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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 | |
