff ergodic theorem) and are recurrent. We introduce and compare
some
notion of complexity for orbits in dynamical systems and we prove
that the
complexity of the orbits of random points equals the entropy of the
system.
http://arxiv.org/abs/0801.0209
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