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Probability Abstracts 87
This document contains abstracts 3374-3515.
They have been mailed on July 1, 2005.
Author(s): Julien Bect and Hana Baili and Gilles Fleury
Abstract: We consider a Markov process on a Riemannian manifold, which solves a
stochastic differential equation in the interior of the manifold and jumps
according to a deterministic reset map when it reaches the boundary. We derive
a partial differential equation for the probability density function, involving
a non-local boundary condition which accounts for the jumping behaviour of the
process. This is a generalisation of the usual Fokker-Planck-Kolmogorov
equation for diffusion processes. The result is illustrated with an example in
the field of stochastic hybrid systems.
http://arXiv.org/abs/math/0504583
http://front.math.ucdavis.edu/math.PR/0504583
(alternate)
Author(s): D.A. Dawson (Carleton University) and Zenghu Li (Beijing Normal University)
Abstract: A general affine Markov semigroup is formulated as the convolution of a
homogeneous one with a skew convolution semigroup. We provide some sufficient
conditions for the regularities of the homogeneous affine semigroup and the
skew convolution semigroup. The corresponding affine Markov process is
constructed as the strong solution of a system of stochastic equations with
non-Lipschitz coefficients and Poisson-type integrals over some random sets.
Based on this characterization, it is proved that the affine process arises
naturally in a limit theorem for the difference of a pair of reactant processes
in a catalytic branching system with immigration.
http://arXiv.org/abs/math/0505444
http://front.math.ucdavis.edu/math.PR/0505444
(alternate)
Author(s): D. Bloemker and M. Romito and R. Tribe
Abstract: The solutions to a large class of semi-linear parabolic PDEs are given in
terms of expectations of suitable functionals of a tree of branching particles.
A sufficient, and in some cases necessary, condition is given for the
integrability of the stochastic representation, using a companion scalar PDE.
In cases where the representation fails to be integrable a sequence of pruned
trees is constructed, producing a approximate stochastic representations that
in some cases converge, globally in time, to the solution of the original PDE.
http://arXiv.org/abs/math/0505449
http://front.math.ucdavis.edu/math.PR/0505449
(alternate)
Author(s): Misja Nuyens and Bert Zwart
Abstract: We consider a GI/GI/1 queue with the shortest remaining processing time
discipline (SRPT) and light-tailed service times. Our interest is focused on
the tail behavior of the sojourn-time distribution. We obtain a general
expression for its large-deviations decay rate. The value of this decay rate
critically depends on whether there is mass in the endpoint of the service-time
distribution or not. An auxiliary priority queue, for which we obtain some new
results, plays an important role in our analysis. We apply our SRPT-results to
compare SRPT with FIFO from a large-deviations point of view.
http://arXiv.org/abs/math/0505450
http://front.math.ucdavis.edu/math.PR/0505450
(alternate)
Author(s): Ashkan Nikeghbali
Abstract: This note deals with the question: what remains of the Burkholder-Davis-Gundy
inequalities when stopping times $T$ are replaced by arbitrary random times
$\rho $? We prove that these inequalities still hold when $T$ is a
pseudo-stopping time and never holds for ends of predictable sets.
http://arXiv.org/abs/math/0505483
http://front.math.ucdavis.edu/math.PR/0505483
(alternate)
Author(s): Pierluigi Contucci and Cristian Giardina'
Abstract: If the variance of a Gaussian spin-glass Hamiltonian grows like the volume
the model fulfills the Ghirlanda-Guerra identities in terms of the normalized
Hamiltonian covariance.
http://arXiv.org/abs/math-ph/0505055
http://front.math.ucdavis.edu/math-ph/0505055
(alternate)
Author(s): V.I.Bakhtin
Abstract: In the present paper we introduce positive flows and processes, which
generalize the ordinary dynamical systems and stochastic processes. We develop
a branch of theory of positive operators based on the concepts of phase and
positive algebras, the spectral potential, the dual entropy, equilibrium
measures, the action functional, sensitive states, empirical measures and prove
within it the law of large numbers with respect to the sensitive states and
calculate asymptotics for probabilities of large deviations in terms of the
action functional.
http://arXiv.org/abs/math/0505446
http://front.math.ucdavis.edu/math.DS/0505446
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: The paper studies a closed queueing network containing two types of node. The
first type (server station) is an infinite server queueing system, and the
second type (client station) is a single server queueing system with autonomous
service, i.e. every client station serves customers (units) only at random
instants generated by strictly stationary and ergodic sequence of random
variables. It is assumed that there are $r$ server stations. At the initial
time moment all units are distributed in the server stations, and the $i$th
server station contains $N_i$ units, $i=1,2,...,r$, where all the values $N_i$
are large numbers of the same order. The total number of client stations is
equal to $k$. The expected times between departures in the client stations are
small values of the order $O(N^{-1})$ ~ $(N=N_1+N_2+...+N_r)$. After service
completion in the $i$th server station a unit is transmitted to the $j$th
client station with probability $p_{i,j}$ ~ ($j=1,2,...,k$), and being served
in the $j$th client station the unit returns to the $i$th server station. Under
the assumption that only one of the client stations is a bottleneck node, i.e.
the expected number of arrivals per time unit to the node is greater than the
expected number of departures from that node, the paper derives the
representation for non-stationary queue-length distributions in non-bottleneck
client stations.
http://arXiv.org/abs/math/0505489
http://front.math.ucdavis.edu/math.PR/0505489
(alternate)
Author(s): Franck Barthe (LSProba) and Patrick Cattiaux (MODAL'X and CMAP) and Cyril Roberto (LAMA)
Abstract: If a random variable is not exponentially integrable, it is known that no
concentration inequality holds for an infinite sequence of independent copies.
Under mild conditions, we establish concentration inequalities for finite
sequences of $n$ independent copies, with good dependence in $n$.
http://arXiv.org/abs/math/0505492
http://front.math.ucdavis.edu/math.PR/0505492
(alternate)
Author(s): Cristopher Moore and Gabriel Istrate and Demetrios Demopoulos and and Moshe Y. Vardi
Abstract: We compute the probability of satisfiability of a class of random Horn-SAT
formulae, motivated by a connection with the nonemptiness problem of finite
tree automata. In particular, when the maximum clause length is 3, this model
displays a curve in its parameter space along which the probability of
satisfiability is discontinuous, ending in a second-order phase transition
where it becomes continuous. This is the first case in which a phase transition
of this type has been rigorously established for a random constraint
satisfaction problem.
http://arXiv.org/abs/math/0505032
http://front.math.ucdavis.edu/math.PR/0505032
(alternate)
Author(s): Victor Rivero (MODAL'X)
Abstract: We prove that the upward ladder height subordinator $H$ associated to a real
valued L\'{e}vy process $\xi$ has Laplace exponent $\phi$ that varies regularly
at $\infty$ (resp. at 0) if and only if the underlying L\'{e}vy process $\xi$
satisfies Sinai's condition at 0 (resp. at $\infty$). Sinai's condition for
real valued L\'{e}vy processes is the continuous time analogue of Sinai's
condition for random walks. We provide several criteria in terms of the
characteristics of $\xi$ to determine whether or not it satisfies Sinai's
condition. Some of these criteria are deduced from tail estimates of the
L\'{e}vy measure of $H,$ here obtained, and which are analogous to the
estimates of the tail distribution of the ladder height random variable of a
random walk which are due to Veraverbeke and Gr\"{u}bel
http://arXiv.org/abs/math/0505495
http://front.math.ucdavis.edu/math.PR/0505495
(alternate)
Author(s): Bernardo Lafuerza-Guillen and Jose L. Rodriguez
Abstract: In this paper we consider probabilistic normed spaces as defined by Alsina,
Sklar, and Schweizer, but equipped with non necessarily continuous triangle
functions. Such spaces endow a generalized topology that is
Fr\'echet-separable, translation-invariant and countably generated by radial
and circled 0-neighborhoods. Conversely, we show that such generalized
topologies are probabilistically normable.
http://arXiv.org/abs/math/0505484
http://front.math.ucdavis.edu/math.GN/0505484
(alternate)
Author(s): Ashkan Nikeghbali
Abstract: In this paper, we consider the special class of positive local submartingales
$(X_{t})$ of the form: $X_{t}=N_{t}+A_{t}$, where the measure $(dA_{t})$ is
carried by the set ${t: X_{t}=0}$. We show that many examples of stochastic
processes studied in the literature are in this class and propose a unified
approach based on martingale techniques to study them. In particular, we
establish some martingale characterizations for these processes and compute
explicitly some distributions involving the pair $(X_{t},A_{t})$. We also
associate with $X$ a solution to the Skorokhod's stopping problem for
probability measures on the positive half-line.
http://arXiv.org/abs/math/0505515
http://front.math.ucdavis.edu/math.PR/0505515
(alternate)
Author(s): Pedro J. Fernandez and Pablo A. Ferrari and Sebastian Grynberg
Abstract: A "coupling from the past" construction of the Gibbs sampler process is used
to perfectly simulate a random vector in a box B, a Cartesian product of
bounded intervals. An algorithm to sample vectors with multinormal distribution
truncated to B is implemented.
http://arXiv.org/abs/math/0505522
http://front.math.ucdavis.edu/math.PR/0505522
(alternate)
Author(s): Svante Janson and Joel Spencer
Abstract: We study a point process describing the asymptotic behavior of sizes of the
largest components of the random graph G(n,p) in the critical window
p=n^{-1}+lambda n^{-4/3}. In particular, we show that this point process has a
surprising rigidity. Fluctuations in the large values will be balanced by
opposite fluctuations in the small values such that the sum of the values
larger than a small epsilon is almost constant.
http://arXiv.org/abs/math/0505529
http://front.math.ucdavis.edu/math.PR/0505529
(alternate)
Author(s): Anne-Severine Boudou and Pietro Caputo and Paolo Dai Pra and Gustavo Posta
Abstract: We develop a general technique, based on the Bakry-Emery approach, to
estimate spectral gaps of a class of Markov operators. We apply this technique
to various interacting particle systems. In particular, we give a simple and
short proof of the diffusive scaling of the spectral gap of the Kawasaki model
at high temperature. Similar results are derived for Kawasaki-type dynamics in
the lattice without exclusion, and in the continuum. New estimates for
Glauber-type dynamics are also obtained.
http://arXiv.org/abs/math/0505533
http://front.math.ucdavis.edu/math.PR/0505533
(alternate)
Author(s): Gordon Blower and Fran\c{c}ois Bolley (UMPA-ENSL)
Abstract: For a stochastic process with state space some Polish space, this paper gives
sufficient conditions on the initial and conditional distributions for the
joint law to satisfy Gaussian concentration inequalities, transportation
inequalities and also logarithmic Sobolev inequalities in the case of the
Euclidean space. In several cases, the obtained constants are of optimal order
of growth with respect to the number of variables, or are independent of this
number. These results extend results known for mutually independent variables
to weakly dependent variables under Dobrushin-Shlosman type conditions.
http://arXiv.org/abs/math/0505536
http://front.math.ucdavis.edu/math.PR/0505536
(alternate)
Author(s): Joshua N. Cooper and Fan Chung
Abstract: What is the length of the shortest sequence $S$ of reals so that the set of
consecutive $n$-words in $S$ form a covering code for permutations on $\{1,2,
>..., n\}$ of radius $R$ ? (The distance between two $n$-words is the number of
transpositions needed to have the same order type.) The above problem can be
viewed as a special case of finding a De Bruijn covering code for a rooted
hypergraph. Each edge of a rooted hypergraph contains a special vertex, called
the {\it root} of the edge, and each vertex is the root of a unique edge,
called its {\it ball}. A De Bruijn covering code is a subset of the roots such
that every vertex is in some edge containing a chosen root. Under some mild
conditions, we obtain an upper bound for the shortest length of a De Bruijn
covering code of a rooted hypergraph, a bound which is within a factor of $\log
n$ of the lower bound.
http://arXiv.org/abs/math/0505528
http://front.math.ucdavis.edu/math.CO/0505528
(alternate)
Author(s): Janko Gravner and David Griffeath
Abstract: We consider discrete time random perturbations of monotone cellular automata
(CA) in two dimensions. Under general conditions, we prove the existence of
half--space velocities, and then establish the validity of the Wulff
construction for asymptotic shapes arising from finite initial seeds. Such a
shape converges to the polygonal invariant shape of the corresponding
deterministic model as the perturbation decreases. In many cases, exact
stability is observed. That is, for small perturbations, the shapes of the
deterministic and random processes agree exactly. We give a complete
characterization of such cases, and show that they are prevalent among
threshold growth CA with box neighborhood. We also design a nontrivial family
of CA in which the shape is exactly computable for all values of its
probability parameter.
http://arXiv.org/abs/math/0505039
http://front.math.ucdavis.edu/math.PR/0505039
(alternate)
Author(s): S. V. Lototsky and B. L. Rozovskii
Abstract: Space-only noise is a natural random perturbation in equations without time
evolution. Even the simplest equations driven by this noise often do not have a
square-integrable solution and must be solved in special weighted spaces. The
Cameron-Martin version of the Wiener chaos decomposition is an effective tool
to study both stationary and evolution equations driven by space-only noise.
The paper presents the main results about solvability of such equations in
weighted Wiener chaos spaces and studies the long-time behavior of the
solutions of evolution equations with space-only noise.
http://arXiv.org/abs/math/0505551
http://front.math.ucdavis.edu/math.PR/0505551
(alternate)
Author(s): Peter J. Forrester and Eric M. Rains
Abstract: In a recent work Killip and Nenciu gave random recurrences for the
characteristic polynomials of certain unitary and real orthogonal upper
Hessenberg matrices. The corresponding eigenvalue p.d.f.'s are
beta-generalizations of the classical groups. Left open was the direct
calculation of certain Jacobians. We provide the sought direct calculation.
Furthermore, we show how a multiplicative rank 1 perturbation of the unitary
Hessenberg matrices provides a joint eigenvalue p.d.f generalizing the circular
beta-ensemble, and we show how this joint density is related to known
inter-relations between circular ensembles. Projecting the joint density onto
the real line leads to the derivation of a random three-term recurrence for
polynomials with zeros distributed according to the circular Jacobi
beta-ensemble.
http://arXiv.org/abs/math/0505552
http://front.math.ucdavis.edu/math.PR/0505552
(alternate)
Author(s): A. A. Dorogovtsev
Abstract: The random measures on the space of continuous functions are considered.
Stationary random measures are described. The weak solutions of the stochastic
equations are substituted by the strong measure-valued solutions.
http://arXiv.org/abs/math/0505569
http://front.math.ucdavis.edu/math.PR/0505569
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: The goal of the paper is to study asymptotic behavior of the number of lost
messages. Long messages are assumed to be divided into a random number of
packets which are transmitted independently of one another. An error in
transmission of a packet results in the loss of the entire message. Messages
arrive to the $M/GI/1$ finite buffer model and can be lost in two cases as
either at least one of its packets is corrupted or the buffer is overflowed.
With the parameters of the system typical for models of information
transmission in real networks, we obtain theorems on asymptotic behavior of the
number of lost messages. We also study how the loss probability changes if
redundant packets are added. Our asymptotic analysis approach is based on
Tauberian theorems with remainder.
http://arXiv.org/abs/math/0505596
http://front.math.ucdavis.edu/math.PR/0505596
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: This paper provides the asymptotic analysis of the loss probability in the
$GI/M/1/n$ queueing system as $n$ increases to infinity. The approach of this
paper is alternative to that of the recent papers of Choi and Kim [2000] and
Choi et al [2000] and based on application of modern Tauberian theorems with
remainder. This enables us to simplify the proofs of the results on asymptotic
behavior of the loss probability of the abovementioned paper of Choi and Kim
[2000] as well as to obtain some new results.
http://arXiv.org/abs/math/0505597
http://front.math.ucdavis.edu/math.PR/0505597
(alternate)
Author(s): Emilio De Santis and Carlo Marinelli
Abstract: We introduce and study a class of infinite-horizon non-zero-sum
non-cooperative stochastic games with infinitely many interacting agents using
ideas of statistical mechanics. First we show, in the general case of
asymmetric interactions, the existence of a strategy that allows any player to
eliminate losses after a finite random time. In the special case of symmetric
interactions, we also prove that, as time goes to infinity, the game converges
to a Nash equilibrium. Moreover, assuming that all agents adopt the same
strategy, using arguments related to those leading to perfect simulation
algorithms, spatial mixing and ergodicity are proved. In turn, ergodicity
allows us to prove ``fixation'', i.e. that players will adopt a constant
strategy after a finite time. The resulting dynamics is related to
zero-temperature Glauber dynamics on random graphs of possibly infinite volume.
http://arXiv.org/abs/math/0505608
http://front.math.ucdavis.edu/math.PR/0505608
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: The paper establishes necessary and sufficient conditions for the stability
of different join-the-shortest-queue models including the load-balanced network
with general input and output processes. It is shown that the necessary and
sufficient condition for the stability of the load-balanced network is related
to the solution of the linear programming problem precisely formulated in the
paper. It is proved that if the minimum of the objective function of that
linear programming problem is less than 1, then the associated load-balanced
network is stable.
http://arXiv.org/abs/math/0505040
http://front.math.ucdavis.edu/math.PR/0505040
(alternate)
Author(s): Michael Blank and Leonid Bunimovich
Abstract: We show that under certain simple assumptions on the topology (structure) of
networks of strongly interacting chaotic elements a phenomenon of long range
action takes place, namely that the asymptotic (as time goes to infinity)
dynamics of an arbitrary large network is completely determined by its boundary
conditions. This phenomenon takes place under very mild and robust assumptions
on local dynamics with short range interactions. However, we show that it is
unstable with respect to arbitrarily weak local random perturbations.
http://arXiv.org/abs/math/0505610
http://front.math.ucdavis.edu/math.DS/0505610
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: The paper studies a multiserver retrial queueing system with $m$ servers.
Arrival process is a point process with strictly stationary and ergodic
increments. A customer arriving to the system occupies one of the free servers.
If upon arrival all servers are busy, then the customer goes to the secondary
queue, orbit, and after some random time retries more and more to occupy a
server. A service time of each customer is exponentially distributed random
variable with parameter $\mu_1$. A time between retrials is exponentially
distributed with parameter $\mu_2$ for each customer. Using a martingale
approach the paper provides an analysis of this system. The paper establishes
the stability condition and studies a behavior of the limiting queue-length
distributions as $\mu_2$ increases to infinity. As $\mu_2\to\infty$, the paper
also proves the convergence of appropriate queue-length distributions to those
of the associated `usual' multiserver queueing system without retrials. An
algorithm for numerical solution of the equations, associated with the limiting
queue-length distribution of retrial systems, is provided.
http://arXiv.org/abs/math/0505046
http://front.math.ucdavis.edu/math.PR/0505046
(alternate)
Author(s): Elizabeth S. Meckes and Mark W. Meckes
Abstract: Motivated by the central limit problem for convex bodies, we study normal
approximation of linear functionals of high-dimensional random vectors with
various types of symmetries. In particular, we obtain results for distributions
which are coordinatewise symmetric, uniform in a regular simplex, or
spherically symmetric. Our proofs are based on Stein's method of exchangeable
pairs; as far as we know, this approach has not previously been used in convex
geometry and we give a brief introduction to the classical method. The
spherically symmetric case is treated by a variation of Stein's method which is
adapted for continuous symmetries.
http://arXiv.org/abs/math/0505618
http://front.math.ucdavis.edu/math.PR/0505618
(alternate)
Author(s): Ashkan Nikeghbali
Abstract: Az\'{e}ma associated with an honest time $L$ the supermartingale
$Z_{t}^{L}=\mathbb{P}[L>t|\mathcal{F}_{t}]$ and established some of its
important properties. This supermartingale plays a central role in the general
theory of stochastic processes and in particular in the theory of progressive
enlargements of filtrations. In this paper, we shall give an additive
characterization for these supermartingales, which in turn will naturally
provide many examples of enlargements of filtrations. In particular, we use
this characterization to establish some path decomposition results, closely
related to or reminiscent of Williams' path decomposition results.
http://arXiv.org/abs/math/0505623
http://front.math.ucdavis.edu/math.PR/0505623
(alternate)
Author(s): Luis Baez-Duarte
Abstract: Let $X$ be a measure space and $T:X\to X$ a measurable transformation. For
any measurable $E\subseteq X$ and $x\in E$, the possibly infinite return time
is $n_E(x):=\inf\{n>0: T^n x\in E\}$. If $T$ is an ergodic tranformation of the
probability space $X$, and $\mu(E)>0$, then a theorem of M. Kac states that
$\int_E n_E d\mu=1$. We generalize this to any invertible measure preserving
transformation $T$ on a finite measure space $X$, by proving independently, and
nearly trivially that for any measurable $E\subseteq X$ one has $\int_E n_E
d\mu=\mu(I_E)$, where $I_E$ is the smallest invariant set containing $E$. In
particular this also provides a simpler proof of Poincar\'{e}'s recurrence
theorem.
http://arXiv.org/abs/math/0505625
http://front.math.ucdavis.edu/math.PR/0505625
(alternate)
Author(s): Hermine Bierm\'e and Anne Estrade
Abstract: We study a random field obtained by counting the number of balls containing
each point, when overlapping balls are thrown at random according to a Poisson
random measure. We are particularly interested in the local asymptotical
self-similarity (lass) properties of the field, as well as the action of X-ray
transforms. We exhibit two different lass properties when considering the
asymptotic either "in law" or "on the second order moment" and prove a
relationship between the lass behavior of the field and the lass behavior of
its X-ray transform. These results can be exploited to modelize and analyze
granular media, images or connections network.
http://arXiv.org/abs/math/0505635
http://front.math.ucdavis.edu/math.PR/0505635
(alternate)
Author(s): Gustavo Posta
Abstract: An unbounded one-dimensional solid-on-solid model with integer heights is
studied. Unbounded here means that there is no a priori restrictions on the
discret e gradient of the interface. The interaction Hamiltonian of the
interface is given by a finite range part, pr oportional to the sum of height
differences, plus a part of exponentially decaying long range potentials. The
evolution of the interface is a reversible Markov process. We prove that if
this system is started in the center of a box of size L after a time of order
L^3 it reaches, with a very large probability, the top or the bottom of the
box.
http://arXiv.org/abs/math/0505643
http://front.math.ucdavis.edu/math.PR/0505643
(alternate)
Author(s): B. T. Graham and G. R. Grimmett
Abstract: The influence theorem for product measures on the discrete space {0,1}^N may
be extended to probability measures with the property of monotonicity (which is
equivalent to `strong positive-association'). Corresponding results are valid
for probability measures on the cube [0,1]^N that are absolutely continuous
with respect to Lebesgue measure. These results lead to a sharp-threshold
theorem for measures of random-cluster type, and this may be applied to
box-crossings in the two-dimensional random-cluster model.
http://arXiv.org/abs/math/0505057
http://front.math.ucdavis.edu/math.PR/0505057
(alternate)
Author(s): T. Komorowski and L. Ryzhik
Abstract: We consider the motion of a particle in a two-dimensional spatially
homogeneous mixing potential and show that its momentum converges to the
Brownian motion on a circle. This complements the limit theorem of Kesten and
Papanicolaou \cite{KP} proved in dimensions $d\ge 3$.
http://arXiv.org/abs/math-ph/0505083
http://front.math.ucdavis.edu/math-ph/0505083
(alternate)
Author(s): Sergey Agievich and Oleg Solovey
Abstract: We consider a self-decimated generator of pseudorandom numbers and examine
the preperiod $\lambda$ and the period $\mu$ of its state sequence. We obtain
the expectations and variances of $\lambda$ and $\mu$ for the case when
decimation steps are chosen randomly and independently from the set {1,2}.
http://arXiv.org/abs/math/0505660
http://front.math.ucdavis.edu/math.CO/0505660
(alternate)
Author(s): Benoit Collins and Piotr Sniady
Abstract: We study asymptotics of the Itzykson-Zuber integrals in the scaling when one
of the matrices has a small rank compared to the full rank. We show that the
result is basically the same as in the case when one of the matrices has a
fixed rank. In this way we extend the recent results of Guionnet and Maida who
showed that for a latter scaling the Itzykson-Zuber integral is given in terms
of the Voiculescu's R-transform of the full rank matrix.
http://arXiv.org/abs/math/0505664
http://front.math.ucdavis.edu/math.PR/0505664
(alternate)
Author(s): Christopher Hoffman and Alexander E. Holroyd and Yuval Peres
Abstract: Let $\Xi$ be a discrete set in $\rd$. Call the elements of $\Xi$ centers. The
well-known Voronoi tessellation partitions $\rd$ into polyhedral regions (of
varying sizes) by allocating each site of $\rd$ to the closest center. Here we
study "fair" allocations of $\rd$ to $\Xi$ in which the regions allocated to
different centers have equal volumes.
We prove that if $\Xi$ is obtained from a translation-invariant ergodic point
process, then there is a unique fair allocation which is stable in the sense of
the Gale-Shapley marriage problem. (That is, sites and centers both prefer to
be allocated as close as possible, and an allocation is said to be unstable if
some site and center both prefer each other over their current allocations.)
We show that the region allocated to each center $\xi$ is a union of finitely
many bounded connected sets. However, in the case of a Poisson process, an
infinite volume of sites are allocated to a centers further away than $\xi$. We
prove power law lower bounds on the allocation distance of a typical site. It
is an open problem to prove any upper bound in $d>1$.
http://arXiv.org/abs/math/0505668
http://front.math.ucdavis.edu/math.PR/0505668
(alternate)
Author(s): S. Malefaki and G. Iliopoulos
Abstract: We consider importance sampling as well as other properly weighted samples
with respect to a target distribution $\pi$ from a different point of view. By
considering the associated weights as sojourn times until the next jump, we
define appropriate jump processes. When the original sample sequence forms an
ergodic Markov chain, the associated jump process is an ergodic semi--Markov
process with stationary distribution $\pi$. Hence, the type of convergence of
properly weighted samples may be stronger than that of weighted means. In
particular, when the samples are independent and the mean weight is bounded
above, we describe a slight modification in order to achieve exact (weighted)
samples from the target distribution.
http://arXiv.org/abs/math/0505045
http://front.math.ucdavis.edu/math.ST/0505045
(alternate)
Author(s): P. Mathieu and A. L. Piatnitski
Abstract: We prove the almost sure ('quenched') invariance principle for a random
walker on an infinite Bernoulli percolation cluster in $\Z^d$ where $d$ is
larger or equal than 2.
http://arXiv.org/abs/math/0505672
http://front.math.ucdavis.edu/math.PR/0505672
(alternate)
Author(s): Norio Konno and Naoki Masuda and Rahul Roy and Anish Sarkar
Abstract: We analyze the threshold network model in which a pair of vertices with
random weights are connected by an edge when the summation of the weights
exceeds a threshold. We prove some convergence theorems and central limit
theorems on the vertex degree, degree correlation, and the number of prescribed
subgraphs. We also generalize some results in the spatially extended cases.
http://arXiv.org/abs/math/0505681
http://front.math.ucdavis.edu/math.PR/0505681
(alternate)
Author(s): Klaus Fleischmann and Vitali Wachtel
Abstract: There is a well-known sequence of constants c_n describing the growth of
supercritical Galton-Watson processes Z_n. With 'lower deviation probabilities'
we refer to P(Z_n=k_n) with k_n=o(c_n) as n increases. We give a detailed
picture of the asymptotic behavior of such lower deviation probabilities. This
complements and corrects results known from the literature concerning special
cases. Knowledge on lower deviation probabilities is needed to describe large
deviations of the ratio Z_{n+1}/Z_n. The latter are important in statistical
inference to estimate the offspring mean. For our proofs, we adapt the
well-known Cramer method for proving large deviations of sums of independent
variables to our needs.
http://arXiv.org/abs/math/0505683
http://front.math.ucdavis.edu/math.PR/0505683
(alternate)
Author(s): M. Reiss and M. Riedle and O. van Gaans
Abstract: We consider a stochastic delay differential equation driven by a general Levy
process. Both, the drift and the noise term may depend on the past, but only
the drift term is assumed to be linear. We show that the segment process is
eventually Feller, but in general not eventually strong Feller on the Skorokhod
space. The existence of an invariant measure is shown by proving tightness of
the segments using semimartingale characteristics and the Krylov-Bogoliubov
method. A counterexample shows that the stationary solution in completely
general situations may not be unique, but in more specific cases uniqueness is
established.
http://arXiv.org/abs/math/0505684
http://front.math.ucdavis.edu/math.PR/0505684
(alternate)
Author(s): Alexander Gnedin and Jim Pitman
Abstract: The bijection between composition structures and random closed subsets of the
unit interval implies that the composition structures associated with $S \cap
[0,1]$ for a self-similar random set $S\subset {\mathbb R}_+$ are those which
are consistent with respect to a simple truncation operation. Using the
standard coding of compositions by finite strings of binary digits starting
with a 1, the random composition of $n$ is defined by the first $n$ terms of a
random binary sequence of infinite length. The locations of 1s in the sequence
are the places visited by an increasing time-homogeneous Markov chain on the
positive integers if and only if $S = \exp(-W)$ for some stationary
regenerative random subset $W$ of the real line. Complementing our study in
previous papers, we identify self-similar Markovian composition structures
associated with the two-parameter family of partition structures.
http://arXiv.org/abs/math/0505687
http://front.math.ucdavis.edu/math.PR/0505687
(alternate)
Author(s): Sharad Goel and Ravi Montenegro and Prasad Tetali
Abstract: On complete, non-compact manifolds and infinite graphs, Faber-Krahn
inequalities have been used to estimate the rate of decay of the heat kernel.
We develop this technique in the setting of finite Markov chains, proving upper
and lower mixing time bounds via the spectral profile. This approach lets us
recover and refine previous conductance-based bounds of mixing time (including
the Morris-Peres result), and in general leads to sharper estimates of
convergence rates. We apply this method to several models including groups with
moderate growth, the fractal-like Viscek graphs, and the torus, to obtain tight
bounds on the corresponding mixing times.
http://arXiv.org/abs/math/0505690
http://front.math.ucdavis.edu/math.PR/0505690
(alternate)
Author(s): Alexander Gnedin and Zbigniew Nitecki
Abstract: A rearrangement of $n$ independent uniform $[0,1]$ random variables is a
sequence of $n$ random variables $Y_1,...,Y_n$ whose vector of order statistics
has the same distribution as that for the $n$ uniforms. We consider
rearrangements satisfying the strong rank independence condition, that the rank
of $Y_k$ among $Y_1,...,Y_k$ is independent of the values of $Y_1,...,Y_{k-1}$,
for $k=1,...,n$. Nontrivial examples of such rearrangements are the travellers'
processes defined by Gnedin and Krengel. We show that these are the only
examples when $n=2$, and when certain restrictive assumptions hold for $n\geq
3$; we also construct a new class of examples of such rearrangements for which
the restrictive assumptions do not hold.
http://arXiv.org/abs/math/0505692
http://front.math.ucdavis.edu/math.PR/0505692
(alternate)
Author(s): Matthieu Delescluse (LPC) and Christophe Pouzat (LPC)
Abstract: We demonstrate the efficacy of a new spike-sorting method based on a Markov
Chain Monte Carlo (MCMC) algorithm by applying it to real data recorded from
Purkinje cells (PCs) in young rat cerebellar slices. This algorithm is unique
in its capability to estimate and make use of the firing statistics as well as
the spike amplitude dynamics of the recorded neurons. PCs exhibit multiple
discharge states, giving rise to multimodal interspike interval (ISI)
histograms and to correlations between successive ISIs. The amplitude of the
spikes generated by a PC in an "active" state decreases, a feature typical of
many neurons from both vertebrates and invertebrates. These two features
constitute a major and recurrent problem for all the presently available
spike-sorting methods. We first show that a Hidden Markov Model with 3
log-Normal states provides a flexible and satisfying description of the complex
firing of single PCs. We then incorporate this model into our previous MCMC
based spike-sorting algorithm (Pouzat et al, 2004, J. Neurophys. 91, 2910-2928)
and test this new algorithm on multi-unit recordings of bursting PCs. We show
that our method successfully classifies the bursty spike trains fired by PCs by
using an independent single unit recording from a patch-clamp pipette.
http://arXiv.org/abs/q-bio/0505053
http://front.math.ucdavis.edu/q-bio.QM/0505053
(alternate)
Author(s): Anamaria Savu
Abstract: A fourth-order nonlinear evolution equation is derived from a microscopic
model for surface diffusion, namely, the continuum solid-on-solid model. We use
the method developed by Varadhan for the computation of hydrodynamic scaling
limit of nongradient models. What distinguishes our model from other models
discussed so far is the presence of two conservation laws for the dynamics in a
nonperiodic box and the complex dynamics that is not nearest-neighbor. Along
the way, a few steps has to be adapted to our new context. As a byproduct of
our main result we also derive the hydrodynamic scaling limit of a perturbation
of continuum solid-on-solid model, a model that incorporates both surface
diffusion and surface electromigration.
http://arXiv.org/abs/math/0506001
http://front.math.ucdavis.edu/math.PR/0506001
(alternate)
Author(s): Arnak Dalalyan (PMA) and Markus Reiss (WIAS)
Abstract: Asymptotic local equivalence in the sense of Le Cam is established for
inference on the drift in multidimensional ergodic diffusions and an
accompanying sequence of Gaussian shift experiments. The nonparametric local
neighbourhoods can be attained for any dimension, provided the regularity of
the drift is sufficiently large. In addition, a heteroskedastic Gaussian
regression experiment is given, which is also locally asymptotically equivalent
and which does not depend on the centre of localisation. For one direction of
the equivalence an explicit Markov kernel is constructed.
http://arXiv.org/abs/math/0505053
http://front.math.ucdavis.edu/math.ST/0505053
(alternate)
Author(s): Stephan Lawi
Abstract: We give a necessary and sufficient condition for a homogeneous Markov process
taking values in $\R^n$ to enjoy the time-inversion property of degree
$\alpha$. The condition sets the shape for the semigroup densities of the
process and allows to further extend the class of known processes satisfying
the time-inversion property. As an application we recover the result of
Watanabe in \cite{Wa1975} for continuous and conservative Markov processes on
$\R_+$. As new examples we generalize Dunkl processes and construct a
matrix-valued process with jumps related to the Wishart process by a
skew-product representation.
http://arXiv.org/abs/math/0506013
http://front.math.ucdavis.edu/math.PR/0506013
(alternate)
Author(s): Anamaria Savu
Abstract: For a general vector field we exhibit two Hilbert spaces, namely the space of
so called closed functions and the space of exact functions and we calculate
the codimension of the space of exact functions inside the larger space of
closed functions. In particular we provide a new approach for the known cases:
the Glauber field and the second-order Ginzburg-Landau field, and for the case
of the fourth-order Ginzburg-Landau field.
http://arXiv.org/abs/math/0506002
http://front.math.ucdavis.edu/math.FA/0506002
(alternate)
Author(s): Radoslaw Adamczak
Abstract: We present moment inequalities for completely degenerate Banach space valued
(generalized) U-statistics of arbitrary order. The estimates involve suprema of
empirical processes, which in the real valued case can be replaced by simpler
norms of the kernel matrix (i.e. norms of some multilinear operators associated
with the kernel matrix). As a corollary we derive tail inequalities for
U-statistics with bounded kernels and for some multiple stochastic integrals.
http://arXiv.org/abs/math/0506026
http://front.math.ucdavis.edu/math.PR/0506026
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: The $M/GI/m/n$ queueing system under the assumption that $\lambda = m\mu$ is
considered, where $\lambda$ is the rate of arrivals, $\mu$ is the reciprocal of
the expected service times, $m$ is the number of servers and $n$ is the
maximally possible queue-length. It is proved that the expectation of the
number of losses during a busy period is equal to $m^m/m!$ for all $n\geq 0$.
This result is an extension of the corresponding result for the $M/GI/1/n$
queueing system established originally by the author.
http://arXiv.org/abs/math/0506033
http://front.math.ucdavis.edu/math.PR/0506033
(alternate)
Author(s): Jon Warren
Abstract: We consider stochastic processes indexed by the vertices of an infinite
binary tree having a simple recursive structure. The value at any vertex is
some fixed function of the values at the two daughter vertices together with
some independent innovation. Endogeny means the innovations are generating.
When endogeny does not hold there exist dynamics in which the innovations are
held fixed while some additional randomness on the boundary of the tree is
perturbed.
http://arXiv.org/abs/math/0506038
http://front.math.ucdavis.edu/math.PR/0506038
(alternate)
Author(s): A. M. G. Cox and D. G. Hobson
Abstract: In this paper we consider the Skorokhod embedding problem in Brownian motion.
In particular, we give a solution based on the local time at zero of a variably
skewed Brownian motion related to the underlying Brownian motion. Special cases
of the construction include the Azema-Yor and Vallois embeddings. In turn, the
construction has an interpretation in the Chacon-Walsh framework.
http://arXiv.org/abs/math/0506040
http://front.math.ucdavis.edu/math.PR/0506040
(alternate)
Author(s): Henri Comman
Abstract: Let $(\mu_{\alpha})$ be a net of Radon sub-probability measures on the real
line, and $(t_{\alpha})$ be a net in $]0,+\infty[$ converging to 0. Assuming
that the generalized log-moment generating function $L(\lambda)$ exists for all
$\lambda$ in a nonempty open interval $G$, we give conditions on the left or
right derivatives of $L_{\mid G}$, implying vague (and thus narrow when $0\in
G$) large deviations. The rate function (which can be nonconvex) is obtained as
an abstract Legendre-Fenchel transform. This allows us to strengthen the
G\"{a}rtner-Ellis theorem by removing the usual differentiability assumption. A
related question of R. S. Ellis is solved.
http://arXiv.org/abs/math/0506044
http://front.math.ucdavis.edu/math.PR/0506044
(alternate)
Author(s): Robert O. Bauer and Roland M. Friedrich
Abstract: We give a geometric derivation of SLE($\kappa,\rho$) in terms of conformally
invariant random growing subsets of polygons. We relate the parameters $\rho_j$
to the exterior angles of the polygons. We also show that SLE($\kappa,\rho$)
can be generated by a metric Brownian motion, where metric and Brownian motion
are coupled and the metric ist the pull-back metric of the Euclidean metric of
an evolving polygon.
http://arXiv.org/abs/math/0506062
http://front.math.ucdavis.edu/math.PR/0506062
(alternate)
Author(s): Jiali Liao and Ted Theodosopoulos
Abstract: Over-the-counter derivatives have contributed significantly to the
effectiveness and efficiency of the international financial system but also
entail significant counterparty credit risk. Collateralization is one of the
most important and widespread credit risk mitigation techniques used in
derivatives transactions. However, the relevant decisions are often made in an
ad-hoc manner, without reference to an analytical framework. Very little
academic research has addressed the quantitative analysis of collateralization
for contingent credit risk control. The issue of mark-to-market timing becomes
important for reducing credit exposure of illiquid and long term derivative
contracts due to the difficulty and cost of marking to market. the goal of this
research is to propose a framework for minimizing the potential credit exposure
of collateralized derivative transactions by optimizing mark-to-market timing.
http://arXiv.org/abs/math/0506077
http://front.math.ucdavis.edu/math.PR/0506077
(alternate)
Author(s): Jean Bertoin (PMA) and Jean-Fran\c{c}ois Le Gall (DMA)
Abstract: We prove several limit theorems that relate coalescent processes to
continuous-state branching processes. Some of these theorems are stated in
terms of the so-called generalized Fleming-Viot processes, which describe the
evolution of a population with fixed size, and are duals to the coalescents
with multiple collisions studied by Pitman and others. We first discuss
asymptotics when the initial size of the population tends to infinity. In that
setting, under appropriate hypotheses, we show that a rescaled version of the
generalized Fleming-Viot process converges weakly to a continuous-state
branching process. As a corollary, we get a hydrodynamic limit for certain
sequences of coalescents with multiple collisions: Under an appropriate
scaling, the empirical measure associated with sizes of the blocks converges to
a (deterministic) limit which solves a generalized form of Smoluchowski's
coagulation equation. We also study the behavior in small time of a fixed
coalescent with multiple collisions, under a regular variation assumption on
the tail of the measure $\nu$ governing the coalescence events. Precisely, we
prove that the number of blocks with size less than $\epsilon x$ at time
$(\epsilon\nu([\epsilon,1]))^{-1}$ behaves like
$\epsilon^{-1}\lambda\_1(]0,x[)$ as $\epsilon\to 0$, where $\lambda\_1$ is the
distribution of the size of one cluster at time 1 in a continuous-state
branching process with stable branching mechanism. This generalizes a classical
result for the Kingman coalescent.
http://arXiv.org/abs/math/0506092
http://front.math.ucdavis.edu/math.PR/0506092
(alternate)
Author(s): Yuval Peres and Paul Shields
Abstract: We present two new methods for estimating the order (memory depth) of a
finite alphabet Markov chain from observation of a sample path. One method is
based on entropy estimation via recurrence times of patterns, and the other
relies on a comparison of empirical conditional probabilities. The key to both
methods is a qualitative change that occurs when a parameter (a candidate for
the order) passes the true order. We also present extensions to order
estimation for Markov random fields.
http://arXiv.org/abs/math/0506080
http://front.math.ucdavis.edu/math.ST/0506080
(alternate)
Author(s): Boaz Nadler and Stephane Lafon and Ronald R. Coifman and Ioannis G. Kevrekidis
Abstract: This paper presents a diffusion based probabilistic interpretation of
spectral clustering and dimensionality reduction algorithms that use the
eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency
matrix of all points, we define a diffusion distance between any two data
points and show that the low dimensional representation of the data by the
first few eigenvectors of the corresponding Markov matrix is optimal under a
certain mean squared error criterion. Furthermore, assuming that data points
are random samples from a density $p(\x) = e^{-U(\x)}$ we identify these
eigenvectors as discrete approximations of eigenfunctions of a Fokker-Planck
operator in a potential $2U(\x)$ with reflecting boundary conditions. Finally,
applying known results regarding the eigenvalues and eigenfunctions of the
continuous Fokker-Planck operator, we provide a mathematical justification for
the success of spectral clustering and dimensional reduction algorithms based
on these first few eigenvectors. This analysis elucidates, in terms of the
characteristics of diffusion processes, many empirical findings regarding
spectral clustering algorithms.
http://arXiv.org/abs/math/0506090
http://front.math.ucdavis.edu/math.NA/0506090
(alternate)
Author(s): David Maher
Abstract: We reprove a result concerning certain ruin in the classical problem of the
probability of ruin with risky investments and several of it's generalisations.
We also provide the combined transition density of the risk and investment
processes in the diffusion case.
http://arXiv.org/abs/math/0506127
http://front.math.ucdavis.edu/math.PR/0506127
(alternate)
Author(s): Ariel Yadin
Abstract: We study a Markov chain called the mixer chain, swapping tiles placed on a
graph. If the graph is a Cayley graph, this process is a random walk on a
semidirect product of groups. For the graph Z, we study the rate of escape of
this chain. We show that, with probability tending to 1 as time tends to
infinity, the chain is at distance at least t^{3/4} from its origin, and at
most t^{3/4} log^{5/4}(t).
http://arXiv.org/abs/math/0506129
http://front.math.ucdavis.edu/math.PR/0506129
(alternate)
Author(s): Thomas Lepage (1) and Stephan Lawi (2) and Paul Tupper (1) and David Bryant (1) ((1) McGill University (2) Universit\'e Pierre et Marie Curie)
Abstract: We propose a continuous model for evolutionary rate variation across sites
and over the tree and derive exact transition probabilities under this model.
Changes in rate are modelled using the CIR process, a diffusion widely used in
financial applications. The model directly extends the standard gamma
distributed rates across site model, with one additional parameter governing
changes in rate down the tree. The parameters of the model can be estimated
directly from two well-known statistics: the index of dispersion and the gamma
shape parameter of the rates across sites model. The CIR model can be readily
incorporated into probabilistic models for sequence evolution. We provide here
an exact formula for the likelihood of a three taxa tree. Larger trees can be
evaluated using Monte-Carlo methods.
http://arXiv.org/abs/math/0506145
http://front.math.ucdavis.edu/math.PR/0506145
(alternate)
Author(s): Oded Schramm and Jeffrey E. Steif
Abstract: One goal of this paper is to prove that dynamical critical site percolation
on the planar triangular lattice has exceptional times at which percolation
occurs. In doing so, new quantitative noise sensitivity results for percolation
are obtained. The latter is based on a novel method for controlling the
"level k" Fourier coefficients via the construction of a randomized algorithm
which looks at random bits, outputs the value of a particular function but
looks at any fixed input bit with low probability. We also obtain upper and
lower bounds on the Hausdorff dimension of the set of percolating times. We
then study the problem of exceptional times for certain "k-arm" events on
wedges and cones. As a corollary of this analysis, we prove, among other
things, that there are no times at which there are two infinite "white"
clusters, obtain an upper bound on the Hausdorff dimension of the set of times
at which there are both an infinite white cluster and an infinite black cluster
and prove that for dynamical critical bond percolation on the square grid there
are no exceptional times at which three disjoint infinite clusters are present.
http://arXiv.org/abs/math/0504586
http://front.math.ucdavis.edu/math.PR/0504586
(alternate)
Author(s): Oliver Johnson
Abstract: Define the non-overlapping return time of a random process to be the number
of blocks that we wait before a particular block reappears. We prove a Central
Limit Theorem based on these return times. This result has applications to
entropy estimation, and to the problem of determining if digits have come from
an independent equidistribted sequence. In the case of an equidistributed
sequence, we use an argument based on negative association to prove convergence
under weaker conditions.
http://arXiv.org/abs/math/0506165
http://front.math.ucdavis.edu/math.PR/0506165
(alternate)
Author(s): Alain Comtet and Satya N. Majumdar
Abstract: We consider a discrete time random walk in one dimension. At each time step
the walker jumps by a random distance, independent from step to step, drawn
from an arbitrary symmetric density function. We show that the expected
positive maximum E[M_n] of the walk up to n steps behaves asymptotically for
large n as, E[M_n]/\sigma=\sqrt{2n/\pi}+ \gamma +O(n^{-1/2}), where \sigma^2 is
the variance of the step lengths. While the leading \sqrt{n} behavior is
universal and easy to derive, the leading correction term turns out to be a
nontrivial constant \gamma. For the special case of uniform distribution over
[-1,1], Coffmann et. al. recently computed \gamma=-0.516068...by exactly
enumerating a lengthy double series. Here we present a closed exact formula for
\gamma valid for arbitrary symmetric distributions. We also demonstrate how
\gamma appears in the thermodynamic limit as the leading behavior of the
difference variable E[M_n]-E[|x_n|] where x_n is the position of the walker
after n steps. An application of these results to the equilibrium
thermodynamics of a Rouse polymer chain is pointed out. We also generalize our
results to L\'evy walks.
http://arXiv.org/abs/cond-mat/0506195
http://front.math.ucdavis.edu/cond-mat/0506195
(alternate)
Author(s): Makoto Katori
Abstract: In the paper [7] we studied the temporally inhomogeneous system of
non-colliding Brownian motions and proved that multi-time correlation functions
are generally given by the quaternion determinants in the sense of Dyson and
Mehta. In this report we give another proof of the equivalent statement using
Fredholm determinant and Fredholm pfaffian, and claim that the present system
is a typical example of pfaffian processes.
http://arXiv.org/abs/math/0506186
http://front.math.ucdavis.edu/math.PR/0506186
(alternate)
Author(s): Makoto Katori and Hideki Tanemura
Abstract: Yor's generalized meander is a temporally inhomogeneous modification of the
$2(\nu+1)$-dimensional Bessel process with $\nu > -1$, in which the
inhomogeneity is indexed by $\kappa \in [0, 2(\nu+1))$. We introduce the
non-colliding particle systems of the generalized meanders and prove that they
are the Pfaffian processes, in the sense that any multitime correlation
function is given by a Pfaffian. In the infinite particle limit, we show that
the elements of matrix kernels of the obtained infinite Pfaffian processes are
generally expressed by the Riemann-Liouville differintegrals of functions
comprising the Bessel functions $J_{\nu}$ used in the fractional calculus,
where orders of differintegration are determined by $\nu-\kappa$. As special
cases of the two parameters $(\nu, \kappa)$, the present infinite systems
include the quaternion determinantal processes studied by Forrester, Nagao and
Honner and by Nagao, which exhibit the temporal transitions between the
universality classes of random matrix theory.
http://arXiv.org/abs/math/0506187
http://front.math.ucdavis.edu/math.PR/0506187
(alternate)
Author(s): Hyun Jae Yoo
Abstract: Given a positive definite, bounded linear operator $A$ on the Hilbert space
$\mathcal{H}_0:=l^2(E)$, we consider a reproducing kernel Hilbert space
$\mathcal{H}_+$ with a reproducing kernel $A(x,y)$. Here $E$ is any countable
set and $A(x,y)$, $x,y\in E$, is the representation of $A$ w.r.t. the usual
basis of $\mathcal{H}_0$. Imposing further conditions on the operator $A$, we
also consider another reproducing kernel Hilbert space $\mathcal{H}_-$ with a
kernel function $B(x,y)$, which is the representation of the inverse of $A$ in
a sense, so that $\mathcal{H}_-\supset\mathcal{H}_0\supset\mathcal{H}_+$
becomes a rigged Hilbert space. We investigate a relationship between the
ratios of determinants of some partial matrices related to $A$ and $B$ and the
suitable projections in $\mathcal{H}_-$ and $\mathcal{H}_+$. We also get a
variational principle on the limit ratios of these values. We apply this
relation to show the Gibbsianness of the determinantal point process (or
fermion point process) defined by the operator $A(I+A)^{-1}$ on the set $E$. It
turns out that the class of determinantal point processes that can be
recognized as Gibbs measures for suitable interactions is much bigger than that
obtained by Shirai and Takahashi.
http://arXiv.org/abs/math/0506189
http://front.math.ucdavis.edu/math.PR/0506189
(alternate)
Author(s): Bertrand Deroin & Victor Kleptsyn
Abstract: We consider random dynamical systems such as groups of conformal
transformations with a probability measure, or transversaly conformal
foliations with a Laplace operator along the leaves, in which case we consider
the holonomy pseudo-group. We prove that either there exists a measure
invariant under all the elements of the group (or the pseudo-group), or almost
surely a long composition of maps contracts exponentially a ball. We deduce
some results about the unique ergodicity.
http://arXiv.org/abs/math/0506204
http://front.math.ucdavis.edu/math.DS/0506204
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: In this paper continuity theorems are established for the number of losses
during a busy period of the $M/M/1/n$ queue, when the service time probability
distribution, slightly different in certain sense from the exponential
distribution, is approximated by that exponential distribution. Continuity
theorems are obtained in the form of one or two-side stochastic inequalities.
The paper shows how the bounds of these inequalities are changed if one or
other assumption, associated with specific properties of the service time
distribution (precisely described in the paper), is done. Specifically, some
parametric families of service time distributions are discussed, and the paper
establishes uniform estimations (given for all possible values of the
parameter) and local estimations (where the parameter is fixed and takes only
the given value). The analysis of the paper is based on the level crossing
approach and some characterization properties of exponential distribution.
http://arXiv.org/abs/math/0506227
http://front.math.ucdavis.edu/math.PR/0506227
(alternate)
Author(s): J. E. Yukich and Yu Zhang
Abstract: Let a and b be fixed positive scalars. Assign independently to each edge in
the two-dimensional integer lattice the value a with probability p or the value
b with probability 1-p. For all u and v in the two-dimensional integer lattice,
let T(u,v) denote the first passage time between u and v. We show that there
are points x in the plane such that the `time constant' in the direction of x,
namely lim_{n \to \infty} n^{-1} E_p[T(0, nx)], is not a three times
differentiable function of p.
http://arXiv.org/abs/math/0506241
http://front.math.ucdavis.edu/math.PR/0506241
(alternate)
Author(s): E. Sandhya and S. Sherly and and N. Raju
Abstract: In this paper we discuss the basic properties of a discrete distribution
introduced by Harris in 1948 and obtain a characterization of it. The
divisibility properties of the distribution are also studied. We derive the
moment and maximum likelihood estimators for both the parameters and verify
them by simulated observations.
http://arXiv.org/abs/math/0506220
http://front.math.ucdavis.edu/math.ST/0506220
(alternate)
Author(s): Shahar Mendelson and Alain Pajor and Nicole Tomczak-Jaegermann
Abstract: We present a randomized method to approximate any vector $v$ from some set $T
\subset \R^n$. The data one is given is the set $T$, and $k$ scalar products
$(\inr{X_i,v})_{i=1}^k$, where $(X_i)_{i=1}^k$ are i.i.d. isotropic subgaussian
random vectors in $\R^n$, and $k \ll n$. We show that with high probability,
any $y \in T$ for which $(\inr{X_i,y})_{i=1}^k$ is close to the data vector
$(\inr{X_i,v})_{i=1}^k$ will be a good approximation of $v$, and that the
degree of approximation is determined by a natural geometric parameter
associated with the set $T$.
We also investigate a random method to identify exactly any vector which has
a relatively short support using linear subgaussian measurements as above. It
turns out that our analysis, when applied to $\{-1,1\}$-valued vectors with
i.i.d, symmetric entries, yields new information on the geometry of faces of
random $\{-1,1\}$-polytope; we show that a $k$-dimensional random
$\{-1,1\}$-polytope with $n$ vertices is $m$-neighborly for very large $m\le
{ck/\log (c' n/k)}$. The proofs are based on new estimates on the behavior of
the empirical process $\sup_{f \in F} |k^{-1}\sum_{i=1}^k f^2(X_i) -\E f^2 |$
when $F$ is a subset of the $L_2$ sphere. The estimates are given in terms of
the $\gamma_2$ functional with respect to the $\psi_2$ metric on $F$, and hold
both in exponential probability and in expectation.
http://arXiv.org/abs/math/0506239
http://front.math.ucdavis.edu/math.FA/0506239
(alternate)
Author(s): Marek Biskup and Lincoln Chayes and S. Alex Smith
Abstract: We present a large-deviations/thermodynamic approach to the classic problem
of percolation on the complete graph. Specifically, we determine the
large-deviation rate function for the probability that the giant component
occupies a fixed fraction of the graph. One consequence is an immediate
derivation of the "cavity" formula for the fraction of sites in the giant
component. As a by-product of our analysis we compute also the large-deviation
rate functions for the probabilities of the event that the random graph is
connected, the event that it contains no loops and the event that it contains
only "small" components.
http://arXiv.org/abs/math/0506255
http://front.math.ucdavis.edu/math.PR/0506255
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: The present paper provides some new stochastic inequalities for the
characteristics of the $M/GI/1/n$ and $GI/M/1/n$ loss queueing systems. These
stochastic inequalities are based on the deepen up- and down-crossings method,
and they are stronger than the known stochastic inequalities obtained earlier.
http://arXiv.org/abs/math/0505068
http://front.math.ucdavis.edu/math.PR/0505068
(alternate)
Author(s): Inder Jeet Taneja
Abstract: There are many information and divergence measures exist in the literature on
information theory and statistics. The most famous among them are
Kullback-Leibler (1951) relative information and Jeffreys (1951) J-divergence.
Sibson (1969) Jensen-Shannon divergence has also found its applications in the
literature. The author (1995) studied a new divergence measures based on
arithmetic and geometric means. The measures like harmonic mean divergence and
triangular discrimination are also known in the literature. Recently, Dragomir
et al. (2001) also studies a new measure similar to J-divergence, we call here
the relative J-divergence. Another measures arising due to Jensen-Shannon
divergence is also studied by Lin (1991). Here we call it relative
Jensen-Shannon divergence. Relative arithmetic-geometric divergence (ref.
Taneja, 2004) is also studied here. All these measures can be written as
particular cases of Csiszar's f-divergence. By putting some conditions on the
probability distribution, the aim here is to obtain bounds among the measures.
http://arXiv.org/abs/math/0506256
http://front.math.ucdavis.edu/math.PR/0506256
(alternate)
Author(s): Alexander Teplyaev
Abstract: We define finitely connected fractafolds, which are generalizations of p.c.f.
self-similar sets introduced by Kigami and of fractafolds introduced by
Strichartz. Any self-similarity is not assumed, and countably infinite
ramification is allowed. We prove that if a fractafold has a resistance form in
the sense of Kigami that satisfies certain assumptions, then there exists a
weak Riemannian metric, defined almost everywhere, such that the energy can be
expressed as the integral of the norm of a weak gradient with respect to an
energy measure. This generalizes earlier results by Kusuoka and the author.
Furthermore, we prove that if the fractafold can be homeomorphically
represented in harmonic coordinates, then the weak gradient can be replaced by
the usual gradient for smooth functions, which generalizes an earlier result by
Kigami. We also prove a simple formula for the energy measure Laplacian in
harmonic coordinates.
http://arXiv.org/abs/math/0506261
http://front.math.ucdavis.edu/math.PR/0506261
(alternate)
Author(s): Boris Tsirelson
Abstract: The scaling limit of the critical percolation, is it a black noise? The
answer depends on stability to perturbations concentrated along a line. This
text, containing no proofs, reports experimental results that suggest the
affirmative answer.
http://arXiv.org/abs/math/0506269
http://front.math.ucdavis.edu/math.PR/0506269
(alternate)
Author(s): Alexander Soshnikov
Abstract: We study translation-invariant determinantal random point fields on the real
line. We prove, under quite general conditions, that the smallest nearest
spacings between the particles in a large interval have Poisson statistics as
the length of the interval goes to infinity.
http://arXiv.org/abs/math/0506286
http://front.math.ucdavis.edu/math.PR/0506286
(alternate)
Author(s): K. Fleischmann and J. M. Swart
Abstract: Recently, several authors have studied maps where a function, describing the
local diffusion matrix of a diffusion process with a linear drift towards an
attraction point, is mapped into the average of that function with respect to
the unique invariant measure of the diffusion process, as a function of the
attraction point. Such mappings arise in the analysis of infinite systems of
diffusions indexed by the hierarchical group, with a linear attractive
interaction between the components. In this context, the mappings are called
renormalization transformations. We consider such maps for catalytic
Wright-Fisher diffusions. These are diffusions on the unit square where the
first component (the catalyst) performs an autonomous Wright-Fisher diffusion,
while the second component (the reactant) performs a Wright-Fisher diffusion
with a rate depending on the first component through a catalyzing function. We
determine the limit of rescaled iterates of renormalization transformations
acting on the diffusion matrices of such catalytic Wright-Fisher diffusions.
http://arXiv.org/abs/math/0506311
http://front.math.ucdavis.edu/math.PR/0506311
(alternate)
Author(s): Joseph Najnudel (PMA)
Abstract: In this paper, we construct a family of probability measures, by
penalizations of a Walsh's Brownian motion with a weight dependent on its value
and its local time at a time t. We prove that this family converges to a
probability measure as t tends to infinity, and we study the behaviour of this
limit measure.
http://arXiv.org/abs/math/0506329
http://front.math.ucdavis.edu/math.PR/0506329
(alternate)
Author(s): Michael J. Kozdron (University of Regina)
Abstract: We prove that the scaling limit of two-dimensional simple random walk
excursion measure in any bounded, simply connected Jordan domain with given
inradius is the Brownian excursion measure, a conformally invariant infinite
measure on paths.
http://arXiv.org/abs/math/0506337
http://front.math.ucdavis.edu/math.PR/0506337
(alternate)
Author(s): Javiera Barrera (MAP5) and Thierry Huillet (LPTM) and Christian Paroissin (LMA - PAU)
Abstract: Consider a list of $n$ files whose popularities are random. These files are
updated according to the move-to-front rule and we consider the induced Markov
chain at equilibrium. We give the exact limiting distribution of the
search-cost per item as $n$ tends to infinity. Some examples are supplied.
http://arXiv.org/abs/math/0506343
http://front.math.ucdavis.edu/math.PR/0506343
(alternate)
Author(s): Marek Biskup and Roman Kotecky
Abstract: Existence of first-order phase transitions is often proved with the aid of
reflection positivity and chessboard estimates. The standard approach relies on
estimates of correlations in torus measures which yield the existence of a
transition point where the free energy has a discontinuous derivative with
respect to a suitably chosen variable. In addition, at the transition point,
two distinct translation-invariant Gibbs states are extracted from torus
measures in which the one-sided derivatives of the free energy are realized as
expectations of a local observable $X$. Here we show that (most of) the gap
between these extreme expected values is forbidden: There are no shift-ergodic
Gibbs states for which the expectation of $X$ lies deep inside the gap. We
point out several recent results based on chessboard estimates where our main
theorems provide important additional information concerning the structure of
the set of possible thermodynamic equilibria.
http://arXiv.org/abs/math-ph/0505011
http://front.math.ucdavis.edu/math-ph/0505011
(alternate)
Author(s): Ashkan Nikeghbali
Abstract: In this paper, we establish a multiplicative decomposition formula for
nonnegative local martingales and use it to characterize the set of continuous
local submartingales $Y$ of the form $Y=N+A$, where the measure $dA$ is carried
by the set of zeros of $Y$. In particular, we shall see that in the set of all
local submartingales with the same martingale part in the multiplicative
decomposition, these submartingales are the smallest ones. We also study some
integrability questions in the multiplicative decomposition and interpret the
notion of saturated sets in the light of our results.
http://arXiv.org/abs/math/0506369
http://front.math.ucdavis.edu/math.PR/0506369
(alternate)
Author(s): D. H. U. Marchetti and V. Sidoravicius and M. E. Vares
Abstract: We consider the one-dimensional long-range Fortuin--Kasteleyn random-cluster
model, generated by the edge occupation probabilities p_{} = p if |x-y| =
1, 1 - exp{-beta |x-y|^2} otherwise, and weighting factor kappa \geq 1. We
prove the occurrence of oriented percolation when beta>1 and kappa \geq 1,
provided p is chosen sufficiently close to 1. We also show that the oriented
truncated connectivity tau ^{prime}(x,y) satisfies tau ^{prime}(x,y) \leq C
|x-y|^{-theta} with theta = min(2(beta eta -1),2) where eta = eta(p) \nearrow 1
as p \nearrow 1.
http://arXiv.org/abs/math/0506404
http://front.math.ucdavis.edu/math.PR/0506404
(alternate)
Author(s): P.Okunev
Abstract: We propose a fast algorithm for computing the expected tranche loss in the
Gaussian factor model. We test it on portfolios ranging in size from 25 (the
size of DJ iTraxx Australia) to 100 (the size of DJCDX.NA.HY) with a single
factor Gaussian model and show that the algorithm gives accurate results. The
algorithm proposed here is an extension of the algorithm proposed in \cite{PO}.
The advantage of the new algorithm is that it works well for portfolios of
smaller size for which the normal approximation proposed in \cite{PO} in not
sufficiently accurate. The algorithm is intended as an alternative to the much
slower Fourier transform based methods \cite{MD}.
http://arXiv.org/abs/math/0506378
http://front.math.ucdavis.edu/math.ST/0506378
(alternate)
Author(s): Gautam Iyer
Abstract: We consider a stochastic flow with drift $u$ and diffusion coefficient
$\sqrt{2 \nu}$. We demand that the drift be recovered from the flow map using
the Weber formula, as in the Eulerian-Lagrangian formulation of the Euler
equations. In the absence of diffusion, this will yield the Euler equations. We
first prove the existence of such stochastic flows, and that the expected value
of this process approximates the Navier-Stokes equations (with viscosity $\nu$)
to order $O(t^{3/2})$. As a result of our estimates we also obtain a local
existence and uniqueness results for the Navier-Stokes equations.
http://arXiv.org/abs/math/0505066
http://front.math.ucdavis.edu/math.AP/0505066
(alternate)
Author(s): Richard F. Bass and Xia Chen and and Jay Rosen
Abstract: Let B_n be the number of self-intersections of a symmetric random walk with
finite second moments in the integer planar lattice. We obtain moderate
deviation estimates for B_n - E B_n and E B_n- B_n, which are given in terms of
the best constant of a certain Gagliardo-Nirenberg inequality. We also prove
the corresponding laws of the iterated logarithm.
http://arXiv.org/abs/math/0506414
http://front.math.ucdavis.edu/math.PR/0506414
(alternate)
Author(s): G. Giacomin (1) and F. L. Toninelli (2) ((1) Universite' de Paris 7 and (2) ENS Lyon, UMR--CNRS 5672)
Abstract: We consider general disordered models of pinning of directed polymers on a
defect line. This class contains in particular the disordered
$(1+1)$--dimensional interface wetting model, a version of the Poland--Scheraga
model of DNA denaturation and other $(1+d)$--dimensional polymers in
interaction with flat interfaces. We consider also the case of copolymers with
adsorption at a selective interface.
Under quite general conditions, these models are known to have a
(de)localization transition at some critical line in the phase diagram. In this
work we prove in particular that, as soon as disorder is present, the
transition is at least of second order, in the sense that the free energy is
differentiable at the critical line, so that the order parameter vanishes
continuously at the transition. On the other hand, it is known that the
corresponding non--disordered models can have a first order (de)localization
transition, with a discontinuous first derivative. Our result shows therefore
that the presence of the disorder has really a smoothening effect on the
transition.
http://arXiv.org/abs/math/0506431
http://front.math.ucdavis.edu/math.PR/0506431
(alternate)
Author(s): Jason Levesley and Cem Salp and Sanju Velani
Abstract: Let $K$ denote the middle third Cantor set and ${\cal A}:= \{3^n : n = 0,1,2,
>... \} $. Given a real, positive function $\psi$ let $ W_{\cal A}(\psi)$
denote the set of real numbers $x$ in the unit interval for which there exist
infinitely many $(p,q) \in \Z \times {\cal A} $ such that $ |x - p/q| < \psi(q)
$. The analogue of the Hausdorff measure version of the Duffin-Schaeffer
conjecture is established for $ W_{\cal A}(\psi) \cap K $. One of the
consequences of this is that there exist very well approximable numbers, other
than Liouville numbers, in $K$ -- an assertion attributed to K. Mahler.
http://arXiv.org/abs/math/0505074
http://front.math.ucdavis.edu/math.NT/0505074
(alternate)
Author(s): C. Landim
Abstract: We prove Gaussian tail estimates for the transition probability of $n$
particles evolving as symmetric exclusion processes on $\bb Z^d$, improving
results obtained in \cite{l}. We derive from this result a non-equilibrium
Boltzmann-Gibbs principle for the symmetric simple exclusion process in
dimension 1 starting from a product measure with slowly varying parameter.
http://arXiv.org/abs/math/0505089
http://front.math.ucdavis.edu/math.PR/0505089
(alternate)
Author(s): Bela Bollobas and Svante Janson and Oliver Riordan
Abstract: We introduce a very general model of an inhomogenous random graph with
independence between the edges, which scales so that the number of edges is
linear in the number of vertices. This scaling corresponds to the p=c/n scaling
for G(n,p) used to study the phase transition; also, it seems to be a property
of many large real-world graphs. Our model includes as special cases many
models previously studied.
We show that under one very weak assumption (that the expected number of
edges is `what it should be'), many properties of the model can be determined,
in particular the critical point of the phase transition, and the size of the
giant component above the transition. We do this by relating our random graphs
to branching processes, which are much easier to analyze.
We also consider other properties of the model, showing, for example, that
when there is a giant component, it is `stable': for a typical random graph, no
matter how we add or delete o(n) edges, the size of the giant component does
not change by more than o(n). We believe that this result is new even for the
classical graph G(n,c/n), in which case the proof is much simpler.
http://arXiv.org/abs/math/0504589
http://front.math.ucdavis.edu/math.PR/0504589
(alternate)
Author(s): C. Landim and J. A. Ramirez and H.-T. Yau
Abstract: It was proved \cite{EMYa, QY} that stochastic lattice gas dynamics converge
to the Navier-Stokes equations in dimension $d=3$ in the incompressible limits.
In particular, the viscosity is finite. We proved that, on the other hand, the
viscosity for a two dimensional lattice gas model diverges faster than $\log
\log t$. Our argument indicates that the correct divergence rate is $(\log
t)^{1/2}$. This problem is closely related to the logarithmic correction of the
time decay rate for the velocity auto-correlation function of a tagged
particle.
http://arXiv.org/abs/math/0505090
http://front.math.ucdavis.edu/math.PR/0505090
(alternate)
Author(s): M. D. Jara and C. Landim
Abstract: We prove a nonequilibirum central limit theorem for the position of a tagged
particle in the one-dimensional nearest-neighbor symmetric simple exclusion
process under diffusive scaling starting from a Bernoulli product measure
associated to a smooth profile $\rho_0:\bb R\to [0,1]$.
http://arXiv.org/abs/math/0505091
http://front.math.ucdavis.edu/math.PR/0505091
(alternate)
Author(s): Claudio Landim and Glauco Valle
Abstract: We study a class of one-dimensional interacting particle systems with random
boundaries as a microscopic model for Stefan's melting and freezing problem. We
prove that under diffusive rescaling these particle systems exhibit a
hydrodynamic behavior described by the solution of a Cauchy-Stefan problem.
http://arXiv.org/abs/math/0505092
http://front.math.ucdavis.edu/math.PR/0505092
(alternate)
Author(s): Patrik L. Ferrari (1) and Herbert Spohn (1) ((1) TU-Muenchen)
Abstract: Investigating the long time asymptotics of the totally asymmetric simple
exclusion process, Sasamoto obtains rather indirectly a formula for the GOE
Tracy-Widom distribution. We establish that his novel formula indeed agrees
with more standard expressions.
http://arXiv.org/abs/math-ph/0505012
http://front.math.ucdavis.edu/math-ph/0505012
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: The paper studies asymptotic behavior of the loss probability for the
$GI/M/m/n$ queueing system as $n$ increases to infinity. The approach of the
paper is based on applications of classic results of Tak\'acs (1967) and the
Tauberian theorem with remainder of Postnikov (1979-1980) associated with the
recurrence relation of convolution type. The main result of the paper is
associated with asymptotic behavior of the loss probability. Specifically it is
shown that in some cases (precisely described in the paper) where the load of
the system approaches 1 from the left and $n$ increases to infinity, the loss
probability of the $GI/M/m/n$ queue becomes asymptotically independent of the
parameter $m$.
http://arXiv.org/abs/math/0505127
http://front.math.ucdavis.edu/math.PR/0505127
(alternate)
Author(s): Mireille Chaleyat-Maurel (PMA and MAP5) and Valentine Genon-Catalot (MAP5)
Abstract: Let us consider a pair signal-observation ((xn,yn),n 0) where the unobserved
signal (xn) is a Markov chain and the observed component is such that, given
the whole sequence (xn), the random variables (yn) are independent and the
conditional distribution of yn only depends on the corresponding state variable
xn. The main problems raised by these observations are the prediction and
filtering of (xn). We introduce sufficient conditions allowing to obtain
computable filters using mixtures of distributions. The filter system may be
finite or infinite dimensional. The method is applied to the case where the
signal xn = Xn is a discrete sampling of a one dimensional diffusion process:
Concrete models are proved to fit in our conditions. Moreover, for these
models, exact likelihood inference based on the observation (y0,...,yn) is
feasable.
http://arXiv.org/abs/math/0505153
http://front.math.ucdavis.edu/math.PR/0505153
(alternate)
Author(s): Davar Khoshnevisan
Abstract: A classical theorem of S. Bochner states that a function
$f:R^n \to C$ is the Fourier transform of a finite Borel measure if and only
if $f$ is positive definite. In 1938, I. Schoenberg found a beautiful converse
to Bochner's theorem.
We present a non-technical derivation of of Schoenberg's theorem that relies
chiefly on the law of large numbers of classical probability theory.
http://arXiv.org/abs/math/0504603
http://front.math.ucdavis.edu/math.PR/0504603
(alternate)
Author(s): Kevin Costello and Terence Tao and Van Vu
Abstract: Let $Q_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal
entries are i.i.d. Bernoulli random variables (which take values 0 and 1 with
probability 1/2). We prove that $Q_n$ is non-singular with probability
$1-O(n^{-1/8+\delta})$ for any fixed $\delta > 0$. The proof uses a quadratic
version of Littlewood-Offord type results concerning the concentration
functions of random variables and can be extended for more general models of
random matrices.
http://arXiv.org/abs/math/0505156
http://front.math.ucdavis.edu/math.PR/0505156
(alternate)
Author(s): Andrew D. Barbour and Alexander V. Gnedin
Abstract: For $S$ a subordinator and $\Pi_n$ an independent Poisson process of
intensity $ne^{-x}, x>0,$ we are interested in the number $K_n$ of gaps in the
range of $S$ that are hit by at least one point of $\Pi_n$. Extending previous
studies in \cite{Bernoulli, GPYI, GPYII} we focus on the case when the tail of
the L{\'e}vy measure of $S$ is slowly varying. We view $K_n$ as the terminal
value of a random process ${\cal K}_n$, and provide an asymptotic analysis of
the fluctuations of ${\cal K}_n$, as $n\to\infty$, for a wide spectrum of
situations.
http://arXiv.org/abs/math/0505171
http://front.math.ucdavis.edu/math.PR/0505171
(alternate)
Author(s): Radoslaw Adamczak
Abstract: We prove logarithmic Sobolev inequalities and concentration results for
convex functions and a class of product random vectors. The results are used to
derive tail and moment inequalities for chaos variables (in spirit of Talagrand
and Arcones, Gine). We also show that the same proof may be used for chaoses
generated by log-concave random variables, recovering results by Lochowski and
present an application to exponential integrability of Rademacher chaos.
http://arXiv.org/abs/math/0505175
http://front.math.ucdavis.edu/math.PR/0505175
(alternate)
Author(s): K.D. Elworthy and A. Truman and H.Z. Zhao
Abstract: Generalised Ito formulae are proved for time dependent functions of
continuous real valued semi-martingales.The conditions involve left space and
time first derivatives, with the left space derivative required to have locally
bounded 2-dimensional variation. In particular a class of functions with
discontinuous first derivative is included. An estimate of Krylov allows
further weakening of these conditions when the semi-martingale is a diffusion.
http://arXiv.org/abs/math/0505195
http://front.math.ucdavis.edu/math.PR/0505195
(alternate)
Author(s): Chunrong Feng and Huaizhong Zhao
Abstract: A generalized It${\hat {\rm o}}$ formula for time dependent functions of
two-dimensional continuous semi-martingales is proved. The formula uses the
local time of each coordinate process of the semi-martingale, left space and
time first derivatives and second derivative $\nabla_1^- \nabla_2^-f$ only
which are assumed to be of locally bounded variation in certain variables, and
stochastic Lebesgue-Stieltjes integrals of two parameters.The two-parameter
integral is defined as a natural generalization of the It${\hat {\rm o}}$
integral and Lebesgue-Stieltjes integral through a type of It${\hat {\rm o}}$
isometry formula.
http://arXiv.org/abs/math/0505196
http://front.math.ucdavis.edu/math.PR/0505196
(alternate)
Author(s): Inder Jeet Taneja
Abstract: There are many information and divergence measures exist in the literature on
information theory and statistics. The most famous among them are
Kullback-Leiber's (1951)relative information and Jeffreys (1946) J-divergence,
Information radius or Jensen difference divergence measure due to Sibson (1969)
also known in the literature. Burbea and Rao (1982) has also found its
applications in the literature. Taneja (1995) studied another kind of
divergence measure based on arithmetic and geometric means. These three
divergence measures bear a good relationship among each other. But there are
another measures arising due to J-divergence, JS-divergence and AG-divergence.
These measures we call here relative divergence measures or non-symmetric
divergence measures. Here our aim is to obtain bounds on symmetric and
non-symmetric divergence measures in terms of relative information of type s
using properties of Csiszar's f-divergence.
http://arXiv.org/abs/math/0505204
http://front.math.ucdavis.edu/math.PR/0505204
(alternate)
Author(s): Jinho Baik
Abstract: In a recent study of large non-null sample covariance matrices, a new
sequence of functions generalizing the GUE Tracy-Widom distribution of random
matrix theory was obtained. This paper derives Painlev\'e formulas of these
functions and use them to prove that they are indeed distribution functions.
Applications of these new distribution functions to last passage percolation,
queues in tandem and totally asymmetric simple exclusion process are also
discussed. As a part of the proof, a representation of orthogonal polynomials
on the unit circle in terms of an operator on a discrete set is presented.
http://arXiv.org/abs/math/0504606
http://front.math.ucdavis.edu/math.PR/0504606
(alternate)
Author(s): Dirk Becherer and Martin Schweizer
Abstract: We use probabilistic methods to study classical solutions for systems of
interacting semilinear parabolic partial differential equations. In a modeling
framework for a financial market with interacting Ito and point processes, such
PDEs are shown to provide a natural description for the solution of hedging and
valuation problems for contingent claims with a recursive payoff structure.
http://arXiv.org/abs/math/0505208
http://front.math.ucdavis.edu/math.PR/0505208
(alternate)
Author(s): Bar Ata and J. M. Harrison and L. A. Shepp
Abstract: A system manager dynamically controls a diffusion process Z that lives in a
finite interval [0,b]. Control takes the form of a negative drift rate \theta
that is chosen from a fixed set A of available values. The controlled process
evolves according to the differential relationship dZ=dX-\theta(Z) dt+dL-dU,
where X is a (0,\sigma) Brownian motion, and L and U are increasing processes
that enforce a lower reflecting barrier at Z=0 and an upper reflecting barrier
at Z=b, respectively. The cumulative cost process increases according to the
differential relationship d\xi =c(\theta(Z)) dt+p dU, where c(\cdot) is a
nondecreasing cost of control and p>0 is a penalty rate associated with
displacement at the upper boundary. The objective is to minimize long-run
average cost. This problem is solved explicitly, which allows one to also solve
the following, essentially equivalent formulation: minimize the long-run
average cost of control subject to an upper bound constraint on the average
rate at which U increases. The two special problem features that allow an
explicit solution are the use of a long-run average cost criterion, as opposed
to a discounted cost criterion, and the lack of state-related costs other than
boundary displacement penalties. The application of this theory to power
control in wireless communication is discussed.
http://arXiv.org/abs/math/0505210
http://front.math.ucdavis.edu/math.PR/0505210
(alternate)
Author(s): Michel Mandjes and Miranda van Uitert
Abstract: This paper considers Gaussian flows multiplexed in a queueing network. A
single node being a useful but often incomplete setting, we examine more
advanced models. We focus on a (two-node) tandem queue, fed by a large number
of Gaussian inputs. With service rates and buffer sizes at both nodes scaled
appropriately, Schilder's sample-path large-deviations theorem can be applied
to calculate the asymptotics of the overflow probability of the second queue.
More specifically, we derive a lower bound on the exponential decay rate of
this overflow probability and present an explicit condition for the lower bound
to match the exact decay rate. Examples show that this condition holds for a
broad range of frequently used Gaussian inputs. The last part of the paper
concentrates on a model for a single node, equipped with a priority scheduling
policy. We show that the analysis of the tandem queue directly carries over to
this priority queueing system.
http://arXiv.org/abs/math/0505214
http://front.math.ucdavis.edu/math.PR/0505214
(alternate)
Author(s): Thomas Mountford and Herve Guiol
Abstract: We prove a strong law of large numbers for the location of the second class
particle in a totally asymmetric exclusion process when the process is started
initially from a decreasing shock. This completes a study initiated in Ferrari
and Kipnis [Ann. Inst. H. Poincare Probab. Statist. 13 (1995) 143-154].
http://arXiv.org/abs/math/0505216
http://front.math.ucdavis.edu/math.PR/0505216
(alternate)
Author(s): Houman Owhadi and Lei Zhang
Abstract: Heterogeneous multi-scale structures can be found everywhere in nature. Can
these structures be accurately simulated at a coarse level? Homogenization
theory allows us to do so under the assumptions of ergodicity and scale
separation by transferring bulk (averaged) information from sub-grid scales to
computational scales. Can we get rid of these assumptions? can we compress a
PDE with arbitrary coefficients? Surprisingly the answer is yes, is rigorous
and based on a new form of compensation. We will consider divergence form
elliptic operators in dimension $n\geq 2$ to introduce this method. Although
solutions of these operators are only H\"{o}lder continuous, we show that their
regularity with respect to Harmonic mappings is $C^{1,\alpha}$. It follows that
these PDEs can be up-scaled by transferring a new metric in addition to
traditional bulk quantities from small scales into coarse scales and error
bounds can be given.
http://arXiv.org/abs/math/0505223
http://front.math.ucdavis.edu/math.NA/0505223
(alternate)
Author(s): P. Collet and D. Duarte and A. Galves
Abstract: We present a new approach to the bootstrap for chains of infinite order
taking values on a finite alphabet. It is based on a sequential Bootstrap
Central Limit Theorem for the sequence of canonical Markov approximations of
the chain of infinite order. Combined with previous results on the rate of
approximation this leads to a Central Limit Theorem for the bootstrapped
estimator of the sample mean which is the main result of this paper.
http://arXiv.org/abs/math/0505232
http://front.math.ucdavis.edu/math.PR/0505232
(alternate)
Author(s): Inder Jeet Taneja
Abstract: There are many information and divergence measures exist in the literature on
information theory and statistics. The most famous among them are
Kullback-Leiber relative information and Jeffreys J-divergence. The measures
like, Bhattacharya distance, Hellinger discrimination, Chi-square divergence,
triangular discrimination and harmonic mean divergence are also famous in the
literature on statistics. In this paper we have obtained bounds on triangular
discrimination and symme | |
