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Probability Abstracts 86
This document contains abstracts 3205-3373.
They have been mailed on May 2, 2005.
Author(s): Remco van der Hofstad and Gerard Hooghiemstra and Dmitri Znamenski
Abstract: In this paper we study distances and connectivity properties of random graphs
with an arbitrary i.i.d. degree sequence. When the tail of the degree
distribution is regularly varying with exponent $1-\tau$ there are three
distinct cases: (i) $\tau>3$, where the degrees have finite variance, (ii)
$\tau\in (2,3)$, where the degrees have infinite variance, but finite mean, and
(iii) $\tau\in (1,2)$, where the degrees have infinite mean. These random
graphs can serve as models for complex networks where degree power laws are
observed. The distances between pairs of nodes in the three cases mentioned
above have been studied in three previous publications, and we survey the
results obtained there. Apart from the critical cases $\tau=1$, $\tau=2$ and
$\tau=3$, this completes the scaling picture. We explain the results
heuristically and describe related work and open problems. We also compare the
behavior in this model to Internet data, where a degree power law with exponent
$\tau\approx 2.2$ is observed.
Furthermore, in this paper we derive results concerning the connected
components and the diameter. We give a criterion when there exists a unique
largest connected component of size proportional to the size of the graph, and
study sizes of the other connected components. Also, we show that for $\tau\in
(2,3)$, which is most often observed in real networks, the diameter in this
model grows much faster than the typical distance between two arbitrary nodes.
http://arXiv.org/abs/math/0502580
http://front.math.ucdavis.edu/math.PR/0502580
(alternate)
Author(s): Moez Draief
Abstract: We consider the coupling of a single server queue and a storage model defined
as a Queue/Store model in Draief et al. 2004. We establish that if the input
variables both arrivals to the queue and to the store satisfy large deviations
principles and are linked through an {\em exponential tilting} than the output
variables (departures from each system) satisfy large deviations principles
with the same rate function. This generalizes to the context of large
deviations the extension of Burke's Theorem derived in Draief et al. 2004.
http://arXiv.org/abs/math/0503016
http://front.math.ucdavis.edu/math.PR/0503016
(alternate)
Author(s): Bert Zwart and Sem Borst and Krzystof Debicki
Abstract: We investigate the tail asymptotics of the supremum of X(t)+Y(t)-ct, where
X={X(t),t\geq 0} and Y={Y(t),t\geq 0} are two independent stochastic processes.
We assume that the process Y has subexponential characteristics and that the
process X is more regular in a certain sense than Y. A key issue examined in
earlier studies is under what conditions the process X contributes to large
values of the supremum only through its average behavior. The present paper
studies various scenarios where the latter is not the case, and the process X
shows some form of ``atypical'' behavior as well. In particular, we consider a
fluid model fed by a Gaussian process X and an (integrated) On-Off process Y.
We show that, depending on the model parameters, the Gaussian process may
contribute to the tail asymptotics by its moderate deviations, large
deviations, or oscillatory behavior.
http://arXiv.org/abs/math/0503482
http://front.math.ucdavis.edu/math.PR/0503482
(alternate)
Author(s): J.-R. Chazottes and P. Collet and C. Kuelske and F. Redig
Abstract: We present a new and simple approach to deviation inequalities for
non-product measures, i.e., for dependent random variables. Our method is based
on coupling. We illustrate our abstract results with chains with complete
connections and Gibbsian random fields, both at high and low temperature.
http://arXiv.org/abs/math/0503483
http://front.math.ucdavis.edu/math.PR/0503483
(alternate)
Author(s): A. M. Etheridge and P. Pfaffelhuber and A. Wakolbinger
Abstract: For a genetic locus carrying a strongly beneficial allele which has just
fixed in a large population we study the ancestry at a linked neutral locus.
During this ''selective sweep'' the linkage between the two loci is broken up
by recombination, and the ancestry at the neutral locus is modelled by a
structured coalescent in a random background. For large selection coefficients
$\alpha$ and under an appropriate scaling of the recombination rate, we derive
a sampling formula with an order of accuracy of $O((\log\alpha)^{-2})$ in
probability. In particular we see that, with this order of accuracy, in a
sample of fixed size there are at most two non-singleton families of
individuals which are identi cal by descent at the neutral locus from the
beginning of the sweep. This refines a formula going back to the work of
Maynard Smith and Haigh, and co mplements recent work of Schweinsberg and
Durrett on selective sweeps in the Moran model.
http://arXiv.org/abs/math/0503485
http://front.math.ucdavis.edu/math.PR/0503485
(alternate)
Author(s): Robert D. Foley and David R. McDonald
Abstract: Consider a modified, stable, two node Jackson network where server 2 helps
server 1 when server 2 is idle. The probability of a large deviation of the
number of customers at node one can be calculated using the flat boundary
theory of Schwartz and Weiss [Large Deviations Performance Analysis (1994),
Chapman and Hall, New York]. Surprisingly, however, these calculations show
that the proportion of time spent on the boundary, where server 2 is idle, may
be zero. This is in sharp contrast to the unmodified Jackson network which
spends a nonzero proportion of time on this boundary.
http://arXiv.org/abs/math/0503487
http://front.math.ucdavis.edu/math.PR/0503487
(alternate)
Author(s): Robert D. Foley and David R. McDonald
Abstract: We extend the Markov additive methodology developed in [Ann. Appl. Probab. 9
(1999) 110-145, Ann. Appl. Probab. 11 (2001) 596-607] to obtain the sharp
asymptotics of the steady state probability of a queueing network when one of
the nodes gets large. We focus on a new phenomenon we call a bridge. The bridge
cases occur when the Markovian part of the twisted Markov additive process is
one null recurrent or one transient, while the jitter cases treated in [Ann.
Appl. Probab. 9 (1999) 110-145, Ann. Appl. Probab. 11 (2001) 596-607] occur
when the Markovian part is (one) positive recurrent. The asymptotics of the
steady state is an exponential times a polynomial term in the bridge case, but
is purely exponential in the jitter case. We apply this theory to a modified,
stable, two node Jackson network where server two helps server one when server
two is idle. We derive the sharp asymptotics of the steady state distribution
of the number of customers queued at each node as the number of customers
queued at the server one grows large. In so doing we get an intuitive
understanding of the companion paper [Ann. Appl. Probab. 15 (2005) 519-541]
which gives a large deviation analysis of this problem using the flat boundary
theory in the book by Shwartz and Weiss. Unlike the (unscaled) large deviation
path of a Jackson network which jitters along the boundary, the unscaled large
deviation path of the modified network tries to avoid the boundary where server
two helps server one (and forms a bridge).
http://arXiv.org/abs/math/0503488
http://front.math.ucdavis.edu/math.PR/0503488
(alternate)
Author(s): Dominic Schuhmacher
Abstract: We consider the behavior of spatial point processes when subjected to a class
of linear transformations indexed by a variable T. It was shown in Ellis [Adv.
in Appl. Probab. 18 (1986) 646-659] that, under mild assumptions, the
transformed processes behave approximately like Poisson processes for large T.
In this article, under very similar assumptions, explicit upper bounds are
given for the d_2-distance between the corresponding point process
distributions. A number of related results, and applications to kernel density
estimation and long range dependence testing are also presented. The main
results are proved by applying a generalized Stein-Chen method to discretized
versions of the point processes.
http://arXiv.org/abs/math/0503491
http://front.math.ucdavis.edu/math.PR/0503491
(alternate)
Author(s): Elchanan Mossel and Ryan O'Donnell and Krzysztof Oleszkiewicz
Abstract: In this paper we study functions with low influences on product probability
spaces. The analysis of boolean functions with low influences has become a
central problem in discrete Fourier analysis. It is motivated by fundamental
questions arising from the construction of probabilistically checkable proofs
in theoretical computer science and from problems in the theory of social
choice in economics.
We prove an invariance principle for multilinear polynomials with low
influences and bounded degree; it shows that under mild conditions the
distribution of such polynomials is essentially invariant for all product
spaces. Ours is one of the very few known non-linear invariance principles. It
has the advantage that its proof is simple and that the error bounds are
explicit. We also show that the assumption of bounded degree can be eliminated
if the polynomials are slightly ``smoothed''; this extension is essential for
our applications to ``noise stability''-type problems.
In particular, as applications of the invariance principle we prove two
conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer
science, which was the original motivation for this work, and the ``It Ain't
Over Till It's Over'' conjecture from social choice theory.
http://arXiv.org/abs/math/0503503
http://front.math.ucdavis.edu/math.PR/0503503
(alternate)
Author(s): Ivan Gentil
Abstract: We develop in this paper an amelioration of the method given by S. Bobkov and
M. Ledoux in GAFA (2000). We prove by Prekopa-Leindler Theorem an optimal
modified logarithmic Sobolev inequality adapted for all log-concave measure on
$\dR^n$. This inequality implies results proved by Bobkov and Ledoux, the
Euclidean Logarithmic Sobolev inequality generalized in the last years and it
also implies some convex logarithmic Sobolev inequalities for large entropy.
http://arXiv.org/abs/math/0503476
http://front.math.ucdavis.edu/math.FA/0503476
(alternate)
Author(s): Yu. G. Kondratiev and E. Lytvynov and M. R\"ockner
Abstract: We construct two types of equilibrium dynamics of infinite particle systems
in a Riemannian manifold $X$. These dynamics are analogs of the Glauber,
respectively Kawasaki dynamics of lattice spin systems. The Glauber dynamics
now is a process where interacting particles randomly appear and disappear,
i.e., it is a birth-and-death process in $X$, while in the Kawasaki dynamics
interacting particles randomly jump over $X$. We establish conditions on a
priori explicitly given symmetrizing measures and generators of both dynamics
under which corresponding conservative Markov processes exist.
http://arXiv.org/abs/math/0503042
http://front.math.ucdavis.edu/math.PR/0503042
(alternate)
Author(s): Iljana Zahle and J. Theodore Cox and Richard Durrett
Abstract: This paper extends earlier work by Cox and Durrett, who studied the
coalescence times for two lineages in the stepping stone model on the
two-dimensional torus. We show that the genealogy of a sample of size n is
given by a time change of Kingman's coalescent. With DNA sequence data in mind,
we investigate mutation patterns under the infinite sites model, which assumes
that each mutation occurs at a new site. Our results suggest that the spatial
structure of the human population contributes to the haplotype structure and a
slower than expected decay of genetic correlation with distance revealed by
recent studies of the human genome.
http://arXiv.org/abs/math/0503512
http://front.math.ucdavis.edu/math.PR/0503512
(alternate)
Author(s): Peter H. Baxendale
Abstract: We give computable bounds on the rate of convergence of the transition
probabilities to the stationary distribution for a certain class of
geometrically ergodic Markov chains. Our results are different from earlier
estimates of Meyn and Tweedie, and from estimates using coupling, although we
start from essentially the same assumptions of a drift condition toward a
``small set.'' The estimates show a noticeable improvement on existing results
if the Markov chain is reversible with respect to its stationary distribution,
and especially so if the chain is also positive. The method of proof uses the
first-entrance-last-exit decomposition, together with new quantitative versions
of a result of Kendall from discrete renewal theory.
http://arXiv.org/abs/math/0503515
http://front.math.ucdavis.edu/math.PR/0503515
(alternate)
Author(s): Gordan Zitkovic
Abstract: We introduce a linear space of finitely additive measures to treat the
problem of optimal expected utility from consumption under a stochastic clock
and an unbounded random endowment process. In this way we establish existence
and uniqueness for a large class of utility-maximization problems including the
classical ones of terminal wealth or consumption, as well as the problems that
depend on a random time horizon or multiple consumption instances. As an
example we explicitly treat the problem of maximizing the logarithmic utility
of a consumption stream, where the local time of an Ornstein-Uhlenbeck process
acts as a stochastic clock.
http://arXiv.org/abs/math/0503516
http://front.math.ucdavis.edu/math.PR/0503516
(alternate)
Author(s): Heinrich Matzinger
Abstract: Let {\xi (n)}_{n\in Z} be a two-color random scenery, that is, a random
coloring of Z in two colors, such that the \xi (i)'s are i.i.d. Bernoulli
variables with parameter \tfrac12. Let {S(n)}_{n\in N} be a symmetric random
walk starting at 0. Our main result shows that a.s., \xi \circ S (the
composition of \xi and S) determines \xi up to translation and reflection. In
other words, by observing the scenery \xi along the random walk path S, we can
a.s. reconstruct \xi up to translation and reflection. This result gives a
positive answer to the question of H. Kesten of whether one can a.s. detect a
single defect in almost every two-color random scenery by observing it only
along a random walk path.
http://arXiv.org/abs/math/0503517
http://front.math.ucdavis.edu/math.PR/0503517
(alternate)
Author(s): Rami Atar
Abstract: This paper studies a diffusion model that arises as the limit of a queueing
system scheduling problem in the asymptotic heavy traffic regime of Halfin and
Whitt. The queueing system consists of several customer classes and many
servers working in parallel, grouped in several stations. Servers in different
stations offer service to customers of each class at possibly different rates.
The control corresponds to selecting what customer class each server serves at
each time. The diffusion control problem does not seem to have explicit
solutions and therefore a characterization of optimal solutions via the
Hamilton-Jacobi-Bellman equation is addressed. Our main result is the existence
and uniqueness of solutions of the equation. Since the model is set on an
unbounded domain and the cost per unit time is unbounded, the analysis requires
estimates on the state process that are subexponential in the time variable. In
establishing these estimates, a key role is played by an integral formula that
relates queue length and idle time processes, which may be of independent
interest.
http://arXiv.org/abs/math/0503518
http://front.math.ucdavis.edu/math.PR/0503518
(alternate)
Author(s): Mathew D. Penrose and Aidan Sudbury
Abstract: We consider random sequential adsorption processes where the initially empty
sites of a graph are irreversibly occupied, in random order, either by monomers
which block neighboring sites, or by dimers. We also consider a process where
initially occupied sites annihilate their neighbors at random times. We verify
that these processes are well defined on infinite graphs, and derive forward
equations governing joint vacancy/occupation probabilities. Using these, we
derive exact formulae for occupation probabilities and pair correlations in
Bethe lattices. For the blocking and annihilation processes we also prove
positive correlations between sites an even distance apart, and for blocking we
derive rigorous lower bounds for the site occupation probability in lattices,
including a lower bound of 1/3 for Z^2. We also give normal approximation
results for the number of occupied sites in a large finite graph.
http://arXiv.org/abs/math/0503519
http://front.math.ucdavis.edu/math.PR/0503519
(alternate)
Author(s): Istvan Berkes and Lajos Horvath and Piotr Kokoszka
Abstract: Motivated by regularities observed in time series of returns on speculative
assets, we develop an asymptotic theory of GARCH(1,1) processes {y_k} defined
by the equations y_k=\sigma_k\epsilon_k, \sigma_k^2=\omega +\alpha
y_{k-1}^2+\beta \sigma_{k-1}^2 for which the sum \alpha +\beta approaches unity
as the number of available observations tends to infinity. We call such
sequences near-integrated. We show that the asymptotic behavior of
near-integrated GARCH(1,1) processes critically depends on the sign of \gamma
:=\alpha +\beta -1. We find assumptions under which the solutions exhibit
increasing oscillations and show that these oscillations grow approximately
like a power function if \gamma \leq 0 and exponentially if \gamma >0. We
establish an additive representation for the near-integrated GARCH(1,1)
processes which is more convenient to use than the traditional multiplicative
Volterra series expansion.
http://arXiv.org/abs/math/0503520
http://front.math.ucdavis.edu/math.PR/0503520
(alternate)
Author(s): Zhi-Dong Bai and Feifang Hu
Abstract: This paper studies a very general urn model stimulated by designs in clinical
trials, where the number of balls of different types added to the urn at trial
n depends on a random outcome directed by the composition at trials
1,2,...,n-1. Patient treatments are allocated according to types of balls. We
establish the strong consistency and asymptotic normality for both the urn
composition and the patient allocation under general assumptions on random
generating matrices which determine how balls are added to the urn. Also we
obtain explicit forms of the asymptotic variance-covariance matrices of both
the urn composition and the patient allocation. The conditions on the
nonhomogeneity of generating matrices are mild and widely satisfied in
applications. Several applications are also discussed.
http://arXiv.org/abs/math/0503521
http://front.math.ucdavis.edu/math.PR/0503521
(alternate)
Author(s): Pierre Del Moral and Samy Tindel
Abstract: In this paper we investigate the speed of convergence of the fluctuations of
a general class of Feynman-Kac particle approximation models. We design an
original approach based on new Berry-Esseen type estimates for abstract
martingale sequences combined with original exponential concentration estimates
of interacting processes. These results extend the corresponding statements in
the classical theory and apply to a class of branching and genealogical
path-particle models arising in nonlinear filtering literature as well as in
statistical physics and biology.
http://arXiv.org/abs/math/0503522
http://front.math.ucdavis.edu/math.PR/0503522
(alternate)
Author(s): Erwin Bolthausen and Giambattista Giacomin
Abstract: We analyze a (1+1)-dimension directed random walk model of a polymer dipped
in a medium constituted by two immiscible solvents separated by a flat
interface. The polymer chain is heterogeneous in the sense that a single
monomer may energetically favor one or the other solvent. We focus on the case
in which the polymer types are periodically distributed along the chain or, in
other words, the polymer is constituted of identical stretches of fixed length.
The phenomenon that one wants to analyze is the localization at the interface:
energetically favored configurations place most of the monomers in the
preferred solvent and this can be done only if the polymer sticks close to the
interface. We investigate, by means of large deviations, the energy-entropy
competition that may lead, according to the value of the parameters (the
strength of the coupling between monomers and solvents and an asymmetry
parameter), to localization. We express the free energy of the system in terms
of a variational formula that we can solve. We then use the result to analyze
the phase diagram.
http://arXiv.org/abs/math/0503523
http://front.math.ucdavis.edu/math.PR/0503523
(alternate)
Author(s): T. Byczkowski and M. Ryznar
Abstract: Let $\tau$ be the first hitting time of the point 1 by the geometric Brownian
motion $X(t)= x \exp(B(t)-2\mu t)$ with drift $\mu \geq 0$ starting from $x>1$.
Here $B(t)$ is the Brownian motion starting from 0 with $E^0 B^2(t) = 2t$. We
provide an integral formula for the density function of the stopped exponential
functional $A(\tau)=\int_0^\tau X^2(t) dt$ and determine its asymptotic
behaviour at infinity. Although we basically rely on methods developed in
\cite{BGS}, the present paper also covers the case of arbitrary drifts $\mu
\geq 0$ and provides a significant unification and extension of results of the
above-mentioned paper. As a corollary we provide an integral formula and give
asymptotic behaviour at infinity of the Poisson kernel for half-spaces for
Brownian motion with drift in real hyperbolic spaces of arbitrary dimension.
http://arXiv.org/abs/math/0503060
http://front.math.ucdavis.edu/math.PR/0503060
(alternate)
Author(s): Rinaldo B. Schinazi
Abstract: We introduce a spatial stochastic process on the lattice Z^d to model mass
extinctions. Each site of the lattice may host a flock of up to N individuals.
Each individual may give birth to a new individual at the same site at rate
\phi until the maximum of N individuals has been reached at the site. Once the
flock reaches N individuals, then, and only then, it starts giving birth on
each of the 2d neighboring sites at rate \lambda(N). Finally, disaster strikes
at rate 1, that is, the whole flock disappears. Our model shows that, at least
in theory, there is a critical maximum flock size above which a species is
certain to disappear and below which it may survive.
http://arXiv.org/abs/math/0503525
http://front.math.ucdavis.edu/math.PR/0503525
(alternate)
Author(s): Benoite de Saporta and Jian-Feng Yao
Abstract: Let Y be an Ornstein-Uhlenbeck diffusion governed by a stationary and ergodic
Markov jump process X: dY_t=a(X_t)Y_t dt+\sigma(X_t) dW_t, Y_0=y_0. Ergodicity
conditions for Y have been obtained. Here we investigate the tail propriety of
the stationary distribution of this model. A characterization of either heavy
or light tail case is established. The method is based on a renewal theorem for
systems of equations with distributions on R.
http://arXiv.org/abs/math/0503527
http://front.math.ucdavis.edu/math.PR/0503527
(alternate)
Author(s): Lorens A. Imhof
Abstract: Fudenberg and Harris' stochastic version of the classical replicator dynamics
is considered. The behavior of this diffusion process in the presence of an
evolutionarily stable strategy is investigated. Moreover, extinction of
dominated strategies and stochastic stability of strict Nash equilibria are
studied. The general results are illustrated in connection with a discrete war
of attrition. A persistence result for the maximum effort strategy is obtained
and an explicit expression for the evolutionarily stable strategy is derived.
http://arXiv.org/abs/math/0503529
http://front.math.ucdavis.edu/math.PR/0503529
(alternate)
Author(s): Thomas Muller-Gronbach
Abstract: We study pathwise approximation of scalar stochastic differential equations
at a single point. We provide the exact rate of convergence of the minimal
errors that can be achieved by arbitrary numerical methods that are based (in a
measurable way) on a finite number of sequential observations of the driving
Brownian motion. The resulting lower error bounds hold in particular for all
methods that are implementable on a computer and use a random number generator
to simulate the driving Brownian motion at finitely many points. Our analysis
shows that approximation at a single point is strongly connected to an
integration problem for the driving Brownian motion with a random weight.
Exploiting general ideas from estimation of weighted integrals of stochastic
processes, we introduce an adaptive scheme, which is easy to implement and
performs asymptotically optimally.
http://arXiv.org/abs/math/0503531
http://front.math.ucdavis.edu/math.PR/0503531
(alternate)
Author(s): R. Douc and E. Moulines and Jeffrey S. Rosenthal
Abstract: Convergence rates of Markov chains have been widely studied in recent years.
In particular, quantitative bounds on convergence rates have been studied in
various forms by Meyn and Tweedie [Ann. Appl. Probab. 4 (1994) 981-1101],
Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566], Roberts and Tweedie
[Stochastic Process. Appl. 80 (1999) 211-229], Jones and Hobert [Statist. Sci.
16 (2001) 312-334] and Fort [Ph.D. thesis (2001) Univ. Paris VI]. In this
paper, we extend a result of Rosenthal [J. Amer. Statist. Assoc. 90 (1995)
558-566] that concerns quantitative convergence rates for time-homogeneous
Markov chains. Our extension allows us to consider f-total variation distance
(instead of total variation) and time-inhomogeneous Markov chains. We apply our
results to simulated annealing.
http://arXiv.org/abs/math/0503532
http://front.math.ucdavis.edu/math.PR/0503532
(alternate)
Author(s): Tomasz Komorowski and Grzegorz Krupa
Abstract: We study the transport of a passive tracer particle in a steady strongly
mixing flow with a nonzero mean velocity. We show that there exists a
probability measure under which the particle Lagrangian velocity process is
stationary. This measure is absolutely continuous with respect to the
underlying probability measure for the Eulerian flow.
http://arXiv.org/abs/math/0503534
http://front.math.ucdavis.edu/math.PR/0503534
(alternate)
Author(s): Alexander Cox
Abstract: In this paper we consider the Skorokhod embedding problem for general
starting and target measures. In particular, we provide necessary and
sufficient conditions for a stopping time to be minimal in the sense of
Monroe(1972). The resulting conditions have a nice interpretation in the
graphical picture of Chacon and Walsh. Further, we demonstrate how the
construction of Chacon and Walsh can be extended to any (integrable) starting
and target distributions, allowing the constructions of Azema-Yor, Vallois and
Jacka to be viewed in this context, and thus extended easily to general
starting and target distributions. In particular, we describe in detail the
extension of the Azema-Yor embedding in this context, and show that it retains
its optimality property.
http://arXiv.org/abs/math/0503535
http://front.math.ucdavis.edu/math.PR/0503535
(alternate)
Author(s): Garud Iyengar and Karl Sigman
Abstract: We introduce penalty-function-based admission control policies to
approximately maximize the expected reward rate in a loss network. These
control policies are easy to implement and perform well both in the transient
period as well as in steady state. A major advantage of the penalty approach is
that it avoids solving the associated dynamic program. However, a disadvantage
of this approach is that it requires the capacity requested by individual
requests to be sufficiently small compared to total available capacity. We
first solve a related deterministic linear program (LP) and then translate an
optimal solution of the LP into an admission control policy for the loss
network via an exponential penalty function. We show that the penalty policy is
a target-tracking policy--it performs well because the optimal solution of the
LP is a good target. We demonstrate that the penalty approach can be extended
to track arbitrarily defined target sets. Results from preliminary simulation
studies are included.
http://arXiv.org/abs/math/0503536
http://front.math.ucdavis.edu/math.PR/0503536
(alternate)
Author(s): Mark Jerrum and Jung-Bae Son and Prasad Tetali and Eric Vigoda
Abstract: We consider finite-state Markov chains that can be naturally decomposed into
smaller ``projection'' and ``restriction'' chains. Possibly this decomposition
will be inductive, in that the restriction chains will be smaller copies of the
initial chain. We provide expressions for Poincare (resp. log-Sobolev)
constants of the initial Markov chain in terms of Poincare (resp. log-Sobolev)
constants of the projection and restriction chains, together with further a
parameter. In the case of the Poincare constant, our bound is always at least
as good as existing ones and, depending on the value of the extra parameter,
may be much better. There appears to be no previously published decomposition
result for the log-Sobolev constant. Our proofs are elementary and
self-contained.
http://arXiv.org/abs/math/0503537
http://front.math.ucdavis.edu/math.PR/0503537
(alternate)
Author(s): Claudia Kluppelberg and Andreas E. Kyprianou and Ross A. Maller
Abstract: We formulate the insurance risk process in a general Levy process setting,
and give general theorems for the ruin probability and the asymptotic
distribution of the overshoot of the process above a high level, when the
process drifts to -\infty a.s. and the positive tail of the Levy measure, or of
the ladder height measure, is subexponential or, more generally, convolution
equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64
(1996) 103-125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207-226]
for ruin probabilities and the overshoot in random walk and compound Poisson
models are shown to have analogues in the general setup. The identities we
derive open the way to further investigation of general renewal-type properties
of Levy processes.
http://arXiv.org/abs/math/0503539
http://front.math.ucdavis.edu/math.PR/0503539
(alternate)
Author(s): Alice Guionnet and \'Edouard Maurel-Segala
Abstract: We show that under reasonably general assumptions, the first order
asymptotics of the free energy of matrix models are generating functions for
colored planar maps. This is based on the fact that solutions of the
differential Schwinger-Dyson equations are, by nature, generating functions for
enumerating planar maps, a remark which bypasses the use of Gaussian calculus.
http://arXiv.org/abs/math/0503064
http://front.math.ucdavis.edu/math.PR/0503064
(alternate)
Author(s): Rabi Bhattacharya and Mukul Majumdar
Abstract: Iteration of randomly chosen quadratic maps defines a Markov process:
X_{n+1}=\epsilon_{n+1}X_n(1-X_n), where \epsilon_n are i.i.d. with values in
the parameter space [0,4] of quadratic maps F_{\theta}(x)=\theta x(1-x). Its
study is of significance as an important Markov model, with applications to
problems of optimization under uncertainty arising in economics. In this
article a broad criterion is established for positive Harris recurrence of X_n.
http://arXiv.org/abs/math/0503540
http://front.math.ucdavis.edu/math.PR/0503540
(alternate)
Author(s): Tahir Choulli and Michael Taksar and Xun Yu Zhou
Abstract: We study a model of a corporation which has the possibility to choose various
production/business policies with different expected profits and risks. In the
model there are restrictions on the dividend distribution rates as well as
restrictions on the risk the company can undertake. The objective is to
maximize the expected present value of the total dividend distributions. We
outline the corresponding Hamilton-Jacobi-Bellman equation, compute explicitly
the optimal return function and determine the optimal policy. As a consequence
of these results, the way the dividend rate and business constraints affect the
optimal policy is revealed. In particular, we show that under certain
relationships between the constraints and the exogenous parameters of the
random processes that govern the returns, some business activities might be
redundant, that is, under the optimal policy they will never be used in any
scenario.
http://arXiv.org/abs/math/0503541
http://front.math.ucdavis.edu/math.PR/0503541
(alternate)
Author(s): Dmitrii S. Silvestrov and Jozef L. Teugels
Abstract: This article is devoted to the investigation of limit theorems for mixed
max-sum processes with renewal type stopping indexes. Limit theorems of weak
convergence type are obtained as well as functional limit theorems.
http://arXiv.org/abs/math/0503543
http://front.math.ucdavis.edu/math.PR/0503543
(alternate)
Author(s): Paul Balister and Bela Bollobas and Mark Walters
Abstract: Let A be the annulus in R^2 centered at the origin with inner and outer radii
r(1-\epsilon) and r, respectively. Place points {x_i} in R^2 according to a
Poisson process with intensity 1 and let G_A be the random graph with vertex
set {x_i} and edges x_ix_j whenever x_i-x_j\in A. We show that if the area of A
is large, then G_A almost surely has an infinite component. Moreover, if we fix
\epsilon, increase r and let n_c=n_c(\epsilon) be the area of A when this
infinite component appears, then n_c\to1 as \epsilon \to 0. This is in contrast
to the case of a ``square'' annulus where we show that n_c is bounded away from
1.
http://arXiv.org/abs/math/0503544
http://front.math.ucdavis.edu/math.PR/0503544
(alternate)
Author(s): Nicolas Fournier and Sylvie Meleard
Abstract: We consider a discrete model that describes a locally regulated spatial
population with mortality selection. This model was studied in parallel by
Bolker and Pacala and Dieckmann, Law and Murrell. We first generalize this
model by adding spatial dependence. Then we give a pathwise description in
terms of Poisson point measures. We show that different normalizations may lead
to different macroscopic approximations of this model. The first approximation
is deterministic and gives a rigorous sense to the number density. The second
approximation is a superprocess previously studied by Etheridge. Finally, we
study in specific cases the long time behavior of the system and of its
deterministic approximation.
http://arXiv.org/abs/math/0503546
http://front.math.ucdavis.edu/math.PR/0503546
(alternate)
Author(s): Daren B. H. Cline and Huay-min H. Pu
Abstract: The Lyapounov exponent and sharp conditions for geometric ergodicity are
determined of a time series model with both a threshold autoregression term and
threshold autoregressive conditional heteroscedastic (ARCH) errors.
The conditions require studying or simulating the behavior of a bounded,
ergodic Markov chain. The method of proof is based on a new approach, called
the piggyback method, that exploits the relationship between the time series
and the bounded chain. The piggyback method also provides a means for
evaluating the Lyapounov exponent by simulation and provides a new perspective
on moments, illuminating recent results for the distribution tails of GARCH
models.
http://arXiv.org/abs/math/0503547
http://front.math.ucdavis.edu/math.PR/0503547
(alternate)
Author(s): Larry Goldstein
Abstract: Given F:[a,b]^k\to [a,b] and a nonconstant X_0 with P(X_0\in [a,b])=1, define
the hierarchical sequence of random variables {X_n}_{n\ge 0} by
X_{n+1}=F(X_{n,1},...,X_{n,k}), where X_{n,i} are i.i.d. as X_n. Such sequences
arise from hierarchical structures which have been extensively studied in the
physics literature to model, for example, the conductivity of a random medium.
Under an averaging and smoothness condition on nontrivial F, an upper bound of
the form C\gamma^n for 0<\gamma<1 is obtained on the Wasserstein distance
between the standardized distribution of X_n and the normal. The results apply,
for instance, to random resistor networks and, introducing the notion of strict
averaging, to hierarchical sequences generated by certain compositions. As an
illustration, upper bounds on the rate of convergence to the normal are derived
for the hierarchical sequence generated by the weighted diamond lattice which
is shown to exhibit a full range of convergence rate behavior.
http://arXiv.org/abs/math/0503549
http://front.math.ucdavis.edu/math.PR/0503549
(alternate)
Author(s): Sara Biagini and Marco Frittelli
Abstract: In an incomplete market the price of a claim f in general cannot be uniquely
identified by no arbitrage arguments. However, the ``classical'' super
replication price is a sensible indicator of the (maximum selling) value of the
claim. When f satisfies certain pointwise conditions (e.g., f is bounded from
below), the super replication price is equal to sup_QE_Q[f], where Q varies on
the whole set of pricing measures. Unfortunately, this price is often too high:
a typical situation is here discussed in the examples. We thus define the less
expensive weak super replication price and we relax the requirements on f by
asking just for ``enough'' integrability conditions. By building up a proper
duality theory, we show its economic meaning and its relation with the
investor's preferences. Indeed, it turns out that the weak super replication
price of f coincides with sup_{Q\in M_{\Phi}}E_Q[f], where M_{\Phi} is the
class of pricing measures with finite generalized entropy (i.e., E[\Phi
(\frac{dQ}{dP})]<\infty) and where \Phi is the convex conjugate of the utility
function of the investor.
http://arXiv.org/abs/math/0503550
http://front.math.ucdavis.edu/math.PR/0503550
(alternate)
Author(s): Zhiyi Chi
Abstract: We consider the asymptotics of various estimators based on a large sample of
branching trees from a critical multi-type Galton-Watson process, as the sample
size increases to infinity. The asymptotics of additive functions of trees,
such as sizes of trees and frequencies of types within trees, a higher-order
asymptotic of the ``relative frequency'' estimator of the left eigenvector of
the mean matrix, a higher-order joint asymptotic of the maximum likelihood
estimators of the offspring probabilities and the consistency of an estimator
of the right eigenvector of the mean matrix, are established.
http://arXiv.org/abs/math/0503552
http://front.math.ucdavis.edu/math.PR/0503552
(alternate)
Author(s): J. M. P. Albin
Abstract: Under a complex technical condition, similar to such used in extreme value
theory, we find the rate q(\epsilon)^{-1} at which a stochastic process with
stationary increments \xi should be sampled, for the sampled process
\xi(\lfloor\cdot /q(\epsilon)\rfloor q(\epsilon)) to deviate from \xi by at
most \epsilon, with a given probability, asymptotically as \epsilon
\downarrow0. The canonical application is to discretization errors in computer
simulation of stochastic processes.
http://arXiv.org/abs/math/0503554
http://front.math.ucdavis.edu/math.PR/0503554
(alternate)
Author(s): Christopher Hoffman
Abstract: Benjamini, Haggstrom, Peres and Steif introduced the concept of a dynamical
random walk. This is a continuous family of random walks, {S_n(t)}. Benjamini
et. al. proved that if d=3 or d=4 then there is an exceptional set of t such
that {S_n(t)} returns to the origin infinitely often. In this paper we consider
a dynamical random walk on Z^2. We show that with probability one there exists
t such that {S_n(t)} never returns to the origin. This exceptional set of times
has dimension one. This proves a conjecture of Benjamini et. al.
http://arXiv.org/abs/math/0503065
http://front.math.ucdavis.edu/math.PR/0503065
(alternate)
Author(s): D. P. Kroese and W. R. W. Scheinhardt and P. G. Taylor
Abstract: Quasi-birth-and-death (QBD) processes with infinite ``phase spaces'' can
exhibit unusual and interesting behavior. One of the simplest examples of such
a process is the two-node tandem Jackson network, with the ``phase'' giving the
state of the first queue and the ``level'' giving the state of the second
queue. In this paper, we undertake an extensive analysis of the properties of
this QBD. In particular, we investigate the spectral properties of Neuts's
R-matrix and show that the decay rate of the stationary distribution of the
``level'' process is not always equal to the convergence norm of R. In fact, we
show that we can obtain any decay rate from a certain range by controlling only
the transition structure at level zero, which is independent of R. We also
consider the sequence of tandem queues that is constructed by restricting the
waiting room of the first queue to some finite capacity, and then allowing this
capacity to increase to infinity. We show that the decay rates for the finite
truncations converge to a value, which is not necessarily the decay rate in the
infinite waiting room case. Finally, we show that the probability that the
process hits level n before level 0 given that it starts in level 1 decays at a
rate which is not necessarily the same as the decay rate for the stationary
distribution.
http://arXiv.org/abs/math/0503555
http://front.math.ucdavis.edu/math.PR/0503555
(alternate)
Author(s): Paul Glasserman and Bin Yu
Abstract: An American option grants the holder the right to select the time at which to
exercise the option, so pricing an American option entails solving an optimal
stopping problem. Difficulties in applying standard numerical methods to
complex pricing problems have motivated the development of techniques that
combine Monte Carlo simulation with dynamic programming. One class of methods
approximates the option value at each time using a linear combination of basis
functions, and combines Monte Carlo with backward induction to estimate optimal
coefficients in each approximation. We analyze the convergence of such a method
as both the number of basis functions and the number of simulated paths
increase. We get explicit results when the basis functions are polynomials and
the underlying process is either Brownian motion or geometric Brownian motion.
We show that the number of paths required for worst-case convergence grows
exponentially in the degree of the approximating polynomials in the case of
Brownian motion and faster in the case of geometric Brownian motion.
http://arXiv.org/abs/math/0503556
http://front.math.ucdavis.edu/math.PR/0503556
(alternate)
Author(s): M. A. Aprodu and T. Bouziane
Abstract: The aim of this paper is to relate the theory of Harmonicity in sense
Korevaar-Schoen and Eells-Fuglede to the notion of a Brownian motion in
riemannian polyhedra achieved by the second author. Firstly, we prove that
Brownian motions is stochastically continuous Markov processes and consequently
it has a unique infinitesimal generator on some Banach space. Secondly, we show
that in some sense, the Brownian motion in Riemannian polyhedra has as an
infinitesimal generator the "Laplacian". Finally, we show that harmonic maps,
with target smooth Riemannian manifolds, in the sense of Eells-Fuglede, are
exactly those which maps Brownian motion in Riemannian polyhedron into a
martingale, while harmonic morphisms are exactly the maps which are Brownian
preserving paths
http://arXiv.org/abs/math/0503557
http://front.math.ucdavis.edu/math.PR/0503557
(alternate)
Author(s): Van Vu
Abstract: Let $K$ be a smooth convex set with volume one in $\BBR^d$. Choose $n$ random
points in $K$ independently according to the uniform distribution. The convex
hull of these points, denoted by $K_n$, is called a {\it random polytope}. We
prove that several key functionals of $K_n$ satisfy the central limit theorem
as $n$ tends to infinity.
http://arXiv.org/abs/math/0503559
http://front.math.ucdavis.edu/math.PR/0503559
(alternate)
Author(s): Noam Berger and Marek Biskup
Abstract: We consider the simple random walk on a two-dimensional super-critical
infinite percolation cluster and prove that for almost every configuration it
scales to Brownian motion.
http://arXiv.org/abs/math/0503576
http://front.math.ucdavis.edu/math.PR/0503576
(alternate)
Author(s): Lea Popovic
Abstract: Consider a continuous-time binary branching process conditioned to have
population size n at some time t, and with a chance p for recording each
extinct individual in the process. Within the family tree of this process, we
consider the smallest subtree containing the genealogy of the extant
individuals together with the genealogy of the recorded extinct individuals. We
introduce a novel representation of such subtrees in terms of a point-process,
and provide asymptotic results on the distribution of this point-process as the
number of extant individuals increases. We motivate the study within the scope
of a coherent analysis for an a priori model for macroevolution.
http://arXiv.org/abs/math/0503577
http://front.math.ucdavis.edu/math.PR/0503577
(alternate)
Author(s): Ali Lazrak
Abstract: This paper develops, in a Brownian information setting, an approach for
analyzing the preference for information, a question that motivates the
stochastic differential utility (SDU) due to Duffie and Epstein [Econometrica
60 (1992) 353-394]. For a class of backward stochastic differential equations
(BSDEs) including the generalized SDU [Lazrak and Quenez Math. Oper. Res. 28
(2003) 154-180], we formulate the information neutrality property as an
invariance principle when the filtration is coarser (or finer) and characterize
it. We also provide concrete examples of heterogeneity in information that
illustrate explicitly the nonneutrality property for some GSDUs. Our results
suggest that, within the GSDUs class of intertemporal utilities, risk aversion
or ambiguity aversion are inflexibly linked to the preference for information.
http://arXiv.org/abs/math/0503579
http://front.math.ucdavis.edu/math.PR/0503579
(alternate)
Author(s): Robert C. Dalang and M.-O. Hongler
Abstract: We consider the problem of finding the optimal time to sell a stock, subject
to a fixed sales cost and an exponential discounting rate \rho. We assume that
the price of the stock fluctuates according to the equation dY_t=Y_t(\mu
dt+\sigma\xi(t) dt), where (\xi(t)) is an alternating Markov renewal process
with values in {\pm1}, with an exponential renewal time. We determine the
critical value of \rho under which the value function is finite. We examine the
validity of the ``principle of smooth fit'' and use this to give a complete and
essentially explicit solution to the problem, which exhibits a surprisingly
rich structure. The corresponding result when the stock price evolves according
to the Black and Scholes model is obtained as a limit case.
http://arXiv.org/abs/math/0503580
http://front.math.ucdavis.edu/math.PR/0503580
(alternate)
Author(s): Sergey G. Bobkov
Abstract: For noncorrelated random variables, we study a concentration property of the
family of distributions of normalized sums formed by sequences of times of a
given large length.
http://arXiv.org/abs/math/0503583
http://front.math.ucdavis.edu/math.PR/0503583
(alternate)
Author(s): Lancelot F. James
Abstract: In a series of recent papers Barndorff-Nielsen and Shephard introduce an
attractive class of continuous time stochastic volatility models for financial
assets where the volatility processes are functions of positive
Ornstein-Uhlenbeck(OU) processes. This models are known to be substantially
more flexible than Gaussian based models. One current problem of this approach
is the unavailability of a tractable exact analysis of likelihood based
stochastic volatility models for the returns of log prices of stocks.
With this point in mind, the likelihood models of Barndorff-Nielsen and
Shephard are viewed as members of a much larger class of models. That is
likelihoods based on n conditionally independent Normal random variables whose
mean and variance are representable as linear functionals of a common
unobserved Poisson random measure. The analysis of these models is facilitated
by applying the methods in James (2005, 2002), in particular an Esscher type
transform of Poisson random measures; in conjunction with a special case of the
Weber-Sonine formula. It is shown that the marginal likelihood may be expressed
in terms of a multidimensional Fourier-cosine transform. This yields tractable
forms of the likelihood and also allows a full Bayesian posterior analysis of
the integrated volatility process. A general formula for the posterior density
of the log price given the observed data is derived, which could potentially
have applications to option pricing. We also identify tractable subclasses,
where inference can be based on a finite number of independent random
variables. It is shown that inference does not necessarily require simulation
of random measures. Rather, classical numerical integration can be used in the
most general cases.
http://arXiv.org/abs/math/0503055
http://front.math.ucdavis.edu/math.ST/0503055
(alternate)
Author(s): Ivan Gentil and Arnaud Guillin and Laurent Miclo
Abstract: We present a logarithmic Sobolev inequality adapted to a log-concave measure.
Assume that $\Phi$ is a symmetric convex function on $\dR$ satisfying
$(1+\e)\Phi(x)\leq {x}\Phi'(x)\leq(2-\e)\Phi(x)$ for $x\geq0$ large enough and
with $\e\in]0,1/2]$. We prove that the probability measure on $\dR$
$\mu_\Phi(dx)=e^{-\Phi(x)}/Z_\Phi dx$ satisfies a modified and adapted
logarithmic Sobolev inequality : there exist three constant $A,B,D>0$ such that
for all smooth $f>0$, \begin{equation*}
\ent{\mu_\Phi}{f^2}\leq A\int H_{\Phi}\PAR{{\frac{f'}{f}}}f^2d\mu_\Phi,
\text{with} H_{\Phi}(x)= {\begin{array}{rl} \Phi^*\PAR{Bx} &\text{if
}\ABS{x}\geq D, x^2 &\text{if}\ABS{x}\leq D. \end{array} . \end{equation*}
http://arXiv.org/abs/math/0503585
http://front.math.ucdavis.edu/math.PR/0503585
(alternate)
Author(s): Krzysztof Burdzy and Haya Kaspi
Abstract: We consider a stochastic flow in which individual particles follow skew
Brownian motions, with each one of these processes driven by the same Brownian
motion. One does not have uniqueness for the solutions of the corresponding
stochastic differential equation simultaneously for all real initial
conditions. Due to this lack of the simultaneous strong uniqueness for the
whole system of stochastic differential equations, the flow contains lenses,
that is, pairs of skew Brownian motions which start at the same point,
bifurcate, and then coalesce in a finite time. The paper contains qualitative
and quantitative (distributional) results on the geometry of the flow and
lenses.
http://arXiv.org/abs/math/0503586
http://front.math.ucdavis.edu/math.PR/0503586
(alternate)
Author(s): Shigeki Aida
Abstract: We prove weak Poincare inequalities on domains which are inverse images of
open sets in Wiener spaces under continuous functions of Brownian rough paths.
The result is applicable to Dirichlet forms on loop groups and connected open
subsets of path spaces over compact Riemannian manifolds.
http://arXiv.org/abs/math/0503587
http://front.math.ucdavis.edu/math.PR/0503587
(alternate)
Author(s): Masatoshi Fukushima and Ping He and Jiangang Ying
Abstract: We extend the classical Douglas integral, which expresses the Dirichlet
integral of a harmonic function on the unit disk in terms of its value on
boundary, to the case of conservative symmetric diffusion in terms of Feller
measure, by using the approach of time change of Markov processes.
http://arXiv.org/abs/math/0503588
http://front.math.ucdavis.edu/math.PR/0503588
(alternate)
Author(s): Holger Kosters
Abstract: Let X_1,X_2,... be a sequence of [0,1]-valued i.i.d. random variables, let
c\geq 0 be a sampling cost for each observation and let Y_i=X_i-ic, i=1,2,....
For n=1,2,..., let M(Y_1,...,Y_n)=E(max_{1\leq i\leq n}Y_i) and
V(Y_1,...,Y_n)=sup_{\tau \in C^n}E(Y_{\tau}), where C^n denotes the set of all
stopping rules for Y_1,...,Y_n. Sharp upper bounds for the difference
M(Y_1,...,Y_n)-V(Y_1,...,Y_n) are given under various restrictions on c and n.
http://arXiv.org/abs/math/0503589
http://front.math.ucdavis.edu/math.PR/0503589
(alternate)
Author(s): Dante DeBlassie
Abstract: For continuous \gamma, g:[0,1]\to(0,\infty), consider the degenerate
stochastic differential equation dX_t=[1-|X_t|^2]^{1/2}\gamma(|X_t|)
dB_t-g(|X_t|)X_t dt in the closed unit ball of R^n. We introduce a new idea to
show pathwise uniqueness holds when \gamma and g are Lipschitz and
\frac{g(1)}{\gamma^2(1)}>\sqrt2-1. When specialized to a case studied by Swart
[Stochastic Process. Appl. 98 (2002) 131-149] with \gamma=\sqrt2 and g\equiv c,
this gives an improvement of his result. Our method applies to more general
contexts as well. Let D be a bounded open set with C^3 boundary and suppose
h:\barD\to R Lipschitz on \barD, as well as C^2 on a neighborhood of \partial D
with Lipschitz second partials there. Also assume h>0 on D, h=0 on \partial D
and |\nabla h|>0 on \partial D. An example of such a function is
h(x)=d(x,\partial D). We give conditions which ensure pathwise uniqueness holds
for dX_t=h(X_t)^{1/2}\sigma(X_t) dB_t+b(X_t) dt in \barD.
http://arXiv.org/abs/math/0503590
http://front.math.ucdavis.edu/math.PR/0503590
(alternate)
Author(s): Yueyun Hu and Zhan Shi
Abstract: We present precise moderate deviation probabilities, in both quenched and
annealed settings, for a recurrent diffusion process with a Brownian potential.
Our method relies on fine tools in stochastic calculus, including Kotani's
lemma and Lamperti's representation for exponential functionals. In particular,
our result for quenched moderate deviations is in agreement with a recent
theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571-609]
who studied the corresponding problem for Sinai's random walk in random
environment.
http://arXiv.org/abs/math/0503591
http://front.math.ucdavis.edu/math.PR/0503591
(alternate)
Author(s): Richard F. Bass and Xia Chen
Abstract: If \beta_t is renormalized self-intersection local time for planar Brownian
motion, we characterize when Ee^{\gamma\beta_1} is finite or infinite in terms
of the best constant of a Gagliardo-Nirenberg inequality. We prove large
deviation estimates for \beta_1 and -\beta_1. We establish lim sup and lim inf
laws of the iterated logarithm for \beta_t as t\to\infty.
http://arXiv.org/abs/math/0503592
http://front.math.ucdavis.edu/math.PR/0503592
(alternate)
Author(s): Xia Chen
Abstract: Let \alpha ([0,1]^p) denote the intersection local time of p independent
d-dimensional Brownian motions running up to the time 1. Under the conditions
p(d-2)
http://arXiv.org/abs/math/0503593
http://front.math.ucdavis.edu/math.PR/0503593
(alternate)
Author(s): Anna Karczewska and Carlos Lizama
Abstract: The paper gives necessary and sufficient conditions providing regularity of
solutions to stochastic Volterra equations with infinite delay on a
$d$-dimensional torus. The harmonic analysis techniques and stochastic
integration in function spaces are used.
http://arXiv.org/abs/math/0503595
http://front.math.ucdavis.edu/math.PR/0503595
(alternate)
Author(s): Lancelot F. James and John W. Lau
Abstract: This paper discusses and analyzes a class of likelihood models which are
based on two distributional innovations in financial models for stock returns.
That is, the notion that the marginal distribution of aggregate returns of
log-stock prices are well approximated by generalized hyperbolic distributions,
and that volatility clustering can be handled by specifying the integrated
volatility as a random process such as that proposed in a recent series of
papers by Barndorff-Nielsen and Shephard (BNS). The BNS models produce
likelihoods for aggregate returns which can be viewed as a subclass of latent
regression models where one has n conditionally independent Normal random
variables whose mean and variance are representable as linear functionals of a
common unobserved Poisson random measure. James (2005b) recently obtains an
exact analysis for such models yielding expressions of the likelihood in terms
of quite tractable Fourier-Cosine integrals. Here, our idea is to analyze a
class of likelihoods, which can be used for similar purposes, but where the
latent regression models are based on n conditionally independent models with
distributions belonging to a subclass of the generalized hyperbolic
distributions and whose corresponding parameters are representable as linear
functionals of a common unobserved Poisson random measure. Our models are
perhaps most closely related to the Normal inverse Gaussian/GARCH/A-PARCH
models of Brandorff-Nielsen (1997) and Jensen and Lunde (2001), where in our
case the GARCH component is replaced by quantities such as INT-OU processes. It
is seen that, importantly, such likelihood models exhibit quite different
features structurally. One nice feature of the model is that it allows for more
flexibility in terms of modelling of external regression parameters.
http://arXiv.org/abs/math/0503056
http://front.math.ucdavis.edu/math.ST/0503056
(alternate)
Author(s): Vincent Vargas (PMA)
Abstract: In this article, we consider two models of directed polymers in random
environment: a discrete model and a continuous model. We consider these models
in dimension greater or equal to 3 and we suppose that the normalized partition
function is bounded in L^2. Under these assumptions, Sinai proved a local limit
theorem for the discrete model, using a perturbation expansion. In this
article, we give a new method for proving Sinai's local limit theorem. This new
method can be transposed to the continuous setting in which we prove a similar
local limit theorem.
http://arXiv.org/abs/math/0503596
http://front.math.ucdavis.edu/math.PR/0503596
(alternate)
Author(s): R. Mikulevicius and B. L. Rozovskii
Abstract: This paper concerns the Cauchy problem in R^d for the stochastic
Navier-Stokes equation \partial_tu=\Delta u-(u,\nabla)u-\nabla p+f(u)+
[(\sigma,\nabla)u-\nabla \tilde p+g(u)]\circ \dot W, u(0)=u_0,\qquad divu=0,
driven by white noise \dot W. Under minimal assumptions on regularity of the
coefficients and random forces, the existence of a global weak (martingale)
solution of the stochastic Navier-Stokes equation is proved. In the
two-dimensional case, the existence and pathwise uniqueness of a global strong
solution is shown. A Wiener chaos-based criterion for the existence and
uniqueness of a strong global solution of the Navier-Stokes equations is
established.
http://arXiv.org/abs/math/0503597
http://front.math.ucdavis.edu/math.PR/0503597
(alternate)
Author(s): David Nualart and Giovanni Peccati
Abstract: We characterize the convergence in distribution to a standard normal law for
a sequence of multiple stochastic integrals of a fixed order with variance
converging to 1. Some applications are given, in particular to study the
limiting behavior of quadratic functionals of Gaussian processes.
http://arXiv.org/abs/math/0503598
http://front.math.ucdavis.edu/math.PR/0503598
(alternate)
Author(s): Jean-Francois Le Gall and Leonid Mytnik
Abstract: This paper studies the regularity properties of the density of the exit
measure for super-Brownian motion with (1+\beta)-stable branching mechanism. It
establishes the continuity of the density in dimension d=2 and the
unboundedness of the density in all other dimensions where the density exists.
An alternative description of the exit measure and its density is also given
via a stochastic integral representation. Results are applied to the
probabilistic representation of nonnegative solutions of the partial
differential equation \Delta u=u^{1+\beta}.
http://arXiv.org/abs/math/0503599
http://front.math.ucdavis.edu/math.PR/0503599
(alternate)
Author(s): Michael Eckhoff
Abstract: We investigate the close connection between metastability of the reversible
diffusion process X defined by the stochastic differential equation
dX_t=-\nabla F(X_t) dt+\sqrt2\epsilon dW_t,\qquad \epsilon >0, and the spectrum
near zero of its generator -L_{\epsilon}\equiv \epsilon \Delta -\nabla
F\cdot\nabla, where F:R^d\to R and W denotes Brownian motion on R^d. For
generic F to each local minimum of F there corresponds a metastable state. We
prove that the distribution of its rescaled relaxation time converges to the
exponential distribution as \epsilon \downarrow 0 with optimal and uniform
error estimates. Each metastable state can be viewed as an eigenstate of
L_{\epsilon} with eigenvalue which converges to zero exponentially fast in
1/\epsilon. Modulo errors of exponentially small order in 1/\epsilon this
eigenvalue is given as the inverse of the expected metastable relaxation time.
The eigenstate is highly concentrated in the basin of attraction of the
corresponding trap.
http://arXiv.org/abs/math/0503600
http://front.math.ucdavis.edu/math.PR/0503600
(alternate)
Author(s): Sergio Albeverio and Song Liang
Abstract: Let X_i, i\in N, be i.i.d. B-valued random variables, where B is a real
separable Banach space. Let \Phi be a smooth enough mapping from B into R. An
asymptotic evaluation of Z_n=E(\exp (n\Phi (\sum_{i=1}^nX_i/n))), up to a
factor (1+o(1)), has been gotten in Bolthausen [Probab. Theory Related Fields
72 (1986) 305-318] and Kusuoka and Liang [Probab. Theory Related Fields 116
(2000) 221-238]. In this paper, a detailed asymptotic expansion of Z_n as n\to
\infty is given, valid to all orders, and with control on remainders. The
results are new even in finite dimensions.
http://arXiv.org/abs/math/0503601
http://front.math.ucdavis.edu/math.PR/0503601
(alternate)
Author(s): Uwe Franz
Abstract: Recently, Bercovici has introduced multiplicative convolutions based on
Muraki's monotone independence and shown that these convolution of probability
measures correspond to the composition of some function of their Cauchy
transforms. We provide a new proof of this fact based on the combinatorics of
moments. We also give a new characterisation of the probability measures that
can be embedded into continuous monotone convolution semigroups of probability
measures on the unit circle and briefly discuss a relation to Galton-Watson
processes.
http://arXiv.org/abs/math/0503602
http://front.math.ucdavis.edu/math.PR/0503602
(alternate)
Author(s): Tailen Hsing and Holger Rootzen
Abstract: This paper considers the asymptotic distribution of the longest edge of the
minimal spanning tree and nearest neighbor graph on X_1,...,X_{N_n} where
X_1,X_2,... are i.i.d. in \Re^2 with distribution F and N_n is independent of
the X_i and satisfies N_n/n\to_p1. A new approach based on spatial blocking and
a locally orthogonal coordinate system is developed to treat cases for which F
has unbounded support. The general results are applied to a number of special
cases, including elliptically contoured distributions, distributions with
independent Weibull-like margins and distributions with parallel level curves.
http://arXiv.org/abs/math/0503603
http://front.math.ucdavis.edu/math.PR/0503603
(alternate)
Author(s): Dayue Chen and Fuxi Zhang
Abstract: We consider the simple random walk on the infinite cluster of the Bernoulli
bond percolation of trees, and investigate the relation between the speed of
the simple random walk and the retaining probability $p$ by studying three
classes of trees. A sufficient condition is established for Galton-Watson
trees.
http://arXiv.org/abs/math/0503610
http://front.math.ucdavis.edu/math.PR/0503610
(alternate)
Author(s): Ivan Werner
Abstract: In this paper, we continue development of the theory of contractive Markov
systems (CMSs) initiated in \cite{Wer1}. We extend some results from
\cite{Wer1}, \cite{Wer3}, \cite{Wer5} and \cite{Wer6} to the case of
contractive Markov systems with probabilities which have a square summable
variation by using some ideas of A. Johansson and A. Oeberg \cite{JO}. In
particular, we show that an irreducible CMS has a unique invariant Borel
probability measure if the vertex sets form an open partition of the state
space and the restrictions of the probability functions on their vertex sets
have a square summable variation and are bounded away from zero.
http://arXiv.org/abs/math/0503633
http://front.math.ucdavis.edu/math.PR/0503633
(alternate)
Author(s): Glenn Merlet (IRMAR)
Abstract: Let A(n) be a sequence of i.i.d. topical (i.e. isotone and additively
homogeneous) operators. Let $x(n,x\_0)$ be defined by $x(0,x\_0)=x\_0$ and
$x(n,x\_0)=A(n)x(n-1,x\_0)$. This can modelize a wide range of systems
including, task graphs, train networks, Job-Shop, timed digital circuits or
parallel processing systems. When A(n) has the memory loss property, we use the
spectral gap method to prove limit theorems for $x(n,x\_0)$. Roughly speaking,
we show that $x(n,x\_0)$ behaves like a sum of i.i.d. real variables.
Precisely, we show that with suitable additional conditions, it satisfies a
central limit theorem with rate, a local limit theorem, a renewal theorem and a
large deviations principle, and we give an algebraic condition to ensure the
positivity of the variance in the CLT. When A(n) are defined by matrices in the
\mp semi-ring, we give more effective statements and show that the additional
conditions and the positivity of the variance in the CLT are generic.
http://arXiv.org/abs/math/0503634
http://front.math.ucdavis.edu/math.PR/0503634
(alternate)
Author(s): Franck Barthe and Olivier Guedon and Shahar Mendelson and Assaf Naor
Abstract: This article investigates, by probabilistic methods, various geometric
questions on B_p^n, the unit ball of \ell_p^n. We propose realizations in terms
of independent random variables of several distributions on B_p^n, including
the normalized volume measure. These representations allow us to unify and
extend the known results of the sub-independence of coordinate slabs in B_p^n.
As another application, we compute moments of linear functionals on B_p^n,
which gives sharp constants in Khinchine's inequalities on B_p^n and determines
the \psi_2-constant of all directions on B_p^n. We also study the extremal
values of several Gaussian averages on sections of B_p^n (including mean width
and \ell-norm), and derive several monotonicity results as p varies.
Applications to balancing vectors in \ell_2 and to covering numbers of
polyhedra complete the exposition.
http://arXiv.org/abs/math/0503650
http://front.math.ucdavis.edu/math.PR/0503650
(alternate)
Author(s): Stephane Boucheron and Olivier Bousquet and Gabor Lugosi and Pascal Massart
Abstract: A general method for obtaining moment inequalities for functions of
independent random variables is presented. It is a generalization of the
entropy method which has been used to derive concentration inequalities for
such functions [Boucheron, Lugosi and Massart Ann. Probab. 31 (2003)
1583-1614], and is based on a generalized tensorization inequality due to
Latala and Oleszkiewicz [Lecture Notes in Math. 1745 (2000) 147-168]. The new
inequalities prove to be a versatile tool in a wide range of applications. We
illustrate the power of the method by showing how it can be used to
effortlessly re-derive classical inequalities including Rosenthal and
Kahane-Khinchine-type inequalities for sums of independent random variables,
moment inequalities for suprema of empirical processes and moment inequalities
for Rademacher chaos and U-statistics. Some of these corollaries are apparently
new. In particular, we generalize Talagrand's exponential inequality for
Rademacher chaos of order 2 to any order. We also discuss applications for
other complex functions of independent random variables, such as suprema of
Boolean polynomials which include, as special cases, subgraph counting problems
in random graphs.
http://arXiv.org/abs/math/0503651
http://front.math.ucdavis.edu/math.PR/0503651
(alternate)
Author(s): Samy Tindel
Abstract: In this note we show how to generalize the stochastic calculus method
introduced by Comets and Neveu [Comm. Math. Phys. 166 (1995) 549-564] for two
models of spin glasses, namely, the SK model with external field and the
perceptron model. This method allows to derive quite easily some fluctuation
results for the free energy in those two cases.
http://arXiv.org/abs/math/0503652
http://front.math.ucdavis.edu/math.PR/0503652
(alternate)
Author(s): Imre Csiszar and Frantisek Matus
Abstract: The variation distance closure of an exponential family with a convex set of
canonical parameters is described, assuming no regularity conditions. The tools
are the concepts of convex core of a measure and extension of an exponential
family, introduced previously by the authors, and a new concept of accessible
faces of a convex set. Two other closures related to the information divergence
are also characterized.
http://arXiv.org/abs/math/0503653
http://front.math.ucdavis.edu/math.PR/0503653
(alternate)
Author(s): Erik I. Broman
Abstract: Given a trigonometric polynomial f:[0,1]\to[0,1] of degree m, one can define
a corresponding stationary process {X_i}_{i\in Z} via determinants of the
Toeplitz matrix for f. We show that for m=1 this process, which is trivially
one-dependent, is a two-block-factor.
http://arXiv.org/abs/math/0503654
http://front.math.ucdavis.edu/math.PR/0503654
(alternate)
Author(s): M. Kupsa and Y. Lacroix
Abstract: In this paper we characterize possible asymptotics for hitting times in
aperiodic ergodic dynamical systems: asymptotics are proved to be the
distribution functions of subprobability measures on the line belonging to the
functional class {6pt} {-3mm}(A){6mm}F={F:R\to [0,1]:\left\lbrack \matrixF is
increasing, null on ]-\infty, 0]; \noalignF is continuous and concave;
\noalignF(t)\le t for t\ge 0.\right.}. {6pt} Note that all possible asymptotics
are absolutely continuous.
http://arXiv.org/abs/math/0503655
http://front.math.ucdavis.edu/math.PR/0503655
(alternate)
Author(s): Kacha Dzhaparidze and Harry van Zanten
Abstract: In this paper we develop the spectral theory of the fractional Brownian
motion (fBm) using the ideas of Krein's work on continuous analogous of
orthogonal polynomials on the unit circle. We exhibit the functions which are
orthogonal with respect to the spectral measure of the fBm and obtain an
explicit reproducing kernel in the frequency domain. We use these results to
derive an extension of the classical Paley-Wiener expansion of the ordinary
Brownian motion to the fractional case.
http://arXiv.org/abs/math/0503656
http://front.math.ucdavis.edu/math.PR/0503656
(alternate)
Author(s): V. I. Afanasyev and J. Geiger and G. Kersting and V. A. Vatutin
Abstract: We study branching processes in an i.i.d. random environment, where the
associated random walk is of the oscillating type. This class of processes
generalizes the classical notion of criticality. The main properties of such
branching processes are developed under a general assumption, known as
Spitzer's condition in fluctuation theory of random walks, and some additional
moment condition. We determine the exact asymptotic behavior of the survival
probability and prove conditional functional limit theorems for the generation
size process and the associated random walk. The results rely on a stimulating
interplay between branching process theory and fluctuation theory of random
walks.
http://arXiv.org/abs/math/0503657
http://front.math.ucdavis.edu/math.PR/0503657
(alternate)
Author(s): A. Guillin} and R. Liptser
Abstract: Taking into account some likeness of moderate deviations (MD) and central
limit theorems (CLT), we develop an approach, which made a good showing in CLT,
for MD analysis of a family $$ S^\kappa_t=\frac{1}{t^\kappa}\int_0^tH(X_s)ds, \
t\to\infty $$ for an ergodic diffusion process $X_t$ under $0.5<\kappa<1$ and
appropriate $H$. We mean a decomposition with ``corrector'': $$
\frac{1}{t^\kappa}\int_0^tH(X_s)ds={\rm
corrector}+\frac{1}{t^\kappa}\underbrace{M_t}_{\rm martingale}. $$ and show
that, as in the CLT analysis, the corrector is negligible but in the MD scale,
and the main contribution in the MD brings the family ``$
\frac{1}{t^\kappa}M_t, t\to\infty. $'' Starting from Bayer and Freidlin,
\cite{BF}, and finishing by Wu's papers \cite{Wu1}-\cite{WuH}, in the MD study
Laplace's transform dominates. In the paper, we replace the Laplace technique
by one, admitting to give the conditions, providing the MD, in terms of
``drift-diffusion'' parameters and $H$. However, a verification of these
conditions heavily depends on a specificity of a diffusion model. That is why
the paper is named ``Examples ...''.
http://arXiv.org/abs/math/0503070
http://front.math.ucdavis.edu/math.PR/0503070
(alternate)
Author(s): Didier Piau
Abstract: We extend in two directions our previous results about the sampling and the
empirical measures of immortal branching Markov processes. Direct applications
to molecular biology are rigorous estimates of the mutation rates of polymerase
chain reactions from uniform samples of the population after the reaction.
First, we consider nonhomogeneous processes, which are more adapted to real
reactions. Second, recalling that the first moment estimator is analytically
known only in the infinite population limit, we provide rigorous confidence
intervals for this estimator that are valid for any finite population. Our
bounds are explicit, nonasymptotic and valid for a wide class of nonhomogeneous
branching Markov processes that we describe in detail. In the setting of
polymerase chain reactions, our results imply that enlarging the size of the
sample becomes useless for surprisingly small sizes. Establishing confidence
intervals requires precise estimates of the second moment of random samples.
The proof of these estimates is more involved than the proofs that allowed us,
in a previous paper, to deal with the first moment. On the other hand, our
method uses various, seemingly new, monotonicity properties of the harmonic
moments of sums of exchangeable random variables.
http://arXiv.org/abs/math/0503659
http://front.math.ucdavis.edu/math.PR/0503659
(alternate)
Author(s): Anda Gadidov
Abstract: In this note we show that almost sure convergence to zero of symmetrized
U-statistics indexed by a linear sector in Z^d_+ is equivalent to convergence
along the diagonal of Z^d_+, as it is considered in Lata\la and Zinn [Ann.
Probab. 28 (2000) 1908-1924]. Comparisons with similar results for sums of
multi-indexed i.i.d. random variables are also made.
http://arXiv.org/abs/math/0503660
http://front.math.ucdavis.edu/math.PR/0503660
(alternate)
Author(s): Raluca M. Balan
Abstract: In this paper we generalize Yu's [Ann. Probab. 24 (1996) 2079-2097] strong
invariance principle for associated sequences to the multi-parameter case,
under the assumption that the covariance coefficient u(n) decays exponentially
as n\to \infty. The main tools that we use are the following: the Berkes and
Morrow [Z. Wahrsch. Verw. Gebiete 57 (1981) 15-37] multi-parameter blocking
technique, the Csorgo and Revesz [Z. Wahrsch. Verw. Gebiete 31 (1975) 255-260]
quantile transform method and the Bulinski [Theory Probab. Appl. 40 (1995)
136-144] rate of convergence in the CLT.
http://arXiv.org/abs/math/0503661
http://front.math.ucdavis.edu/math.PR/0503661
(alternate)
Author(s): B. Delyon and A. Juditsky and R. Liptser
Abstract: For ${1/2}<\alpha<1$, we propose the MDP analysis for family $$
S^\alpha_n=\frac{1}{n^\alpha}\sum_{i=1}^nH(X_{i-1}), n\ge 1, $$ where
$(X_n)_{n\ge 0}$ be a homogeneous ergodic Markov chain, $X_n\in \mathbb{R}^d$,
when the spectrum of operator $P_x$ is continuous. The vector-valued function
$H$ is not assumed to be bounded but the Lipschitz continuity of $H$ is
required. The main helpful tools in our approach are Poisson's equation and
Stochastic Exponential; the first enables to replace the original family by
$\frac{1}{n^\alpha}M_n$ with a martingale $M_n$ while the second to avoid the
direct Laplace transform analysis.
http://arXiv.org/abs/math/0503071
http://front.math.ucdavis.edu/math.PR/0503071
(alternate)
Author(s): Remco van der Hofstad and Gerard Hooghiemstra and Dmitri Znamenski
Abstract: In this paper we study random graphs with independent and identically
distributed degrees of which the tail of the distribution function is regularly
varying with exponent $\tau\in (2,3)$.
The number of edges between two arbitrary nodes, also called the graph
distance or hopcount, in a graph with $N$ nodes is investigated when $N\to
\infty$. When $\tau\in (2,3)$, this graph distance grows like $2\frac{\log\log
N}{|\log(\tau-2)|}$. In different papers, the cases $\tau>3$ and $\tau\in
(1,2)$ have been studied. We also study the fluctuations around these
asymptotic means, and describe their distributions. The results presented here
improve upon results of Reittu and Norros, who prove an upper bound only.
http://arXiv.org/abs/math/0502581
http://front.math.ucdavis.edu/math.PR/0502581
(alternate)
Author(s): R. Liptser and A. Novikov
Abstract: We extend some known results relating the distribution tails of a continuous
local martingale supremum and its quadratic variation to the case of locally
square integrable martingales with bounded jumps. The predictable and optional
quadratic variations are involved in the main result.
http://arXiv.org/abs/math/0503072
http://front.math.ucdavis.edu/math.PR/0503072
(alternate)
Author(s): Ole E. Barndorff-Nielsen (DEPT Math Sci) and Svend E. Graversen (DEPT Math Sci), Jean Jacod (PMA), Neil Shephard (NUFFIELD College)
Abstract: In this paper we provide an asymptotic analysis of generalised bipower
measures of the variation of price processes in financial economics. These
measures encompass the usual quadratic variation, power variation and bipower
variations which have been highlighted in recent years in financial
econometrics. The analysis is carried out under some rather general Brownian
semimartingale assumptions, which allow for standard leverage effects.
http://arXiv.org/abs/math/0503711
http://front.math.ucdavis.edu/math.PR/0503711
(alternate)
Author(s): Nathana\"el Enriquez and Christophe Sabot
Abstract: This paper states a law of large numbers for a random walk in a random iid
environment on ${\mathbb Z}^d$, where the environment follows some Dirichlet
distribution. Moreover, we give explicit bounds for the asymptotic velocity of
the process and also an asymptotic expansion of this velocity at low disorder.
http://arXiv.org/abs/math/0503713
http://front.math.ucdavis.edu/math.PR/0503713
(alternate)
Author(s): S R S Varadhan
Abstract: Random walks as well as diffusions in random media are considered. Methods
are developed that allow one to establish large deviation results for both the
`quenched' and the `averaged' case.
http://arXiv.org/abs/math/0503089
http://front.math.ucdavis.edu/math.PR/0503089
(alternate)
Author(s): Anna Rudas and Balint Toth and Benedek Valko
Abstract: We consider a model of random tree growth, where at each time unit a new
vertex is added and attached to an already existing vertex chosen at random.
The probability with which a vertex with degree $k$ is chosen is proportional
to $w(k)$, where the weight function $w$ is the parameter of the model.
In the papers of B. Bollobas, O. Riordan, J. Spencer, G. Tusnady, and,
independently, Mori, the asymptotic degree distribution is obtained for a model
that is equivalent to the special case of ours, when the weight function is
linear. The proof therein strongly relies on the linear choice of $w$.
We give the asymptotical degree distribution for a wide range of weight
functions. Moreover, we provide the asymptotic distribution of the tree itself
as seen from a randomly selected vertex. The latter approach is new and gives
full insight to the limiting structure of the tree.
Our proof relies on the fact that considering the evolution of the random
tree in continuous time, the process may be viewed as a general branching
process, this way classical results can be applied.
http://arXiv.org/abs/math/0503728
http://front.math.ucdavis.edu/math.PR/0503728
(alternate)
Author(s): Rudolf Grubel and Nikolce Stefanoski
Abstract: We investigate the distribution of the depth of a node containing a specific
key or, equivalently, the number of steps needed to retrieve an item stored in
a randomly grown binary search tree. Using a representation in terms of mixed
and compounded standard distributions, we derive approximations by Poisson and
mixed Poisson distributions; these lead to asymptotic normality results. We are
particularly interested in the influence of the key value on the distribution
of the node depth. Methodologically our message is that the explicit
representation may provide additional insight if compared to the standard
approach that is based on the recursive structure of the trees. Further, in
order to exhibit the influence of the key on the distributional asymptotics, a
suitable choice of distance of probability distributions is important. Our
results are also applicable in connection with the number of recursions needed
in Hoare's [Comm. ACM 4 (1961) 321-322] selection algorithm Find.
http://arXiv.org/abs/math/0503738
http://front.math.ucdavis.edu/math.PR/0503738
(alternate)
Author(s): C. Houdr\'e and R. Kawai
Abstract: Fractional tempered stable motion (fTSm)} is defined and studied. FTSm has
the same covariance structure as fractional Brownian motion, while having tails
heavier than Gaussian but lighter than stable. Moreover, in short time it is
close to fractional stable L\'evy motion, while it is approximately fractional
Brownian motion in long time. A series representation of fTSm is derived and
used for simulation and to study some of its sample path properties.
http://arXiv.org/abs/math/0503741
http://front.math.ucdavis.edu/math.PR/0503741
(alternate)
Author(s): C. Houdr\'e and R. Kawai
Abstract: Layered stable (multivariate) distributions and processes are defined and
studied. A layered stable process combines stable trends of two different
indices, one of them possibly Gaussian. More precisely, in short time, it is
close to a stable process while, in long time, it approximates another stable
(possibly Gaussian) process. We also investigate the absolute continuity of a
layered stable process with respect to its short time limiting stable process.
A series representation of layered stable processes is derived, giving insights
into both the structure of the sample paths and of the short and long time
behaviors. This series is further used for sample paths simulation.
http://arXiv.org/abs/math/0503742
http://front.math.ucdavis.edu/math.PR/0503742
(alternate)
Author(s): Rajeeva L Karandikar and M G Nadkarni
Abstract: We give a necessary and sufficient condition on a sequence of functions on a
set $\Omega$ under which there is a measure on $\Omega$ which renders the given
sequence of functions a martingale. Further such a measure is unique if we
impose a natural maximum entropy condition on the conditional probabilities.
http://arXiv.org/abs/math/0503099
http://front.math.ucdavis.edu/math.PR/0503099
(alternate)
Author(s): Carlos G. Moreira (IMPA-Brazil) Daniel Smania (ICMC-USP-Brazil)
Abstract: Consider deterministic random walks F: I x Z -> I x Z, defined by
F(x,n)=(f(x), K(x)+n), where f is an expanding Markov map on the interval I and
K: I->Z. We study the universality (stability) of ergodic (for instance,
recurrence and transience), geometric and multifractal properties in the class
of perturbations of the type G(x,n)=(f_n(x), L(x,n)+n) which are topologically
conjugate with F and f_n are expanding maps exponentially close to f when |n|
goes to infinity. We give applications of these results in the study of the
regularity of conjugacies between (generalized) infinitely renormalizable maps
of the interval and the existence of wild attractors for one-dimensional maps.
http://arXiv.org/abs/math/0503736
http://front.math.ucdavis.edu/math.DS/0503736
(alternate)
Author(s): Omer Angel and Alexander E Holroyd and James B Martin
Abstract: Initially a car is placed with probability p at each site of the
two-dimensional integer lattice. Each car is equally likely to be East-facing
or North-facing, and different sites receive independent assignments. At odd
time steps, each North-facing car moves one unit North if there is a vacant
site for it to move into. At even time steps, East-facing cars move East in the
same way. We prove that when p is sufficiently close to 1 traffic is jammed, in
the sense that no car moves infinitely many times. The result extends to
several variant settings, including a model with cars moving at random times,
and higher dimensions.
http://arXiv.org/abs/math/0504001
http://front.math.ucdavis.edu/math.PR/0504001
(alternate)
Author(s): Philippe Briand (IRMAR) and Ying Hu (IRMAR)
Abstract: In this paper, we study the existence of solution to BSDE with quadratic
growth and unbounded terminal value. We apply a localization procedure together
with a priori bounds. As a byproduct, we apply the same method to extend a
result on BSDEs with integrable terminal condition.
http://arXiv.org/abs/math/0504002
http://front.math.ucdavis.edu/math.PR/0504002
(alternate)
Author(s): Carl Mueller and Leonid Mytnik and Aurel Stan
Abstract: We study the heat equation with a random potential term. The potential is a
one-sided stable noise, with positive jumps, which does not depend on time. To
avoid singularities, we define the equation in terms of a construction similar
to the Skorokhod integral or Wick product. We give a criterion for existence
based on the dimension of the space variable, and the parameter p of the stable
noise. Our arguments are different for p<1 and p>1.
http://arXiv.org/abs/math/0504027
http://front.math.ucdavis.edu/math.PR/0504027
(alternate)
Author(s): Federico Camia and Charles M. Newman
Abstract: We use SLE(6) paths to construct a process of continuum nonsimple loops in
the plane and prove that this process coincides with the full continuum scaling
limit of 2D critical site percolation on the triangular lattice -- that is, the
scaling limit of the set of all interfaces between different clusters. Some
properties of the loop process, including conformal invariance, are also
proved. In the main body of the paper these results are proved while assuming,
as argued by Schramm and Smirnov, that the percolation exploration path
converges in distribution to the trace of chordal SLE(6). Then, in a lengthy
appendix, a detailed proof is provided for this convergence to SLE(6), which
itself relies on Smirnov's result that crossing probabilities converge to
Cardy's formula.
http://arXiv.org/abs/math/0504036
http://front.math.ucdavis.edu/math.PR/0504036
(alternate)
Author(s): Cristina Butucea (PMA and MODALX) and Madalin Guta and Luis Artiles
Abstract: We estimate the quantum state of a light beam from results of quantum
homodyne measurements performed on identically prepared quantum systems. The
state is represented through the Wigner function, a density on R2 which may
take negative values but must respect intrinsic positivity constraints imposed
by quantum physics. The effect of the losses due to detection inefficiencies
which are always present in a real experiment is the addition to the
tomographic data of independent Gaussian noise. We construct a kernel estimator
for the Wigner function and prove that it is minimax efficient for the
pointwise risk over a class of infinitely differentiable functions. For the L2
risk, we compute the upper bounds of a truncated kernel estimator over the same
classes, restricted to functions with sub-Gaussian asymptotic behaviour. We
construct adaptive estimators, i.e. which do not depend on the smoothness
parameters, and prove that in some set-ups they attain the minimax rates for
the corresponding smoothness class.
http://arXiv.org/abs/math/0504058
http://front.math.ucdavis.edu/math.PR/0504058
(alternate)
Author(s): B. Kaulakys and V. Gontis and and M. Alaburda
Abstract: We present a simple point process model of $1/f^{\beta}$ noise, covering
different values of the exponent $\beta$. The signal of the model consists of
pulses or events. The interpulse, interevent, interarrival, recurrence or
waiting times of the signal are described by the general Langevin equation with
the multiplicative noise and stochastically diffuse in some interval resulting
in the power-law distribution. Our model is free from the requirement of a wide
distribution of relaxation times and from the power-law forms of the pulses. It
contains only one relaxation rate and yields $1/f^ {\beta}$ spectra in a wide
range of frequency. We obtain explicit expressions for the power spectra and
present numerical illustrations of the model. Further we analyze the relation
of the point process model of $1/f$ noise with the Bernamont-Surdin-McWhorter
model, representing the signals as a sum of the uncorrelated components. We
show that the point process model is complementary to the model based on the
sum of signals with a wide-range distribution of the relaxation times. In
contrast to the Gaussian distribution of the signal intensity of the sum of the
uncorrelated components, the point process exhibits asymptotically a power-law
distribution of the signal intensity. The developed multiplicative point
process model of $1/f^{\beta}$ noise may be used for modeling and analysis of
stochastic processes in different systems with the power-law distribution of
the intensity of pulsing signals.
http://arXiv.org/abs/cond-mat/0504025
http://front.math.ucdavis.edu/cond-mat/0504025
(alternate)
Author(s): Jay Rosen
Abstract: For the simple random walk in Z^2 we study those points which are visited an
unusually large number of times, and provide a new proof of the Erdos-Taylor
conjecture describing the number of visits to the most visited point.
http://arXiv.org/abs/math/0503108
http://front.math.ucdavis.edu/math.PR/0503108
(alternate)
Author(s): P. Chigansky and R. Liptser
Abstract: This note addresses certain stability properties of the nonlinear filtering
equation in discrete time. The available positive and negative results indicate
that much depends on the structure of the signal state space, its ergodic
properties and observations regularity. We show that certain predicting
estimates are stable under surprisingly general assumptions.
http://arXiv.org/abs/math/0504094
http://front.math.ucdavis.edu/math.PR/0504094
(alternate)
Author(s): Richard Arratia and Thomas M. Liggett
Abstract: Given i.i.d. positive integer valued random variables D_1,...,D_n, one can
ask whether there is a simple graph on n vertices so that the degrees of the
vertices are D_1,...,D_n. We give sufficient conditions on the distribution of
D_i for the probability that this be the case to be asymptotically 0, {1/2} or
strictly between 0 and {1/2}. These conditions roughly correspond to whether
the limit of nP(D_i\geq n) is infinite, zero or strictly positive and finite.
This paper is motivated by the problem of modeling large communications
networks by random graphs.
http://arXiv.org/abs/math/0504096
http://front.math.ucdavis.edu/math.PR/0504096
(alternate)
Author(s): Remco van der Hofstad and Wolfgang Koenig and Peter Moerters
Abstract: We discuss the long time behaviour of the parabolic Anderson model, the
Cauchy problem for the heat equation with random potential on $\Z^d$. We
consider general i.i.d. potentials and show that exactly \emph{four}
qualitatively different types of intermittent behaviour can occur. These four
universality classes depend on the upper tail of the potential distribution:
(1) tails at $\infty$ that are thicker than the double-exponential tails, (2)
double-exponential tails at $\infty$ studied by G\"artner and Molchanov, (3) a
new class called \emph{almost bounded potentials}, and (4) potentials bounded
from above studied by Biskup and K\"onig. The new class (3), which contains
both unbounded and bounded potentials, is studied in both the annealed and the
quenched setting. We show that intermittency occurs on unboundedly increasing
islands whose diameter is slowly varying in time. The characteristic
variational formulas describing the optimal profiles of the potential and of
the solution are solved explicitly by parabolas, respectively, Gaussian
densities.
http://arXiv.org/abs/math/0504102
http://front.math.ucdavis.edu/math.PR/0504102
(alternate)
Author(s): Jean-Fran\c{c}ois Marckert (LM-Versailles) and Gr\'{e}gory Miermont (LM-Orsay)
Abstract: A class of labeled trees, called mobiles, was introduced by Bouttier-di
Francesco and Guitter in order to generalize the bijective studies of planar
maps initiated by Cori-Vauquelin and Schaeffer. We prove an invariance
principle for rescaled random mobiles associated with bipartite random planar
maps under a Boltzmann distribution. We infer that the latter converge in a
certain sense to the Brownian map introduced by Marckert and Mokkadem, which
encompasses results of Chassaing and Schaeffer on quadrangulations (although in
a slightly different context). These results are derived from a new invariance
principle for a class of two-type Galton-Watson trees coupled with a spatial
motion, which are shown to converge to the Brownian snake.
http://arXiv.org/abs/math/0504110
http://front.math.ucdavis.edu/math.PR/0504110
(alternate)
Author(s): Noureddine El Karoui
Abstract: We study the limiting behavior of the largest eigenvalue of a large class of
complex Wishart matrices. In other words, let X be an n*p matrix, and let its
rows be i.i.d complex normal N_{C}(0,Sigma_p). We denote by H_p the spectral
distribution of Sigma_p, and call lambda_i's its ordered eigenvalues. Let us
call l_i's the ordered eigenvalues of X^*X and c the unique root in
[0,1/lambda_1(Sigma_p)) of the equation
\int ((lambda c)/(1-\lambda c))^2 dH_p(lambda) = n/p. The main result of this
paper is that, under technical conditions on (Sigma_p,n,p), we have, when
n->\infty,
(l_1(X^*X)-n mu)/(n^{1/3} sigma) -> TW_2 .
We give explicit formulas for mu and sigma, that depend non trivially on c.
Here TW_2 denotes the Tracy-Widom law appearing in the study of the Gaussian
Unitary Ensemble.
This theorem applies to a number of covariance models found in applications,
including well-behaved Toeplitz matrices and covariance matrices whose spectral
distribution is a sum of atoms (under some conditions on the mass of the
atoms). Generalizations of the theorem to certain spiked versions of models in
G and a.s statements about l_1/n are given. Most known examples of convergence
of the largest eigenvalue of a complex sample covariance matrix to this
Tracy-Widom law are subcases of this result.
http://arXiv.org/abs/math/0503109
http://front.math.ucdavis.edu/math.PR/0503109
(alternate)
Author(s): J. Ben Hough and Manjunath Krishnapur and Yuval Peres and Balint Virag
Abstract: We give a probabilistic introduction to determinantal and permanental point
processes. Determinantal processes arise in physics (fermions, eigenvalues of
random matrices) and in combinatorics (nonintersecting paths, random spanning
trees). They have the striking property that the number of points in a region
$D$ is a sum of independent Bernoulli random variables, with parameters which
are eigenvalues of the relevant operator on $L^2(D)$. Moreover, any
determinantal process can be represented as a mixture of determinantal
projection processes. We give a simple explanation for these known facts, and
establish analogous representations for permanental processes, with geometric
variables replacing the Bernoulli variables. These representations lead to
simple proofs of existence criteria and central limit theorems, and unify known
results on the distribution of absolute values in certain processes with
radially symmetric distributions.
http://arXiv.org/abs/math/0503110
http://front.math.ucdavis.edu/math.PR/0503110
(alternate)
Author(s): Martin T. Barlow and Takashi Kumagai
Abstract: Let ${\cal G}$ be the incipient infinite cluster (IIC) for percolation on a
homogeneous tree of degree $n_0+1$. We obtain estimates for the transition
density of the continuous time simple random walk $Y$ on ${\cal G}$; the
process satisfies anomalous diffusion and has spectral dimension 4/3.
http://arXiv.org/abs/math/0503118
http://front.math.ucdavis.edu/math.PR/0503118
(alternate)
Author(s): Francois Bolley and Arnaud Guillin and Cedric Villani
Abstract: We establish some quantitative concentration estimates for the empirical
measure of many independent variables, in transportation distances. As an
application, we provide some error bounds for particle simulations in a model
mean field problem. The tools include coupling arguments, as well as regularity
and moments estimates for solutions of certain diffusive partial differential
equations.
http://arXiv.org/abs/math/0503123
http://front.math.ucdavis.edu/math.PR/0503123
(alternate)
Author(s): Dimitris Achlioptas and Aaron Clauset and David Kempe and and Cristopher Moore
Abstract: Understanding the structure of the Internet graph is a crucial step for
building accurate network models and designing efficient algorithms for
Internet applications. Yet, obtaining its graph structure is a surprisingly
difficult task, as edges cannot be explicitly queried. Instead, empirical
studies rely on traceroutes to build what are essentially single-source,
all-destinations, shortest-path trees. These trees only sample a fraction of
the network's edges, and a recent paper by Lakhina et al. found empirically
that the resuting sample is intrinsically biased. For instance, the observed
degree distribution under traceroute sampling exhibits a power law even when
the underlying degree distribution is Poisson.
In this paper, we study the bias of traceroute sampling systematically, and,
for a very general class of underlying degree distributions, calculate the
likely observed distributions explicitly. To do this, we use a continuous-time
realization of the process of exposing the BFS tree of a random graph with a
given degree distribution, calculate the expected degree distribution of the
tree, and show that it is sharply concentrated. As example applications of our
machinery, we show how traceroute sampling finds power-law degree distributions
in both delta-regular and Poisson-distributed random graphs. Thus, our work
puts the observations of Lakhina et al. on a rigorous footing, and extends them
to nearly arbitrary degree distributions.
http://arXiv.org/abs/cond-mat/0503087
http://front.math.ucdavis.edu/cond-mat/0503087
(alternate)
Author(s): Robin Pemantle and Yuval Peres
Abstract: We consider the Ising model on a general tree under various boundary
conditions: all plus, free and spin-glass. In each case, we determine when the
root is influenced by the boundary values in the limit as the boundary recedes
to infinity. We obtain exact capacity criteria that govern behavior at critical
temperatures. For plus boundary conditions, an $L^3$ capacity arises. In
particular, on a spherically symmetric tree that has $n^c b^n$ vertices at
level $n$ (up to bounded factors), we prove that there is a unique Gibbs
measure for the ferromagnetic Ising model if and only if $c$ is at most 1/2.
Our proofs are based on a new link between nonlinear recursions on trees and
$L^p$ capacities.
http://arXiv.org/abs/math/0503137
http://front.math.ucdavis.edu/math.PR/0503137
(alternate)
Author(s): Amir Dembo and Yuval Peres and Jay Rosen
Abstract: We show that the largest disc covered by a simple random walk on the planar
square lattice after $n$ steps has radius $n^{1/4+o(1)}$, thus resolving an
open problem of P. R\'ev\'esz (1990). We also show that almost surely, for
infinitely many values of $n$ it takes about $n^{1/2+o(1)}$ steps after step
$n$ for the random walk to reach the first previously unvisited site (and the
exponent 1/2 is sharp). This resolves a problem raised by P. R\'ev\'esz (1993).
Additional results on multiple covering are obtained as well.
http://arXiv.org/abs/math/0503139
http://front.math.ucdavis.edu/math.PR/0503139
(alternate)
Author(s): Siva R. Athreya and Richard F. Bass and Maria Gordina and Edwin A. Perkins
Abstract: We consider the operator $$\sL f(x)=\tfrac12 \sum_{i,j=1}^\infty
a_{ij}(x)\frac{\del^2 f}{\del x_i \del x_j}(x)-\sum_{i=1}^\infty \lam_i x_i
b_i(x) \frac{\del f}{\del x_i}(x).$$ We prove existence and uniqueness of
solutions to the martingale problem for this operator under appropriate
conditions on the $a_{ij}, b_i$, and $\lam_i$. The process corresponding to
$\sL$ solves an infinite dimensional stochastic differential equation similar
to that for the infinite dimensional Ornstein-Uhlenbeck process.
http://arXiv.org/abs/math/0503165
http://front.math.ucdavis.edu/math.PR/0503165
(alternate)
Author(s): Robert O. Bauer and Roland M. Friedrich
Abstract: We discuss the possible candidates for conformally invariant random
non-self-crossing curves which begin and end on the boundary of a multiply
connected planar domain, and which satisfy a Markovian-type property. We
consider both, the case when the curve connects a boundary component to itself
(chordal), and the case when the curve connects two different boundary
components (bilateral). We establish appropriate extensions of Loewner's
equation to multiply connected domains for the two cases. We show that a curve
in the domain induces a motion on the boundary and that this motion is enough
to first recover the motion of the moduli of the domain and then, second, the
curve in the interior. For random curves in the interior we show that the
induced random motion on the boundary is not Markov if the domain is multiply
connected, but that the random motion on the boundary together with the random
motion of the moduli forms a Markov process. In the chordal case, we show that
this Markov process satisfies Brownian scaling and discuss how this limits the
possible conformally invariant random non-self-crossing curves. We show that
the possible candidates are labeled by a real constant and a function
homogeneous of degree minus one which describes the interaction of the random
curve with the boundary. We show that the random curve has the locality
property if the interaction term vanishes and the real parameter equals six.
http://arXiv.org/abs/math/0503178
http://front.math.ucdavis.edu/math.PR/0503178
(alternate)
Author(s): E. Herbin
Abstract: Multifractional Brownian motion is an extension of the well-known fractional
Brownian motion where the Holder regularity is allowed to vary along the paths.
In this paper, two kind of multi-parameter extensions of mBm are studied: one
is isotropic while the other is not. For each of these processes, a moving
average representation, a harmonizable representation, and the covariance
structure are given. The Holder regularity is then studied. In particular, the
case of an irregular exponent function H is investigated. In this situation,
the almost sure pointwise and local Holder exponents of the multi-parameter mBm
are proved to be equal to the correspondent exponents of H. Eventually, a local
asymptotic self-similarity property is proved. The limit process can be another
process than fBm.
http://arXiv.org/abs/math/0503182
http://front.math.ucdavis.edu/math.PR/0503182
(alternate)
Author(s): Damien Gaboriau (UMPA-ENSL)
Abstract: Measure Equivalence (ME) is the measure theoretic counterpart of
quasi-isometry. This field grew considerably during the last years, developing
tools to distinguish between different ME classes of countable groups. On the
other hand, contructions of ME equivalent groups are very rare. We present a
new method, based on a notion of measurable free-factor, and we apply it to
exhibit a new family of groups that are measure equivalent to the free group.
We also present a quite extensive survey on results about Measure Equivalence
for countable groups.
http://arXiv.org/abs/math/0503181
http://front.math.ucdavis.edu/math.DS/0503181
(alternate)
Author(s): Timothy Kusalik and James A. Mingo and and Roland Speicher
Abstract: In this paper we establish a connection between the fluctuations of Wishart
random matrices, shifted Chebyshev polynomials, and planar diagrams whose
linear span form a basis for the irreducible representations of the annular
Temperly-Lieb algebra.
http://arXiv.org/abs/math/0503169
http://front.math.ucdavis.edu/math.OA/0503169
(alternate)
Author(s): Remco van der Hofstad and Joel Spencer
Abstract: We find the asymptotic number of connected graphs with $k$ vertices and
$k-1+l$ edges when $k,l$ approach infinity, reproving a result of Bender,
Canfield and McKay. We use the {\em probabilistic method}, analyzing
breadth-first search on the random graph $G(k,p)$ for an appropriate edge
probability $p$. Central is analysis of a random walk with fixed beginning and
end which is tilted to the left.
http://arXiv.org/abs/math/0502579
http://front.math.ucdavis.edu/math.CO/0502579
(alternate)
Author(s): Christoph Richard
Abstract: We analyse q-functional equations arising from tree-like combinatorial
structures, which are counted by size, internal path length and certain
generalisations thereof. The corresponding counting parameters are labelled by
an integer k>1. We show the existence of a joint limit distribution for these
parameters in the limit of infinite size, if the size generating function has a
square root as dominant singularity. The limit distribution coincides with that
of integrals of (k-1)th powers of the standard Brownian excursion. Our method
yields a recursion for the moments of the joint distribution and admits an
extension to other types of singularities.
http://arXiv.org/abs/math/0503198
http://front.math.ucdavis.edu/math.CO/0503198
(alternate)
Author(s): E. Herbin and E. Merzbach
Abstract: We define and prove the existence of a fractional Brownian motion indexed by
a collection of closed subsets of a measure space. This process is a
generalization of the set-indexed Brownian motion, when the condition of
independance is relaxed. Relations with the Levy fractional Brownian motion and
with the fractional Brownian sheet are studied. We prove stationarity of the
increments and a property of self-similarity with respect to the action of
solid motions. Regularity conditions are exhibited. Finally, behavior of the
set-indexed fractional Brownian motion along increasing paths is analysed.
http://arXiv.org/abs/math/0503211
http://front.math.ucdavis.edu/math.PR/0503211
(alternate)
Author(s): Dmitry Ioffe and Yvan Velenik (LMRS) and Milos Zahradnik
Abstract: We study a system of rods on the 2d square lattice, with hard-core exclusion.
Each rod has a length between 2 and N. We show that, when N is sufficiently
large, and for suitable fugacity, there are several distinct Gibbs states, with
orientational long-range order. This is in sharp contrast with the case N=2
(the monomer-dimer model), for which Heilmann and Lieb proved absence of phase
transition at any fugacity. This is the first example of a pure hard-core
system with phases displaying orientational order, but not translational order;
this is a fundamental characteristic feature of liquid crystals.
http://arXiv.org/abs/math/0503222
http://front.math.ucdavis.edu/math.PR/0503222
(alternate)
Author(s): Jason Fulman
Abstract: Bolthausen used a variation of Stein's method to give an inductive proof of
the Berry-Esseen theorem for sums of independent, identically distributed
random variables. We modify this technique to prove a Berry-Esseen theorem for
character ratios of a random representation of the symmetric group on
transpositions. An analogous result is proved for Jack measure on partitions.
http://arXiv.org/abs/math/0503227
http://front.math.ucdavis.edu/math.CO/0503227
(alternate)
Author(s): S Satheesh and E Sandhya
Abstract: Methods of construction of Max-semi-selfdecompsable laws are given.
Implications of this method in random time changed extremal processes are
discussed. Max-autoregressive model is introduced and characterized using the
max-semi-selfdecompsable laws and exponential max-semi-stable laws. Some
comments regarding the infinite divisibility of semi-stable and max-semi-stable
laws are given.
http://arXiv.org/abs/math/0503232
http://front.math.ucdavis.edu/math.PR/0503232
(alternate)
Author(s): Romuald Lenczewski and Rafal Salapata
Abstract: We construct a sequence of states called m-monotone product states which give
a discrete interpolation between the monotone product of states of Muraki and
the free product of states of Avitzour and Voiculescu in free probability. We
derive the associated basic limit theorems and develop the combinatorics based
on non-crossing ordered partitions with monotone order starting from depth m.
The Hilbert space representations of the limit mixed moments in the invariance
principle lead to m-monotone Gaussian operators living in m-monotone Fock
spaces, which are truncations of the free Fock space over the square-integrable
functions on the non-negative real line (m=1 gives the monotone Fock space). A
new type of combinatorics of inner blocks leads to explicit formulas for the
mixed moments of m-monotone Gaussian operators, which are new even in the case
of monotone independent Gaussian operators with arcsine distributions.
http://arXiv.org/abs/math/0502570
http://front.math.ucdavis.edu/math.QA/0502570
(alternate)
Author(s): Mark Conger and D. Viswanath
Abstract: By a well-known result of Bayer and Diaconis, the maximum entropy model of
the common riffle shuffle implies that the number of riffle shuffles necessary
to mix a standard deck of 52 cards is either 7 or 11 -- with the former number
applying when the metric used to define mixing is the total variation distance
and the later when it is the separation distance. This and other related
results assume all 52 cards in the deck to be distinct and require all $52!$
permutations of the deck to be almost equally likely for the deck to be
considered well mixed. In many instances, not all cards in the deck are
distinct and only the sets of cards dealt out to players, and not the order in
which they are dealt out to each player, needs to be random. We derive
transition probabilities under riffle shuffles between decks with repeated
cards to cover some instances of the type just described. We focus on decks
with cards all of which are labeled either 1 or 2 and describe the consequences
of having a symmetric starting deck of the form $1,...,1,2...,2$ or $1,2,...,
1,2$. Finally, we consider mixing times for common card games.
http://arXiv.org/abs/math/0503233
http://front.math.ucdavis.edu/math.PR/0503233
(alternate)
Author(s): Frederik S. Herzberg
Abstract: We consider an iterative Bermudan option pricing algorithm based on piecewise
harmonic interpolation and give an explicit constructive characterisation of
the smallest fixed point of the iteration step as the approximate price of the
perpetual Bermudan option. The same arguments work for a related iterative
algorithm based on the approximation of subharmonic functions via the r\'eduite
associated with a given closed $F_{\sigma}$ subset of $\RR^d$.
http://arXiv.org/abs/math/0503234
http://front.math.ucdavis.edu/math.PR/0503234
(alternate)
Author(s): Frederik S. Herzberg
Abstract: The subject of this study is an iterative Bermudan option pricing algorithm
based on (high-dimensional) cubature. We show that the sequence of Bermudan
prices (as functions of the underlying assets' logarithmic start prices)
resulting from the iteration is bounded and increases monotonely to the
approximate perpetual Bermudan option price; the convergence is linear in the
supremum norm with the discount factor being the convergence factor.
Furthermore, we prove a characterisation of this approximated perpetual
Bermudan price as the smallest fixed point of the iteration procedure.
http://arXiv.org/abs/math/0503235
http://front.math.ucdavis.edu/math.PR/0503235
(alternate)
Author(s): Lionel Levine and Yuval Peres
Abstract: The rotor-router model is a deterministic analogue of random walk invented by
Jim Propp. It can be used to define a deterministic aggregation model analogous
to internal diffusion limited aggregation. We prove an isoperimetric inequality
for the exit time of simple random walk from a finite region in Z^d, and use
this to prove that the shape of the rotor-router aggregation model in Z^d,
suitably rescaled, converges to a Euclidean ball in R^d.
http://arXiv.org/abs/math/0503251
http://front.math.ucdavis.edu/math.PR/0503251
(alternate)
Author(s): Catherine Donati-Martin (PMA) and Marc Yor (PMA)
Abstract: We give a representation of the Gamma subordinator as a Krein functional of
Brownian motion, using the known representations for stable subordinators and
Esscher transforms. In particular, we have obtained Krein representations of
the subordinators which govern the two parameter Poisson-Dirichlet family of
distributions.
http://arXiv.org/abs/math/0503254
http://front.math.ucdavis.edu/math.PR/0503254
(alternate)
Author(s): Jean-Francois Le Gall (DMA-ENS Paris)
Abstract: We consider Galton-Watson trees associated with a critical offspring
distribution and conditioned to have exactly $n$ vertices. These trees are
embedded in the real line by affecting spatial positions to the vertices, in
such a way that the increments of the spatial positions along edges of the tree
are independent variables distributed according to a symmetric probability
distribution on the real line. We then condition on the event that all spatial
positions are nonnegative. Under suitable assumptions on the offspring
distribution and the spatial displacements, we prove that these conditioned
spatial trees converge as $n\to\infty$, modulo an appropriate rescaling,
towards the conditioned Brownian tree that was studied in previous work.
Applications are given to asymptotics for random quadrangulations.
http://arXiv.org/abs/math/0503263
http://front.math.ucdavis.edu/math.PR/0503263
(alternate)
Author(s): Marcus Hutter
Abstract: Solomonoff unified Occam's razor and Epicurus' principle of multiple
explanations to one elegant, formal, universal theory of inductive inference,
which initiated the field of algorithmic information theory. His central result
is that the posterior of the universal semimeasure M converges rapidly to the
true sequence generating posterior mu, if the latter is computable. Hence, M is
eligible as a universal predictor in case of unknown mu. The first part of the
paper investigates the existence and convergence of computable universal
(semi)measures for a hierarchy of computability classes: recursive, estimable,
enumerable, and approximable. For instance, M is known to be enumerable, but
not estimable, and to dominate all enumerable semimeasures. We present proofs
for discrete and continuous semimeasures. The second part investigates more
closely the types of convergence, possibly implied by universality: in
difference and in ratio, with probability 1, in mean sum, and for Martin-Loef
random sequences. We introduce a generalized concept of randomness for
individual sequences and use it to exhibit difficulties regarding these issues.
In particular, we show that convergence fails (holds) on generalized-random
sequences in gappy (dense) Bernoulli classes.
http://arXiv.org/abs/cs/0503026
http://front.math.ucdavis.edu/cs.LG/0503026
(alternate)
Author(s): Salah-Eldin A Mohammed and Tusheng Zhang and Huaizhong Zhao
Abstract: The main objective of this work is to characterize the pathwise local
structure of solutions of semilinear stochastic evolution equations (see's) and
stochastic partial differential equations (spde's) near stationary solutions.
Such characterization is realized through the long-term behavior of the
solution field near stationary points. The analysis falls in two parts I, II.
In Part I (this paper), we prove a general existence and compactness theorem
for $C^k$-cocycles of semilinear see's and spde's. Our results cover a large
class of semilinear see's as well as certain semilinear spde's with
non-Lipschitz terms such as stochastic reaction diffusion equations and the
stochastic Burgers equation with additive infinite-dimensional noise. In Part
II of this work ([M-Z-Z]), we establish a local stable manifold theorem for
non-linear see's and spde's.
http://arXiv.org/abs/math/0503320
http://front.math.ucdavis.edu/math.PR/0503320
(alternate)
Author(s): Salah-Eldin A. Mohammed and Tusheng Zhang and Huaizhong Zhao
Abstract: This article is a sequel to [M.Z.Z.1] aimed at completing the
characterization of the pathwise local structure of solutions of semilinear
stochastic evolution equations (see's) and stochastic partial differential
equations (spde's) near stationary solutions. Stationary solution are viewed as
random points in the infinite-dimensional state space, and the characterization
is expressed in terms of the almost sure long-time behavior of trajectories of
the equation in relation to the stationary solution. More specifically, we
establish local stable manifold theorems for semilinear see's and spde's
(Theorems 4.1-4.4). These results give smooth stable and unstable manifolds in
the neighborhood of a hyperbolic stationary solution of the underlying
stochastic equation. The stable and unstable manifolds are stationary, live in
a stationary tubular neighborhood of the stationary solution and are
asymptotically invariant under the stochastic semiflow of the see/spde. The
proof uses infinite-dimensional multiplicative ergodic theory techniques and
interpolation arguments (Theorem 2.1).
http://arXiv.org/abs/math/0503321
http://front.math.ucdavis.edu/math.PR/0503321
(alternate)
Author(s): Andrzej Stos (LMP-Clermont)
Abstract: We prove the Boundary Harnack Principle related to fractional powers of
Laplacian for some natural regions in the two-dimensional Sierpinski carpet.
This is a natual application of a probabilistic method based on the
Ikeda-Watanabe formula
http://arXiv.org/abs/math/0503333
http://front.math.ucdavis.edu/math.PR/0503333
(alternate)
Author(s): Lancelot F. James
Abstract: Recently Carr and Wu (2004, 2005) and also Huang and Wu (2004) show that most
stochastic processes used in traditional option pricing models can be cast as
special cases of time-changed L\'evy processes. In particular these are models
which can be tailored to exhibit correlated jumps in both the log price of
assets and the instantaneous volatility. Naturally similar to a recent work of
Barndorff-Nielsen and Shephard (2001a, b), such models may be used in a
likelihood based framework. These likelihoods are based on the unobserved
integrated volatility, rather than the instantaneous volatility. James (2005)
establishes general results for the likelihood and estimation of a large class
of such models which include possible leverage effects. In this note we show
that exact expressions for likelihood models based on generalizations of Carr
and Wu (2005) and Huang and Wu (2005), follow essentially from the arguments in
Theorem 5.1 in James (2005) with some slight modification. This serves to
formally verify a claim made by James (2005).
http://arXiv.org/abs/math/0503314
http://front.math.ucdavis.edu/math.ST/0503314
(alternate)
Author(s): T. Byczkowski and P. Graczyk and A. Stos
Abstract: We provide an integral formula for the Poisson kernel of half-spaces for
Brownian motion in real hyperbolic space $\H^n$. This enables us to find
asymptotic properties of the kernel. Our starting point is the formula for its
Fourier transform. When $n=3$, 4 or 6 we give an explicit formula for the
Poisson kernel itself. In the general case we give various asymptotics and show
convergence to the Poisson kernel of $\H^n$.
http://arXiv.org/abs/math/0503372
http://front.math.ucdavis.edu/math.PR/0503372
(alternate)
Author(s): A. Nikeghbali and M. Yor
Abstract: In the theory of progressive enlargements of filtrations, the supermartingale
$Z_{t}=\mathbf{P}(g>t\mid \mathcal{F}_{t}) $ associated with an honest time
$g$, and its additive (Doob-Meyer) decomposition, play an essential role. In
this paper, we propose an alternative approach, using a multiplicative
representation for the supermartingale $Z_{t}$, based on Doob's maximal
identity. We thus give new examples of progressive enlargements. Moreover, we
give, in our setting, a proof of the decomposition formula for martingales,
using initial enlargement techniques, and use it to obtain some path
decompositions given the maximum or minimum of some processes.
http://arXiv.org/abs/math/0503386
http://front.math.ucdavis.edu/math.PR/0503386
(alternate)
Author(s): Krzysztof Burdzy and Jeremy Quastel
Abstract: We consider two species of particles performing random walks in a domain in
Euclidean space with reflecting boundary conditions, which annihilate on
contact. In addition there is a conservation law so that the total number of
particles of each type is preserved: When the two particles of different
species annihilate each other, particles of each species, chosen at random,
give birth. We assume initially equal numbers of each species and show that the
system has a diffusive scaling limit in which the densities of the two species
are well approximated by the positive and negative parts of the solution of the
heat equation normalized to have constant $L^1$ norm. In particular, the higher
Neumann eigenfunctions appear as asymptotically stable states at the diffusive
time scale.
http://arXiv.org/abs/math/0503395
http://front.math.ucdavis.edu/math.PR/0503395
(alternate)
Author(s): Dayue Chen and Juxin Liu and Fuxi Zhang
Abstract: In this paper we study a one-parameter family of attractive reversible
nearest particle system on a finite interval. As the length of the interval
increases, the time that the nearest particle system first hits the empty set
increases in different order, from logarithmic to exponential, according to the
intensity of interaction. In particular, at the critical case, the first
hitting time increases in a polynomial order.
http://arXiv.org/abs/math/0503409
http://front.math.ucdavis.edu/math.PR/0503409
(alternate)
Author(s): Julien Barral and Stephane Seuret
Abstract: We evaluate the scale at which the multifractal structure of some random
Gibbs measures becomes discernible. The value of this scale is obtained through
what we call the growth speed in H\"older singularity sets of a Borel measure.
This growth speed yields new information on the multifractal behavior of the
rescaled copies involved in the structure of statistically self-similar Gibbs
measures. Our results are useful to understand the multifractal nature of
various heterogeneous jump processes.
http://arXiv.org/abs/math/0503420
http://front.math.ucdavis.edu/math.PR/0503420
(alternate)
Author(s): Julien Barral and Stephane Seuret
Abstract: This paper investigates new properties concerning the multifractal structure
of a class of statistically self-similar measures. These measures include the
well-known Mandelbrot multiplicative cascades, sometimes called independent
random cascades. We evaluate the scale at which the multifractal structure of
these measures becomes discernible. The value of this scale is obtained through
what we call the growth speed in H\"older singularity sets of a Borel measure.
This growth speed yields new information on the multifractal behavior of the
rescaled copies involved in the structure of statistically self-similar
measures. Our results are useful to understand the multifractal nature of
various heterogeneous jump processes.
http://arXiv.org/abs/math/0503421
http://front.math.ucdavis.edu/math.PR/0503421
(alternate)
Author(s): Tomasz Schreiber
Abstract: The purpose of the paper is to construct a polyhedral Markov field in
${\mathbb R}^3$ in analogy with the planar construction of the original Arak
(1982) polygonal Markov field. We provide a dynamic construction of the process
in terms of evolution of two-dimensional multi-edge systems tracing polyhedral
boundaries of the field in three-dimensional time-space. We also give a general
algorithm for simulating Gibbsian modifications of the constructed polyhedral
field.
http://arXiv.org/abs/math/0503429
http://front.math.ucdavis.edu/math.PR/0503429
(alternate)
Author(s): Lancelot F. James and Antonio Lijoi and Igor Pruenster
Abstract: One of the main research areas in Bayesian Nonparametrics is the proposal and
study of priors which generalize the Dirichlet process. Here we exploit
theoretical properties of Poisson random measures in order to provide a
comprehensive Bayesian analysis of random probabilities which are obtained by
an appropriate normalization. Specifically we achieve explicit and tractable
forms of the posterior and the marginal distributions, including an explicit
and easily used description of generalizations of the important
Blackwell-MacQueen P\'olya urn distribution. Such simplifications are achieved
by the use of a latent variable which admits quite interesting interpretations
which allow to gain a better understanding of the behaviour of these random
probability measures. It is noteworthy that these models are generalizations of
models considered by Kingman (1975) in a non-Bayesian context. Such models are
known to play a significant role in a variety of applications including
genetics, physics, and work involving random mappings and assemblies. Hence our
analysis is of utility in those contexts as well. We also show how our results
may be applied to Bayesian mixture models and describe computational schemes
which are generalizations of known efficient methods for the case of the
Dirichlet process. We illustrate new examples of processes which can play the
role of priors for Bayesian nonparametric inference and finally point out some
interesting connections with the theory of generalized gamma convolutions
initiated by Thorin and further developed by Bondesson.
http://arXiv.org/abs/math/0503394
http://front.math.ucdavis.edu/math.ST/0503394
(alternate)
Author(s): Alexander Plakhov and Pedro Cruz
Abstract: An algorithm of searching a zero of an unknown undimensional function is
considered, measured at a point x with some error. The step sizes are random
positive values and are calculated according to the rule: if two consecutive
iterations are in same direction step is multiplied by u>1, otherwise, it is
multiplied by 01,
divergence. Due to the multiplicative rule of updating of the step, it is
natural to expect that the sequence converges rapidly: like a geometric
progression (if convergence takes place), but the limit value may not coincide
with, but instead, approximates one of zeros of the function. By adjusting the
parameters u and d, one can reach necessary precision of approximation; higher
precision is obtained at the expense of lower convergence rate.
http://arXiv.org/abs/math/0503434
http://front.math.ucdavis.edu/math.ST/0503434
(alternate)
Author(s): J.R. Chazottes and F. Redig and E. Verbitskiy
Abstract: We prove an exponential approximation for the law of approximate occurrence
of typical patterns for a class of Gibbsian sources on the lattice $\mathbb
Z^d$, $d\ge 2$. From this result, we deduce a law of large numbers and a large
deviation result for the the waiting time of distorted patterns.
http://arXiv.org/abs/math/0503008
http://front.math.ucdavis.edu/math.PR/0503008
(alternate)
Author(s): A.M.Vershik and M.I.Graev
Abstract: We give the realization of the representation of the current group O(n,1)^X
where X is a manifold, in the Hilbert space of L^2(F,\nu) of functionals on the
the space F of the generalized functions on the manifold X which are square
integrable over measure \nu which is related to a distinguish Levy process with
values in R^{n-1} which generalized one dimensional gamma process. Unipotent
subgroup of the group O(n,1)^X acts as the group of multiplicators. Measure \nu
is sigma-finite and invariant under the action current group O(n-1)^X. Ther
case of n=2 (SL(2,R^X)) was considered before in the series of papers starting
from the article Vershik-Gel'fand-Graev (1973).
http://arXiv.org/abs/math/0503404
http://front.math.ucdavis.edu/math.RT/0503404
(alternate)
Author(s): Paul Dupuis and Hui Wang
Abstract: Importance sampling is a variance reduction technique for efficient
estimation of rare-event probabilities by Monte Carlo. In standard importance
sampling schemes, the system is simulated using an a priori fixed change of
measure suggested by a large deviation lower bound analysis. Recent work,
however, has suggested that such schemes do not work well in many situations.
In this paper we consider dynamic importance sampling in the setting of
uniformly recurrent Markov chains. By ``dynamic'' we mean that in the course of
a single simulation, the change of measure can depend on the outcome of the
simulation up till that time. Based on a control-theoretic approach to large
deviations, the existence of asymptotically optimal dynamic schemes is
demonstrated in great generality. The implementation of the dynamic schemes is
carried out with the help of a limiting Bellman equation. Numerical examples
are presented to contrast the dynamic and standard schemes.
http://arXiv.org/abs/math/0503454
http://front.math.ucdavis.edu/math.PR/0503454
(alternate)
Author(s): Samuel Herrmann and Peter Imkeller
Abstract: Physical notions of stochastic resonance for potential diffusions in
periodically changing double-well potentials such as the spectral power
amplification have proved to be defective. They are not robust for the passage
to their effective dynamics: continuous-time finite-state Markov chains
describing the rough features of transitions between different domains of
attraction of metastable points. In the framework of one-dimensional diffusions
moving in periodically changing double-well potentials we design a new notion
of stochastic resonance which refines Freidlin's concept of quasi-periodic
motion. It is based on exact exponential rates for the transition probabilities
between the domains of attraction which are robust with respect to the reduced
Markov chains. The quality of periodic tuning is measured by the probability
for transition during fixed time windows depending on a time scale parameter.
Maximizing it in this parameter produces the stochastic resonance points.
http://arXiv.org/abs/math/0503455
http://front.math.ucdavis.edu/math.PR/0503455
(alternate)
Author(s): Sanjeev Arora and Ravi Kannan
Abstract: Mixtures of Gaussian (or normal) distributions arise in a variety of
application areas. Many heuristics have been proposed for the task of finding
the component Gaussians given samples from the mixture, such as the EM
algorithm, a local-search heuristic from Dempster, Laird and Rubin [J. Roy.
Statist. Soc. Ser. B 39 (1977) 1-38]. These do not provably run in polynomial
time. We present the first algorithm that provably learns the component
Gaussians in time that is polynomial in the dimension. The Gaussians may have
arbitrary shape, but they must satisfy a ``separation condition'' which places
a lower bound on the distance between the centers of any two component
Gaussians. The mathematical results at the heart of our proof are ``distance
concentration'' results--proved using isoperimetric inequalities--which
establish bounds on the probability distribution of the distance between a pair
of points generated according to the mixture. We also formalize the more
general problem of max-likelihood fit of a Gaussian mixture to unstructured
data.
http://arXiv.org/abs/math/0503457
http://front.math.ucdavis.edu/math.PR/0503457
(alternate)
Author(s): Serban Nacu and Yuval Peres
Abstract: Let S\subset (0,1). Given a known function f:S\to (0,1), we consider the
problem of using independent tosses of a coin with probability of heads p
(where p\in S is unknown) to simulate a coin with probability of heads f(p). We
prove that if S is a closed interval and f is real analytic on S, then f has a
fast simulation on S (the number of p-coin tosses needed has exponential
tails). Conversely, if a function f has a fast simulation on an open set, then
it is real analytic on that set.
http://arXiv.org/abs/math/0503458
http://front.math.ucdavis.edu/math.PR/0503458
(alternate)
Author(s): R. W. R. Darling and J. R. Norris
Abstract: The theme of this paper is the derivation of analytic formulae for certain
large combinatorial structures. The formulae are obtained via fluid limits of
pure jump-type Markov processes, established under simple conditions on the
Laplace transforms of their Levy kernels. Furthermore, a related Gaussian
approximation allows us to describe the randomness which may persist in the
limit when certain parameters take critical values. Our method is quite
general, but is applied here to vertex identifiability in random hypergraphs. A
vertex v is identifiable in n steps if there is a hyperedge containing v all of
whose other vertices are identifiable in fewer steps.
We say that a hyperedge is identifiable if every one of its vertices is
identifiable. Our analytic formulae describe the asymptotics of the number of
identifiable vertices and the number of identifiable hyperedges for a
Poisson(\beta) random hypergraph \Lambda on a set V of N vertices, in the limit
as N\to \infty. Here \beta is a formal power series with nonnegative
coefficients \beta_0,\beta_1,..., and (\Lambda(A))_{A\subseteq V} are
independent Poisson random variables such that \Lambda(A), the number of
hyperedges on A, has mean N\beta_j/\pmatrixN j whenever |A|=j.
http://arXiv.org/abs/math/0503460
http://front.math.ucdavis.edu/math.PR/0503460
(alternate)
Author(s): Zhiyi Chi
Abstract: We study the asymptotics related to the following matching criteria for two
independent realizations of point processes X\sim X and Y\sim Y. Given l>0,
X\cap [0,l) serves as a template. For each t>0, the matching score between the
template and Y\cap [t,t+l) is a weighted sum of the Euclidean distances from
y-t to the template over all y\in Y\cap [t,t+l). The template matching criteria
are used in neuroscience to detect neural activity with certain patterns. We
first consider W_l(\theta), the waiting time until the matching score is above
a given threshold \theta. We show that whether the score is scalar- or
vector-valued, (1/l)\log W_l(\theta) converges almost surely to a constant
whose explicit form is available, when X is a stationary ergodic process and Y
is a homogeneous Poisson point process. Second, as l\to\infty, a strong
approximation for -\log [\Pr{W_l(\theta)=0}] by its rate function is
established, and in the case where X is sufficiently mixing, the rates, after
being centered and normalized by \sqrtl, satisfy a central limit theorem and
almost sure invariance principle. The explicit form of the variance of the
normal distribution is given for the case where X is a homogeneous Poisson
process as well.
http://arXiv.org/abs/math/0503463
http://front.math.ucdavis.edu/math.PR/0503463
(alternate)
Author(s): Dimitris Achlioptas and Cristopher Moore
Abstract: Many NP-complete constraint satisfaction problems appear to undergo a "phase
transition'' from solubility to insolubility when the constraint density passes
through a critical threshold. In all such cases it is easy to derive upper
bounds on the location of the threshold by showing that above a certain density
the first moment (expectation) of the number of solutions tends to zero. We
show that in the case of certain symmetric constraints, considering the second
moment of the number of solutions yields nearly matching lower bounds for the
location of the threshold. Specifically, we prove that the threshold for both
random hypergraph 2-colorability (Property B) and random Not-All-Equal k-SAT is
2^{k-1} ln 2 -O(1). As a corollary, we establish that the threshold for random
k-SAT is of order Theta(2^k), resolving a long-standing open problem.
http://arXiv.org/abs/cond-mat/0310227
http://front.math.ucdavis.edu/cond-mat/0310227
(alternate)
Author(s): Marcos Kiwi and Martin Loebl
Abstract: We address the following question: When a randomly chosen regular bipartite
multi--graph is drawn in the plane in the ``standard way'', what is the
distribution of its maximum size planar matching (set of non--crossing disjoint
edges) and maximum size planar subgraph (set of non--crossing edges which may
share endpoints)? The problem is a generalization of the Longest Increasing
Sequence (LIS) problem (also called Ulam's problem). We present combinatorial
identities which relate the number of $r$-regular bipartite multi--graphs with
maximum planar matching (maximum planar subgraph)of at most $d$ edges to a
signed sum of restricted lattice walks in $\ZZ^d$, and to the number of pairs
of standard Young tableaux of the same shape and with a ``descend--type''
property. Our results are obtained via generalizations of two combinatorial
proofs through which Gessel's identity can be obtained (an identity that is
crucial in the derivation of a bivariate generating function associated to the
distribution of LISs, and key to the analytic attack on Ulam's problem).
http://arXiv.org/abs/math/0503465
http://front.math.ucdavis.edu/math.CO/0503465
(alternate)
Author(s): Stefan Ankirchner and Steffen Dereich and Peter Imkeller
Abstract: The background for the general mathematical link between utility and
information theory investigated in this paper is a simple financial market
model with two kinds of small traders: less informed traders and insiders,
whose extra information is represented by an enlargement of the other agents'
filtration. The expected logarithmic utility increment, i.e. the difference of
the insider's and the less informed trader's expected logarithmic utility is
described in terms of the information drift, i.e. the drift one has to
eliminate in order to perceive the price dynamics as a martingale from the
insider's perspective. On the one hand, we describe the information drift in a
very general setting by natural quantities expressing the probabilistic better
informed view of the world. This on the other hand allows us to identify the
additional utility by entropy related quantities known from information theory.
In particular, in a complete market in which the insider has some fixed
additional information during the entire trading interval, its utility
increment can be represented by the Shannon information of his extra knowledge.
For general markets, and in some particular examples, we provide estimates of
maximal utility by information inequalities.
http://arXiv.org/abs/math/0503013
http://front.math.ucdavis.edu/math.PR/0503013
(alternate)
Author(s): Boaz Nadler and Stephane Lafon and Ronald R. Coifman and Ioannis G. Kevrekidis
Abstract: A central problem in data analysis is the low dimensional representation of
high dimensional data, and the concise description of its underlying geometry
and density. In the analysis of large scale simulations of complex dynamical
systems, where the notion of time evolution comes into play, important problems
are the identification of slow variables and dynamically meaningful reaction
coordinates that capture the long time evolution of the system. In this paper
we provide a unifying view of these apparently different tasks, by considering
a family of {\em diffusion maps}, defined as the embedding of complex (high
dimensional) data onto a low dimensional Euclidian space, via the eigenvectors
of suitably defined random walks defined on the given datasets. Assuming that
the data is randomly sampled from an underlying general probability
distribution $p(\x)=e^{-U(\x)}$, we show that as the number of samples goes to
infinity, the eigenvectors of each diffusion map converge to the eigenfunctions
of a corresponding differential operator defined on the support of the
probability distribution. Different normalizations of the Markov chain on the
graph lead to different limiting differential operators. One normalization
gives the Fokker-Planck operators with the same potential U(x), best suited for
the study of stochastic differential equations as well as for clustering.
Another normalization gives the Laplace-Beltrami (heat) operator on the
manifold in which the data resides, best suited for the analysis of the
geometry of the dataset, regardless of its possibly non-uniform density.
http://arXiv.org/abs/math/0503445
http://front.math.ucdavis.edu/math.NA/0503445
(alternate)
Author(s): Toshio Fukumi
Abstract: Optimal pricing of European call option is described by linear stochastic
differential equation. Trading strategy given by a twin of stochastic variables
was integrated w.r.t. Black-Scholes formula to adopt optimal pricing to
tarading strategy.
http://arXiv.org/abs/math/0503444
http://front.math.ucdavis.edu/math.OC/0503444
(alternate)
Author(s): Eva Strasser
Abstract: The present paper deals with the characterization of no-arbitrage properties
of a continuous semimartingale. The first main result, Theorem
\refMainTheoremCharNA, extends the no-arbitrage criterion by Levental and
Skorohod [Ann. Appl.
Probab. 5 (1995) 906-925] from diffusion processes to arbitrary continuous
semimartingales. The second main result, Theorem 2.4, is a characterization of
a weaker notion of no-arbitrage in terms of the existence of supermartingale
densities. The pertaining weaker notion of no-arbitrage is equivalent to the
absence of immediate arbitrage opportunities, a concept introduced by Delbaen
and Schachermayer [Ann. Appl. Probab. 5 (1995) 926-945]. Both results are
stated in terms of conditions for any semimartingales starting at arbitrary
stopping times \sigma. The necessity parts of both results are known for the
stopping time \sigma=0 from Delbaen and Schachermayer [Ann. Appl. Probab. 5
(1995) 926-945]. The contribution of the present paper is the proofs of the
corresponding sufficiency parts.
http://arXiv.org/abs/math/0503473
http://front.math.ucdavis.edu/math.PR/0503473
(alternate)
Author(s): Yu. Baryshnikov and J. E. Yukich
Abstract: We establish Gaussian limits for general measures induced by binomial and
Poisson point processes in d-dimensional space. The limiting Gaussian field has
a covariance functional which depends on the density of the point process. The
general results are used to deduce central limit theorems for measures induced
by random graphs (nearest neighbor, Voronoi and sphere of influence graph),
random sequential packing models (ballistic deposition and spatial birth-growth
models) and statistics of germ-grain models.
http://arXiv.org/abs/math/0503474
http://front.math.ucdavis.edu/math.PR/0503474
(alternate)
Author(s): Jean-Marc Azais and Mario Wschebor
Abstract: Let I be a compact d-dimensional manifold, let X:I\to R be a Gaussian process
with regular paths and let F_I(u), u\in R, be the probability distribution
function of sup_{t\in I}X(t). We prove that under certain regularity and
nondegeneracy conditions, F_I is a C^1-function and satisfies a certain
implicit equation that permits to give bounds for its values and to compute its
asymptotic behavior as u\to +\infty. This is a partial extension of previous
results by the authors in the case d=1. Our methods use strongly the so-called
Rice formulae for the moments of the number of roots of an equation of the form
Z(t)=x, where Z:I\to R^d is a random field and x is a fixed point in R^d. We
also give proofs for this kind of formulae, which have their own interest
beyond the present application.
http://arXiv.org/abs/math/0503475
http://front.math.ucdavis.edu/math.PR/0503475
(alternate)
Author(s): Baris Ata and Sunil Kumar
Abstract: We consider a class of open stochastic processing networks, with feedback
routing and overlapping server capabilities, in heavy traffic. The networks we
consider satisfy the so-called complete resource pooling condition and
therefore have one-dimensional approximating Brownian control problems.
We propose a simple discrete review policy for controlling such networks.
Assuming 2+\epsilon moments on the interarrival times and processing times,
we provide a conceptually simple proof of asymptotic optimality of the proposed
policy.
http://arXiv.org/abs/math/0503477
http://front.math.ucdavis.edu/math.PR/0503477
(alternate)
Author(s): Rolando Cavazos-Cadena and Daniel Hernandez-Hernandez
Abstract: This work concerns controlled Markov chains with finite state and action
spaces. The transition law satisfies the simultaneous Doeblin condition, and
the performance of a control policy is measured by the (long-run)
risk-sensitive average cost criterion associated to a positive, but otherwise
arbitrary, risk sensitivity coefficient. Within this context, the optimal
risk-sensitive average cost is characterized via a minimization problem in a
finite-dimensional Euclidean space.
http://arXiv.org/abs/math/0503478
http://front.math.ucdavis.edu/math.PR/0503478
(alternate)
Author(s): Lothar Heinrich
Abstract: We study the existence of the (thermodynamic) limit of the scaled
cumulant-generating function L_n(z)=|W_n|^{-1}\logE\exp{z|\Xi\cap W_n|} of the
empirical volume fraction |\Xi\cap W_n|/|W_n|, where |\cdot| denotes the
d-dimensional Lebesgue measure. Here \Xi=\bigcup_{i\ge1}(\Xi_i+X_i) denotes a
d-dimensional Poisson grain model (also known as a Boolean model) defined by a
stationary Poisson process \Pi_{\lambda}=\sum_{i\ge1}\delta_{X_i} with
intensity \lambda >0 and a sequence of independent copies \Xi_1,\Xi_2,... of a
random compact set \Xi_0. For an increasing family of compact convex sets {W_n,
n\ge1} which expand unboundedly in all directions, we prove the existence and
analyticity of the limit lim_{n\to\infty}L_n(z) on some disk in the complex
plane whenever E\exp{a|\Xi_0|}<\infty for some a>0. Moreover, closely connected
with this result, we obtain exponential inequalities and the exact asymptotics
for the large deviation probabilities of the empirical volume fraction in the
sense of Cram\'er and Chernoff.
http://arXiv.org/abs/math/0503479
http://front.math.ucdavis.edu/math.PR/0503479
(alternate)
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