[Pas] Probabilty Abstracts 87b
pas at www.economia.unimi.it
pas at www.economia.unimi.it
Fri Jul 1 22:56:28 CEST 2005
July 1, 2005
Letter 87b
Apologizes for this second mail.
Previous PAS Letter 87 contained abstracts from Letter 86.
stefano iacus
Probability Abstract Service
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3374. FOKKER-PLANCK-KOLMOGOROV EQUATION FOR STOCHASTIC DIFFERENTIAL
EQUATIONS WITH BOUNDARY HITTING RESETS
Julien Bect and Hana Baili and Gilles Fleury
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://front.math.ucdavis.edu/math.PR/0504583
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3375. SKEW CONVOLUTION SEMIGROUPS AND AFFINE MARKOV PROCESSES
D.A. Dawson (Carleton University) and Zenghu Li (Beijing Normal
University)
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://front.math.ucdavis.edu/math.PR/0505444
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3376. A PROBABILISTIC REPRESENTATION FOR THE SOLUTIONS TO SOME NON-
LINEAR PDES USING PRUNED BRANCHING TREES
D. Bloemker and M. Romito and R. Tribe
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://front.math.ucdavis.edu/math.PR/0505449
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3377. A LARGE-DEVIATIONS ANALYSIS OF THE GI/GI/1 SRPT QUEUE
Misja Nuyens and Bert Zwart
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://front.math.ucdavis.edu/math.PR/0505450
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3378. HOW BADLY ARE THE BURHOLDER-DAVIS-GUNDY INEQUALITIES AFFECTED
BY ARBITRARY RANDOM TIMES?
Ashkan Nikeghbali
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://front.math.ucdavis.edu/math.PR/0505483
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3379. THE GHIRLANDA-GUERRA IDENTITIES
Pierluigi Contucci and Cristian Giardina'
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://front.math.ucdavis.edu/math-ph/0505055
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3380. POSITIVE PROCESSES
V.I.Bakhtin
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://front.math.ucdavis.edu/math.DS/0505446
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3381. A LARGE CLOSED QUEUEING NETWORK CONTAINING TWO TYPES OF NODE
AND MULTIPLE CUSTOMER CLASSES: ONE BOTTLENECK STATION
Vyacheslav M. Abramov
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://front.math.ucdavis.edu/math.PR/0505489
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3382. CONCENTRATION FOR INDEPENDENT RANDOM VARIABLES WITH HEAVY TAILS
Franck Barthe (LSProba) and Patrick Cattiaux (MODAL'X and CMAP)
and Cyril Roberto (LAMA)
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://front.math.ucdavis.edu/math.PR/0505492
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3383. A CONTINUOUS-DISCONTINUOUS SECOND-ORDER TRANSITION IN THE
SATISFIABILITY OF RANDOM HORN-SAT FORMULAS
Cristopher Moore and Gabriel Istrate and Demetrios Demopoulos and
and Moshe Y. Vardi
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://front.math.ucdavis.edu/math.PR/0505032
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3384. SINAI'S CONDITION FOR REAL VALUED L\'{E}VY PROCESSES
Victor Rivero (MODAL'X)
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://front.math.ucdavis.edu/math.PR/0505495
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3385. TRANSLATION-INVARIANT GENERALIZED TOPOLOGIES INDUCED BY
PROBABILISTIC NORMS
Bernardo Lafuerza-Guillen and Jose L. Rodriguez
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://front.math.ucdavis.edu/math.GN/0505484
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3386. A CLASS OF REMARKABLE SUBMARTINGALES (I)
Ashkan Nikeghbali
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://front.math.ucdavis.edu/math.PR/0505515
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3387. PERFECTLY RANDOM SAMPLING OF TRUNCATED MULTINORMAL DISTRIBUTIONS
Pedro J. Fernandez and Pablo A. Ferrari and Sebastian Grynberg
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://front.math.ucdavis.edu/math.PR/0505522
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3388. A POINT PROCESS DESCRIBING THE COMPONENT SIZES IN THE CRITICAL
WINDOW OF THE RANDOM GRAPH EVOLUTION
Svante Janson and Joel Spencer
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://front.math.ucdavis.edu/math.PR/0505529
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3389. SPECTRAL GAP ESTIMATES FOR INTERACTING PARTICLE SYSTEMS VIA A
BAKRY & EMERY-TYPE APPROACH
Anne-Severine Boudou and Pietro Caputo and Paolo Dai Pra and
Gustavo Posta
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://front.math.ucdavis.edu/math.PR/0505533
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3390. CONCENTRATION INEQUALITIES ON PRODUCT SPACES WITH APPLICATIONS
TO MARKOV PROCESSES
Gordon Blower and Fran\c{c}ois Bolley (UMPA-ENSL)
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://front.math.ucdavis.edu/math.PR/0505536
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3391. DE BRUIJN COVERING CODES FOR ROOTED HYPERGRAPHS
Joshua N. Cooper and Fan Chung
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://front.math.ucdavis.edu/math.CO/0505528
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3392. RANDOM GROWTH MODELS WITH POLYGONAL SHAPES
Janko Gravner and David Griffeath
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://front.math.ucdavis.edu/math.PR/0505039
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3393. STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY PURELY SPATIAL NOISE
S. V. Lototsky and B. L. Rozovskii
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://front.math.ucdavis.edu/math.PR/0505551
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3394. JACOBIANS AND RANK 1 PERTURBATIONS RELATING TO UNITARY
HESSENBERG MATRICES
Peter J. Forrester and Eric M. Rains
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://front.math.ucdavis.edu/math.PR/0505552
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3395. ON RANDOM MEASURES ON THE SPACE OF TRAJECTORIES AND STRONG AND
WEAK SOLUTIONS OF STOCHASTIC EQUATIONS
A. A. Dorogovtsev
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://front.math.ucdavis.edu/math.PR/0505569
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3396. ASYMPTOTIC BEHAVIOR OF THE NUMBER OF LOST MESSAGES
Vyacheslav M. Abramov
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://front.math.ucdavis.edu/math.PR/0505596
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3397. ASYMPTOTIC ANALYSIS OF THE GI/M/1/N LOSS SYSTEM AS N INCREASES
TO INFINITY
Vyacheslav M. Abramov
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://front.math.ucdavis.edu/math.PR/0505597
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3398. STOCHASTIC GAMES WITH INFINITELY MANY INTERACTING AGENTS
Emilio De Santis and Carlo Marinelli
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://front.math.ucdavis.edu/math.PR/0505608
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3399. THE STABILITY OF JOIN-THE-SHORTEST-QUEUE MODELS WITH GENERAL
INPUT AND OUTPUT PROCESSES
Vyacheslav M. Abramov
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://front.math.ucdavis.edu/math.PR/0505040
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3400. LONG RANGE ACTION IN NETWORKS OF CHAOTIC ELEMENTS
Michael Blank and Leonid Bunimovich
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://front.math.ucdavis.edu/math.DS/0505610
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3401. ANALYSIS OF MULTISERVER RETRIAL QUEUEING SYSTEM: A MARTINGALE
APPROACH AND AN ALGORITHM OF SOLUTION
Vyacheslav M. Abramov
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://front.math.ucdavis.edu/math.PR/0505046
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3402. THE CENTRAL LIMIT PROBLEM FOR RANDOM VECTORS WITH SYMMETRIES
Elizabeth S. Meckes and Mark W. Meckes
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://front.math.ucdavis.edu/math.PR/0505618
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3403. A CLASS OF REMARKABLE SUBMARTINGALES (II): ENLARGEMENTS OF
FILTRATIONS
Ashkan Nikeghbali
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://front.math.ucdavis.edu/math.PR/0505623
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3404. ON THE SPATIAL MEAN OF THE POINCARE CYCLE
Luis Baez-Duarte
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://front.math.ucdavis.edu/math.PR/0505625
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3405. POISSON MICROBALLS: SELF-SIMILARITY AND DIRECTIONAL ANALYSIS
Hermine Bierm\'e and Anne Estrade
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://front.math.ucdavis.edu/math.PR/0505635
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3406. EQUILIBRIUM FLUCTUATIONS FOR A ONE-DIMENSIONAL INTERFACE IN THE
SOLID ON SOLID APPROXIMATION
Gustavo Posta
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://front.math.ucdavis.edu/math.PR/0505643
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3407. INFLUENCE AND SHARP-THRESHOLD THEOREMS FOR MONOTONIC MEASURES
B. T. Graham and G. R. Grimmett
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://front.math.ucdavis.edu/math.PR/0505057
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3408. THE STOCHASTIC ACCELERATION PROBLEM IN TWO DIMENSIONS
T. Komorowski and L. Ryzhik
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://front.math.ucdavis.edu/math-ph/0505083
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3409. ON THE PERIODIC PROPERTIES OF SELF-DECIMATED GENERATORS OF
PSEUDORANDOM NUMBERS
Sergey Agievich and Oleg Solovey
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://front.math.ucdavis.edu/math.CO/0505660
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3410. NEW SCALING OF ITZYKSON-ZUBER INTEGRALS
Benoit Collins and Piotr Sniady
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://front.math.ucdavis.edu/math.PR/0505664
---------------------------------------------------------------
3411. A STABLE MARRIAGE OF POISSON AND LEBESGUE
Christopher Hoffman and Alexander E. Holroyd and Yuval Peres
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://front.math.ucdavis.edu/math.PR/0505668
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3412. ON CONVERGENCE OF IMPORTANCE SAMPLING AND OTHER PROPERLY
WEIGHTED SAMPLES TO THE TARGET DISTRIBUTION
S. Malefaki and G. Iliopoulos
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://front.math.ucdavis.edu/math.ST/0505045
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3413. QUENCHED INVARIANCE PRINCIPLES FOR RANDOM WALKS ON PERCOLATION
CLUSTERS
P. Mathieu and A. L. Piatnitski
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://front.math.ucdavis.edu/math.PR/0505672
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3414. RIGOROUS RESULTS ON THE THRESHOLD NETWORK MODEL
Norio Konno and Naoki Masuda and Rahul Roy and Anish Sarkar
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://front.math.ucdavis.edu/math.PR/0505681
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3415. LOWER DEVIATION PROBABILITIES FOR SUPERCRITICAL GALTON-WATSON
PROCESSES
Klaus Fleischmann and Vitali Wachtel
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://front.math.ucdavis.edu/math.PR/0505683
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3416. DELAY DIFFERENTIAL EQUATIONS DRIVEN BY LEVY PROCESSES:
STATIONARITY AND FELLER PROPERTIES
M. Reiss and M. Riedle and O. van Gaans
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://front.math.ucdavis.edu/math.PR/0505684
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3417. SELF-SIMILAR AND MARKOV COMPOSITION STRUCTURES
Alexander Gnedin and Jim Pitman
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://front.math.ucdavis.edu/math.PR/0505687
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3418. MIXING TIME BOUNDS VIA THE SPECTRAL PROFILE
Sharad Goel and Ravi Montenegro and Prasad Tetali
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://front.math.ucdavis.edu/math.PR/0505690
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3419. RANK INDEPENDENCE AND REARRANGEMENTS OF RANDOM VARIABLES
Alexander Gnedin and Zbigniew Nitecki
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://front.math.ucdavis.edu/math.PR/0505692
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3420. EFFICIENT SPIKE-SORTING OF MULTI-STATE NEURONS USING INTER-
SPIKE INTERVALS INFORMATION
Matthieu Delescluse (LPC) and Christophe Pouzat (LPC)
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://front.math.ucdavis.edu/q-bio.QM/0505053
---------------------------------------------------------------
3421. HYDRODYNAMIC SCALING LIMIT OF CONTINUUM SOLID-ON-SOLID MODEL
Anamaria Savu
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://front.math.ucdavis.edu/math.PR/0506001
---------------------------------------------------------------
3422. ASYMPTOTIC STATISTICAL EQUIVALENCE FOR ERGODIC DIFFUSIONS: THE
MULTIDIMENSIONAL CASE
Arnak Dalalyan (PMA) and Markus Reiss (WIAS)
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://front.math.ucdavis.edu/math.ST/0505053
---------------------------------------------------------------
3423. A CHARACTERIZATION OF MARKOV PROCESSES ENJOYING THE TIME-
INVERSION PROPERTY
Stephan Lawi
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://front.math.ucdavis.edu/math.PR/0506013
---------------------------------------------------------------
3424. CLOSED AND EXACT FUNCTIONS IN THE CONTEXT OF GINZBURG-LANDAU
MODELS
Anamaria Savu
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://front.math.ucdavis.edu/math.FA/0506002
---------------------------------------------------------------
3425. MOMENT INEQUALITIES FOR U-STATISTICS
Radoslaw Adamczak
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://front.math.ucdavis.edu/math.PR/0506026
---------------------------------------------------------------
3426. LOSSES IN M/GI/M/N QUEUES
Vyacheslav M. Abramov
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://front.math.ucdavis.edu/math.PR/0506033
---------------------------------------------------------------
3427. DYNAMICS AND ENDOGENY FOR RECURSIVE PROCESSES ON TREES
Jon Warren
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://front.math.ucdavis.edu/math.PR/0506038
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3428. A UNIFYING CLASS OF SKOROKHOD EMBEDDINGS: CONNECTING THE AZEMA-
YOR AND VALLOIS EMBEDDINGS
A. M. G. Cox and D. G. Hobson
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://front.math.ucdavis.edu/math.PR/0506040
---------------------------------------------------------------
3429. FREE-DIFFERENTIABILITY CONDITIONS ON THE FREE-ENERGY FUNCTION
IMPLYING LARGE DEVIATIONS
Henri Comman
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://front.math.ucdavis.edu/math.PR/0506044
---------------------------------------------------------------
3430. DIFFUSING POLYGONS AND SLE($\KAPPA,\RHO$)
Robert O. Bauer and Roland M. Friedrich
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://front.math.ucdavis.edu/math.PR/0506062
---------------------------------------------------------------
3431. STUDY ON OPTIMAL TIMING OF MARK-TO-MARKET FOR CONTINGENT CREDIT
RISK CONTROL
Jiali Liao and Ted Theodosopoulos
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://front.math.ucdavis.edu/math.PR/0506077
---------------------------------------------------------------
3432. STOCHASTIC FLOWS ASSOCIATED TO COALESCENT PROCESSES III: LIMIT
THEOREMS
Jean Bertoin (PMA) and Jean-Fran\c{c}ois Le Gall (DMA)
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://front.math.ucdavis.edu/math.PR/0506092
---------------------------------------------------------------
3433. TWO NEW MARKOV ORDER ESTIMATORS
Yuval Peres and Paul Shields
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://front.math.ucdavis.edu/math.ST/0506080
---------------------------------------------------------------
3434. DIFFUSION MAPS, SPECTRAL CLUSTERING AND EIGENFUNCTIONS OF
FOKKER-PLANCK OPERATORS
Boaz Nadler and Stephane Lafon and Ronald R. Coifman and Ioannis
G. Kevrekidis
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://front.math.ucdavis.edu/math.NA/0506090
---------------------------------------------------------------
3435. A NOTE ON THE RUIN PROBLEM WITH RISKY INVESTMENTS
David Maher
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://front.math.ucdavis.edu/math.PR/0506127
---------------------------------------------------------------
3436. RATE OF ESCAPE OF THE MIXER CHAIN
Ariel Yadin
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://front.math.ucdavis.edu/math.PR/0506129
---------------------------------------------------------------
3437. CONTINUOUS AND TRACTABLE MODELS FOR THE VARIATION OF
EVOLUTIONARY RATES
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)
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://front.math.ucdavis.edu/math.PR/0506145
---------------------------------------------------------------
3438. QUANTITATIVE NOISE SENSITIVITY AND EXCEPTIONAL TIMES FOR
PERCOLATION
Oded Schramm and Jeffrey E. Steif
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://front.math.ucdavis.edu/math.PR/0504586
---------------------------------------------------------------
3439. A CENTRAL LIMIT THEOREM FOR NON-OVERLAPPING RETURN TIMES
Oliver Johnson
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://front.math.ucdavis.edu/math.PR/0506165
---------------------------------------------------------------
3440. PRECISE ASYMPTOTICS FOR A RANDOM WALKER'S MAXIMUM
Alain Comtet and Satya N. Majumdar
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://front.math.ucdavis.edu/cond-mat/0506195
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3441. NON-COLLIDING SYSTEM OF BROWNIAN PARTICLES AS PFAFFIAN PROCESS
Makoto Katori
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://front.math.ucdavis.edu/math.PR/0506186
---------------------------------------------------------------
3442. INFINITE SYSTEMS OF NON-COLLIDING GENERALIZED MEANDERS AND
RIEMANN-LIOUVILLE DIFFERINTEGRALS
Makoto Katori and Hideki Tanemura
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://front.math.ucdavis.edu/math.PR/0506187
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3443. A VARIATIONAL PRINCIPLE IN THE DUAL PAIR OF REPRODUCING KERNEL
HILBERT SPACES AND AN APPLICATION
Hyun Jae Yoo
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://front.math.ucdavis.edu/math.PR/0506189
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3444. RANDOM CONFORMAL DYNAMICAL SYSTEMS
Bertrand Deroin & Victor Kleptsyn
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://front.math.ucdavis.edu/math.DS/0506204
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3445. CONTINUITY THEOREMS FOR THE $M/M/1/N$ QUEUEING SYSTEM
Vyacheslav M. Abramov
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://front.math.ucdavis.edu/math.PR/0506227
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3446. SINGULARITY POINTS FOR FIRST PASSAGE PERCOLATION
J. E. Yukich and Yu Zhang
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://front.math.ucdavis.edu/math.PR/0506241
---------------------------------------------------------------
3447. HARRIS FAMILY OF DISCRETE DISTRIBUTIONS
E. Sandhya and S. Sherly and and N. Raju
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://front.math.ucdavis.edu/math.ST/0506220
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3448. RECONSTRUCTION AND SUBGAUSSIAN OPERATORS
Shahar Mendelson and Alain Pajor and Nicole Tomczak-Jaegermann
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://front.math.ucdavis.edu/math.FA/0506239
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3449. LARGE-DEVIATIONS/THERMODYNAMIC APPROACH TO PERCOLATION ON THE
COMPLETE GRAPH
Marek Biskup and Lincoln Chayes and S. Alex Smith
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://front.math.ucdavis.edu/math.PR/0506255
---------------------------------------------------------------
3450. STOCHASTIC INEQUALITIES FOR SINGLE-SERVER LOSS QUEUEING SYSTEMS
Vyacheslav M. Abramov
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://front.math.ucdavis.edu/math.PR/0505068
---------------------------------------------------------------
3451. BOUNDS ON NON-SYMMETRIC DIVERGENCE MEASURES IN TERMS OF
SYMMETRIC DIVERGENCE MEASURES
Inder Jeet Taneja
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://front.math.ucdavis.edu/math.PR/0506256
---------------------------------------------------------------
3452. HARMONIC COORDINATES ON FINITELY CONNECTED FRACTAFOLDS
Alexander Teplyaev
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://front.math.ucdavis.edu/math.PR/0506261
---------------------------------------------------------------
3453. PERCOLATION, BOUNDARY, NOISE: AN EXPERIMENT
Boris Tsirelson
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://front.math.ucdavis.edu/math.PR/0506269
---------------------------------------------------------------
3454. STATISTICS OF EXTREME SPACINGS IN DETERMINANTAL RANDOM POINT
PROCESSES
Alexander Soshnikov
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://front.math.ucdavis.edu/math.PR/0506286
---------------------------------------------------------------
3455. RENORMALIZATION ANALYSIS OF CATALYTIC WRIGHT-FISHER DIFFUSIONS
K. Fleischmann and J. M. Swart
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://front.math.ucdavis.edu/math.PR/0506311
---------------------------------------------------------------
3456. P\'{E}NALISATIONS OF WALSH'S BROWNIAN MOTION
Joseph Najnudel (PMA)
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://front.math.ucdavis.edu/math.PR/0506329
---------------------------------------------------------------
3457. ON THE SCALING LIMIT OF SIMPLE RANDOM WALK EXCURSION MEASURE IN
THE PLANE
Michael J. Kozdron (University of Regina)
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://front.math.ucdavis.edu/math.PR/0506337
---------------------------------------------------------------
3458. LIMITING SEARCH COST DISTRIBUTION FOR THE MOVE-TO-FRONT RULE
WITH RANDOM REQUEST PROBABILITIES
Javiera Barrera (MAP5) and Thierry Huillet (LPTM) and Christian
Paroissin (LMA - PAU)
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://front.math.ucdavis.edu/math.PR/0506343
---------------------------------------------------------------
3459. FORBIDDEN GAP ARGUMENT FOR PHASE TRANSITIONS PROVED BY MEANS
OF CHESSBOARD ESTIMATES
Marek Biskup and Roman Kotecky
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://front.math.ucdavis.edu/math-ph/0505011
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3460. A CLASS OF REMARKABLE SUBMARTINGALES (III): MULTIPLICATIVE
DECOMPOSITIONS AND FREQUENCY OF VANISHING OF NONNEGATIVE SUBMARTINGALES
Ashkan Nikeghbali
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://front.math.ucdavis.edu/math.PR/0506369
---------------------------------------------------------------
3461. ORIENTED PERCOLATION IN ONE-DIMENSIONAL BETA |X-Y|^2, BETA > 1
RANDOM-CLUSTER MODEL
D. H. U. Marchetti and V. Sidoravicius and M. E. Vares
We consider the one-dimensional long-range Fortuin--Kasteleyn random-
cluster
model, generated by the edge occupation probabilities p_{<x,y>} = 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://front.math.ucdavis.edu/math.PR/0506404
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3462. FAST COMPUTATION OF THE EXPECTED LOSS OF A LOAN PORTFOLIO
TRANCHE IN THE GAUSSIAN FACTOR MODEL: USING HERMITE EXPANSIONS FOR
HIGHER ACCURACY
P.Okunev
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://front.math.ucdavis.edu/math.ST/0506378
---------------------------------------------------------------
3463. A STOCHASTIC PERTURBATION OF INVISCID FLOWS
Gautam Iyer
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://front.math.ucdavis.edu/math.AP/0505066
---------------------------------------------------------------
3464. MODERATE DEVIATIONS AND LAWS OF THE ITERATED LOGARITHM FOR THE
RENORMALIZED SELF-INTERSECTION LOCAL TIMES OF PLANAR RANDOM WALKS
Richard F. Bass and Xia Chen and and Jay Rosen
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://front.math.ucdavis.edu/math.PR/0506414
---------------------------------------------------------------
3465. SMOOTHENING EFFECT OF QUENCHED DISORDER ON POLYMER DEPINNING
TRANSITIONS
G. Giacomin (1) and F. L. Toninelli (2) ((1) Universite' de Paris 7
and (2) ENS Lyon, UMR--CNRS 5672)
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://front.math.ucdavis.edu/math.PR/0506431
---------------------------------------------------------------
3466. ON A PROBLEM OF K. MAHLER: DIOPHANTINE APPROXIMATION AND CANTOR
SETS
Jason Levesley and Cem Salp and Sanju Velani
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://front.math.ucdavis.edu/math.NT/0505074
---------------------------------------------------------------
3467. GAUSSIAN ESTIMATES FOR SYMMETRIC SIMPLE EXCLUSION PROCESSES
C. Landim
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://front.math.ucdavis.edu/math.PR/0505089
---------------------------------------------------------------
3468. THE PHASE TRANSITION IN INHOMOGENEOUS RANDOM GRAPHS
Bela Bollobas and Svante Janson and Oliver Riordan
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://front.math.ucdavis.edu/math.PR/0504589
---------------------------------------------------------------
3469. SUPERDIFFUSIVITY OF TWO DIMENSIONAL LATTICE GAS MODELS
C. Landim and J. A. Ramirez and H.-T. Yau
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://front.math.ucdavis.edu/math.PR/0505090
---------------------------------------------------------------
3470. NONEQUILIBRIUM CENTRAL LIMIT THEOREM FOR A TAGGED PARTICLE IN
SYMMETRIC SIMPLE EXCLUSION
M. D. Jara and C. Landim
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://front.math.ucdavis.edu/math.PR/0505091
---------------------------------------------------------------
3471. A MICROSCOPIC MODEL FOR STEFAN'S MELTING AND FREEZING PROBLEM
Claudio Landim and Glauco Valle
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://front.math.ucdavis.edu/math.PR/0505092
---------------------------------------------------------------
3472. A DETERMINANTAL FORMULA FOR THE GOE TRACY-WIDOM DISTRIBUTION
Patrik L. Ferrari (1) and Herbert Spohn (1) ((1) TU-Muenchen)
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://front.math.ucdavis.edu/math-ph/0505012
---------------------------------------------------------------
3473. ASYMPTOTIC ANALYSIS OF LOSSES IN THE $GI/M/M/N$ QUEUEING SYSTEM
AS $N$ INCREASES TO INFINITY
Vyacheslav M. Abramov
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://front.math.ucdavis.edu/math.PR/0505127
---------------------------------------------------------------
3474. COMPUTABLE INFINITE DIMENSIONAL FILTERS WITH APPLICATIONS TO
DISCRETIZED DIFFUSION PROCESSES
Mireille Chaleyat-Maurel (PMA and MAP5) and Valentine Genon-Catalot
(MAP5)
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://front.math.ucdavis.edu/math.PR/0505153
---------------------------------------------------------------
3475. SCHOENBERG'S THEOREM VIA THE LAW OF LARGE NUMBERS
Davar Khoshnevisan
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://front.math.ucdavis.edu/math.PR/0504603
---------------------------------------------------------------
3476. RANDOM SYMMETRIC MATRICES ARE ALMOST SURELY NON-SINGULAR
Kevin Costello and Terence Tao and Van Vu
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://front.math.ucdavis.edu/math.PR/0505156
---------------------------------------------------------------
3477. REGENERATIVE COMPOSITIONS IN THE CASE OF SLOW VARIATION
Andrew D. Barbour and Alexander V. Gnedin
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://front.math.ucdavis.edu/math.PR/0505171
---------------------------------------------------------------
3478. LOGARITHMIC SOBOLEV INEQUALITIES AND CONCENTRATION OF MEASURE
FOR CONVEX FUNCTIONS AND POLYNOMIAL CHAOSES
Radoslaw Adamczak
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://front.math.ucdavis.edu/math.PR/0505175
---------------------------------------------------------------
3479. GENERALIZED ITO FORMULAE AND SPACE-TIME LEBESGUE-STIELTJES
INTEGRALS OF LOCAL TIMES
K.D. Elworthy and A. Truman and H.Z. Zhao
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://front.math.ucdavis.edu/math.PR/0505195
---------------------------------------------------------------
3480. A GENERALIZED IT$\HAT {\RM O}$'S FORMULA IN TWO-DIMENSIONS AND
STOCHASTIC LEBESGUE-STIELTJES INTEGRALS
Chunrong Feng and Huaizhong Zhao
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://front.math.ucdavis.edu/math.PR/0505196
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3481. RELATIVE DIVERGENCE MEASURES AND INFORMATION INEQUALITIES
Inder Jeet Taneja
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://front.math.ucdavis.edu/math.PR/0505204
---------------------------------------------------------------
3482. PAINLEVE FORMULAS OF THE LIMITING DISTRIBUTIONS FOR NON-NULL
COMPLEX SAMPLE COVARIANCE MATRICES
Jinho Baik
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://front.math.ucdavis.edu/math.PR/0504606
---------------------------------------------------------------
3483. CLASSICAL SOLUTIONS TO REACTION-DIFFUSION SYSTEMS FOR HEDGING
PROBLEMS WITH INTERACTING ITO AND POINT PROCESSES
Dirk Becherer and Martin Schweizer
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://front.math.ucdavis.edu/math.PR/0505208
---------------------------------------------------------------
3484. DRIFT RATE CONTROL OF A BROWNIAN PROCESSING SYSTEM
Bar Ata and J. M. Harrison and L. A. Shepp
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://front.math.ucdavis.edu/math.PR/0505210
---------------------------------------------------------------
3485. SAMPLE-PATH LARGE DEVIATIONS FOR TANDEM AND PRIORITY QUEUES
WITH GAUSSIAN INPUTS
Michel Mandjes and Miranda van Uitert
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://front.math.ucdavis.edu/math.PR/0505214
---------------------------------------------------------------
3486. THE MOTION OF A SECOND CLASS PARTICLE FOR THE TASEP STARTING
FROM A DECREASING SHOCK PROFILE
Thomas Mountford and Herve Guiol
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://front.math.ucdavis.edu/math.PR/0505216
---------------------------------------------------------------
3487. METRIC BASED UP-SCALING
Houman Owhadi and Lei Zhang
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://front.math.ucdavis.edu/math.NA/0505223
---------------------------------------------------------------
3488. BOOTSTRAP CENTRAL LIMIT THEOREM FOR CHAINS OF INFINITE ORDER
VIA MARKOV APPROXIMATIONS
P. Collet and D. Duarte and A. Galves
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://front.math.ucdavis.edu/math.PR/0505232
---------------------------------------------------------------
3489. BOUNDS ON TRIANGULAR DISCRIMINATION, HARMONIC MEAN AND
SYMMETRIC CHI-SQUARE DIVERGENCES
Inder Jeet Taneja
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 symmetric chi-square divergence in terms of relative
information of type s using Csiszar's f-divergence. A relationship among
triangular discrimination and harmonic mean divergence is also given.
http://front.math.ucdavis.edu/math.PR/0505238
---------------------------------------------------------------
3490. ASYMPTOTIC BEHAVIOR OF A METAPOPULATION MODEL
A. D. Barbour and A. Pugliese
We study the behavior of an infinite system of ordinary differential
equations modeling the dynamics of a metapopulation, a set of (discrete)
populations subject to local catastrophes and connected via migration
under a
mean field rule; the local population dynamics follow a generalized
logistic
law. We find a threshold below which all the solutions tend to total
extinction
of the metapopulation, which is then the only equilibrium; above the
threshold,
there exists a unique equilibrium with positive population, which,
under an
additional assumption, is globally attractive. The proofs employ
tools from the
theories of Markov processes and of dynamical systems.
http://front.math.ucdavis.edu/math.PR/0505240
---------------------------------------------------------------
3491. ON THE CONVERGENCE FROM DISCRETE TO CONTINUOUS TIME IN AN
OPTIMAL STOPPING PROBLEM
Paul Dupuis and Hui Wang
We consider the problem of optimal stopping for a one-dimensional
diffusion
process. Two classes of admissible stopping times are considered. The
first
class consists of all nonanticipating stopping times that take values in
[0,\infty], while the second class further restricts the set of
allowed values
to the discrete grid {nh:n=0,1,2,...,\infty} for some parameter h>0.
The value
functions for the two problems are denoted by V(x) and V^h(x),
respectively. We
identify the rate of convergence of V^h(x) to V(x) and the rate of
convergence
of the stopping regions, and provide simple formulas for the rate
coefficients.
http://front.math.ucdavis.edu/math.PR/0505241
---------------------------------------------------------------
3492. EXCHANGEABLE, GIBBS AND EQUILIBRIUM MEASURES FOR MARKOV SUBSHIFTS
Jon. Aaronson and Hitoshi Nakada
We study a class of strongly irreducible, multidimensional, topological
Markov shifts, comparing two notions of "symmetric measure":
exchangeability
and the Gibbs property. We show that equilibrium measures for such
shifts
(unique and weak Bernoulli in the one dimensional case) exhibit a
variety of
spectral properties.
http://front.math.ucdavis.edu/math.PR/0505011
---------------------------------------------------------------
3493. ON UTILITY MAXIMIZATION IN DISCRETE-TIME FINANCIAL MARKET MODELS
Miklos Rasonyi and Lukasz Stettner
We consider a discrete-time financial market model with finite time
horizon
and give conditions which guarantee the existence of an optimal
strategy for
the problem of maximizing expected terminal utility. Equivalent
martingale
measures are constructed using optimal strategies.
http://front.math.ucdavis.edu/math.PR/0505243
---------------------------------------------------------------
3494. ACCELERATING DIFFUSIONS
Chii-Ruey Hwang and Shu-Yin Hwang-Ma and Shuenn-Jyi Sheu
Let U be a given function defined on R^d and \pi(x) be a density
function
proportional to \exp -U(x). The following diffusion X(t) is often
used to
sample from \pi(x), dX(t)=-\nabla U(X(t)) dt+\sqrt2 dW(t),\qquad X(0)
=x_0. To
accelerate the convergence, a family of diffusions with \pi(x) as
their common
equilibrium is considered, dX(t)=\bigl(-\nabla U(X(t))+C(X(t))\bigr)
dt+\sqrt2
dW(t),\qquad X(0)=x_0. Let L_C be the corresponding infinitesimal
generator.
The spectral gap of L_C in L^2(\pi) (\lambda (C)), and the
convergence exponent
of X(t) to \pi in variational norm (\rho(C)), are used to describe the
convergence rate, where \lambda(C)= Sup{real part of \mu\dvtx\mu is
in the
spectrum of L_C, \mu is not zero}, {-2.8cm}\rho(C) = Inf\biggl{\rho
\dvtx\int |
p(t,x,y) -\pi(y)| dy \le g(x) e^{\rho t}\biggr}.Roughly speaking, L_C
is a
perturbation of the self-adjoint L_0 by an antisymmetric operator C
\cdot\nabla,
where C is weighted divergence free. We prove that \lambda (C)\le
\lambda (0)
and equality holds only in some rare situations. Furthermore, \rho(C)\le
\lambda (C) and equality holds for C=0. In other words, adding an
extra drift,
C(x), accelerates convergence. Related problems are also discussed.
http://front.math.ucdavis.edu/math.PR/0505245
---------------------------------------------------------------
3495. CRAMER'S ESTIMATE FOR A REFLECTED LEVY PROCESS
R. A. Doney and R. A. Maller
The natural analogue for a Levy process of Cramer's estimate for a
reflected
random walk is a statement about the exponential rate of decay of the
tail of
the characteristic measure of the height of an excursion above the
minimum. We
establish this estimate for any Levy process with finite negative
mean which
satisfies Cramer's condition, and give an explicit formula for the
limiting
constant. Just as in the random walk case, this leads to a Poisson limit
theorem for the number of ``high excursions.''
http://front.math.ucdavis.edu/math.PR/0505246
---------------------------------------------------------------
3496. SUMMATION TEST FOR GAP PENALTIES AND STRONG LAW OF THE LOCAL
ALIGNMENT SCORE
Hock Peng Chan
A summation test is proposed to determine admissible types of gap
penalties
for logarithmic growth of the local alignment score. We also define a
converging sequence of log moment generating functions that provide the
constants associated with the large deviation rate and logarithmic
strong law
of the local alignment score and the asymptotic number of matches in the
optimal local alignment.
http://front.math.ucdavis.edu/math.PR/0505247
---------------------------------------------------------------
3497. THE BRANCHING PROCESS WITH LOGISTIC GROWTH
Amaury Lambert
In order to model random density-dependence in population dynamics, we
construct the random analogue of the well-known logistic process in the
branching process' framework. This density-dependence corresponds to
intraspecific competition pressure, which is ubiquitous in ecology, and
translates mathematically into a quadratic death rate. The logistic
branching
process, or LB-process, can thus be seen as (the mass of) a
fragmentation
process (corresponding to the branching mechanism) combined with
constant
coagulation rate (the death rate is proportional to the number of
possible
coalescing pairs). In the continuous state-space setting, the LB-
process is a
time-changed (in Lamperti's fashion) Ornstein-Uhlenbeck type process.
We obtain
similar results for both constructions: when natural deaths do not
occur, the
LB-process converges to a specified distribution; otherwise, it goes
extinct
a.s. In the latter case, we provide the expectation and the Laplace
transform
of the absorption time, as a functional of the solution of a Riccati
differential equation. We also show that the quadratic regulatory
term allows
the LB-process to start at infinity, despite the fact that births occur
infinitely often as the initial state goes to \infty. This result can
be viewed
as an extension of the pure-death process starting from infinity
associated to
Kingman's coalescent, when some independent fragmentation is added.
http://front.math.ucdavis.edu/math.PR/0505249
---------------------------------------------------------------
3498. THE OSCILLATORY DISTRIBUTION OF DISTANCES IN RANDOM TRIES
Costas A. Christophi and Hosam M. Mahmoud
We investigate \Delta_n, the distance between randomly selected pairs of
nodes among n keys in a random trie, which is a kind of digital tree.
Analytical techniques, such as the Mellin transform and an excursion
between
poissonization and depoissonization, capture small fluctuations in
the mean and
variance of these random distances. The mean increases
logarithmically in the
number of keys, but curiously enough the variance remains O(1), as n
\to\infty.
It is demonstrated that the centered random variable
\Delta_n^*=\Delta_n-\lfloor2\log_2n\rfloor does not have a limit
distribution,
but rather oscillates between two distributions.
http://front.math.ucdavis.edu/math.PR/0505259
---------------------------------------------------------------
3499. SUBGEOMETRIC ERGODICITY OF STRONG MARKOV PROCESSES
G. Fort and G. O. Roberts
We derive sufficient conditions for subgeometric f-ergodicity of
strongly
Markovian processes. We first propose a criterion based on modulated
moment of
some delayed return-time to a petite set. We then formulate a
criterion for
polynomial f-ergodicity in terms of a drift condition on the generator.
Applications to specific processes are considered, including Langevin
tempered
diffusions on R^n and storage models.
http://front.math.ucdavis.edu/math.PR/0505260
---------------------------------------------------------------
3500. ASYMPTOTIC RESULTS ON THE MOMENTS OF THE RATIO OF THE RANDOM
SUM OF SQUARES TO THE SQUARE OF THE RANDOM SUM
S.A. Ladoucette
Let \{X_1, X_2, ...\} be a sequence of positive independent and
identically
distributed random variables of Pareto-type with index \alpha>0 and
let \{N(t);
t\geq 0\} be a mixed Poisson process independent of the X_i's. For t
\geq 0,
define T_{N(t)}:=\frac{X_1^2 + X_2^2 + ... + X_{N(t)}^2} {(X_1 + X_2
+ ... +
X_{N(t)})^2} if N(t)\geq 1 and T_{N(t)}:=0 otherwise.
We derive the limiting behavior of the k-th moment of T_{N(t)},
k\in\mathbb{N}, by using the theory of functions of regular variation
and an
integral representation for \mathbb{E}\{T_{N(t)}^k\}. We also point
out the
connection between T_{N(t)} and the sample coefficient of variation
which is a
popular risk measure in practical applications.
http://front.math.ucdavis.edu/math.PR/0505265
---------------------------------------------------------------
3501. A CAPTURE PROBLEM IN BROWNIAN MOTION AND EIGENVALUES OF
SPHERICAL DOMAINS
Jesse Ratzkin and Andrejs Treibergs
We resolve a question of Bramson and Griffeath by showing that the
expected
capture time of four independent Brownian predators pursuing one
Brownian prey
on a line is finite. Our main tool is an eigenvalue estimate for a
particular
spherical domain, which we obtain by a coning construction and domain
perturbation.
http://front.math.ucdavis.edu/math.PR/0505274
---------------------------------------------------------------
3502. ITERATED BROWNIAN MOTION IN BOUNDED DOMAINS IN R^N
Erkan Nane
Let $\tau_{D}(Z) $ is the first exit time of iterated Brownian motion
from a
domain $D \subset \RR{R}^{n}$ started at $z\in D$ and let $P_{z}[\tau_
{D}(Z)
>t]$ be its distribution. In this paper we establish the exact
asymptotics of
$P_{z}[\tau_{D}(Z) >t]$ over bounded domains as an extension of the
result in
DeBlassie \cite{deblassie}, for $z\in D$ $$ P_{z}[\tau_{D}(Z)>t]
\approx t^{1/2}
\exp(-{3/2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}), as t\to\infty . $$ We
also study
asymptotics of the life time of Brownian-time Brownian motion (BTBM),
$Z^{1}_{t}=z+X(Y(t))$, where $X_{t}$ and $Y_{t}$ are independent
one-dimensional Brownian motions.
http://front.math.ucdavis.edu/math.PR/0505026
---------------------------------------------------------------
3503. SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOND-PERCOLATION GRAPHS
Werner Kirsch and Peter M\"uller
Bond-percolation graphs are random subgraphs of the d-dimensional
integer
lattice generated by a standard bond-percolation process. The
associated graph
Laplacians, subject to Dirichlet or Neumann conditions at cluster
boundaries,
represent bounded, self-adjoint, ergodic random operators with off-
diagonal
disorder. They possess almost surely the non-random spectrum [0,4d]
and a
self-averaging integrated density of states. The integrated density
of states
is shown to exhibit Lifshits tails at both spectral edges in the
non-percolating phase. While the characteristic exponent of the
Lifshits tail
for the Dirichlet (Neumann) Laplacian at the lower (upper) spectral
edge equals
d/2, and thus depends on the spatial dimension, this is not the case
at the
upper (lower) spectral edge, where the exponent equals 1/2.
http://front.math.ucdavis.edu/math-ph/0407047
---------------------------------------------------------------
3504. A NOTE ON MULTITYPE BRANCHING PROCESSES WITH IMMIGRATION IN A
RANDOM ENVIRONMENT
Alexander Roitershtein
We consider a multitype branching process with immigration in a random
environment introduced by Key in [12]. It was shown by Key that the
branching
process is subcritical in the sense that it converges to a proper
limit law. We
complement this result by a strong law of large numbers and a central
limit
theorem for the partial sums of the process. In addition, we study the
asymptotic behavior of oscillations of the branching process, i.e. of
the
random segments between successive times when the extinction occurs
and the
process starts afresh with the next wave of the immigration.
http://front.math.ucdavis.edu/math.PR/0505292
---------------------------------------------------------------
3505. ESTIMATES OF MOMENTS AND TAILS OF GAUSSIAN CHAOSES
Rafal Latala
We derive two sided estimates on moments and tails of homogenous
Gaussian
chaoses of any order. Estimates are exact up to constants depending
only on the
order of chaoses.
http://front.math.ucdavis.edu/math.PR/0505313
---------------------------------------------------------------
3506. NON STOPPING TIMES AND STOPPING THEOREMS
Ashkan Nikeghbali
Given a random time, we characterize the set of martingales for which
the
stopping theorems still hold. We also investigate how the stopping
theorems are
modified when we consider arbitrary random times. To this end, we
introduce
some families of martingales with remarkable properties. We also
investigate,
in the Brownian setting, the relationships between a given random
time and the
underlying Brownian Motion in the progressively enlarged filtration with
respect to this random time.
http://front.math.ucdavis.edu/math.PR/0505316
---------------------------------------------------------------
3507. LIMITING BEHAVIOR OF A DIFFUSION IN AN ASYMPTOTICALLY STABLE
ENVIRONMENT
Arvind Singh (PMA)
Let $V$ be a two sided random walk and let $X$ denote a real valued
diffusion
process with generator ${1/2}e^{V([x])}\frac{d}{dx}(e^{-V([x])}\frac
{d}{dx})$.
This process is known to be the continuous equivalent of the one
dimensional
random walk in random environment with potential $V$. Hu and Shi (1997)
described the L\'evy classes of $X$ in the case where $V$ behaves
approximately
like a Brownian motion. In this paper, based on some fine results on the
fluctuations of random walks and stable processes, we obtain an
accurate image
of the almost sure limiting behavior of $X$ when $V$ behaves
asymptotically
like a stable process. These results also apply for the corresponding
random
walk in random environment.
http://front.math.ucdavis.edu/math.PR/0505332
---------------------------------------------------------------
3508. THE EFFICIENT EVALUATION OF THE HYPERGEOMETRIC FUNCTION OF A
MATRIX ARGUMENT
Plamen Koev and Alan Edelman
We present new algorithms that efficiently approximate the
hypergeometric
function of a matrix argument through its expansion as a series of Jack
functions. Our algorithms exploit the combinatorial properties of the
Jack
function, and have complexity that is only linear in the size of the
matrix.
http://front.math.ucdavis.edu/math.PR/0505344
---------------------------------------------------------------
3509. DETERMINANTAL POINT PROCESSES AND FERMIONIC FOCK SPACE
Neretin Yurii A
We construct a canonical embedding of the space $L^2$ over a
determinantal
point process to the fermionic Fock space. Equivalently, we show that a
determinantal process is the spectral measure for some explicit
commutative
group of Gaussian operators in the fermionic Fock space.
http://front.math.ucdavis.edu/math-ph/0505041
---------------------------------------------------------------
3510. ON THE BEST CONSTANTS IN SOME NON-COMMUTATIVE MARTINGALE
INEQUALITIES
Marius Junge and Quanhua Xu
We determine the optimal orders for the best constants in the non-
commutative
Burkholder-Gundy, Doob and Stein inequalities obtained recently in the
non-commutative martingale theory.
http://front.math.ucdavis.edu/math.OA/0505309
---------------------------------------------------------------
3511. SLE COORDINATE CHANGES
Oded Schramm and David B. Wilson
The purpose of this note is to describe a framework which unifies
radial,
chordal and dipolar SLE. When the definition of SLE(\kappa;\rho) is
extended to
the setting where the force points can be in the interior of the
domain, radial
SLE(\kappa) becomes chordal SLE(\kappa;\rho), with \rho=\kappa-6, and
vice
versa. We also write down the martingales describing the Radon-Nykodim
derivative of SLE(\kappa;\rho_1,...,\rho_n) with respect to SLE(\kappa).
http://front.math.ucdavis.edu/math.PR/0505368
---------------------------------------------------------------
3512. A RESOLUTION OF QUANTUM DYNAMICAL SEMIGROUPS
Anilesh Mohari
We consider a class of quantum dissipative systems governed by a one
parameter completely positive maps on a von-Neumann algebra. We
introduce a
notion of recurrent and metastable projections for the dynamics and
prove that
the unit operator can be decomposed into orthogonal projections where
each
projections are recurrent or metastable for the dynamics.
http://front.math.ucdavis.edu/math.OA/0505384
---------------------------------------------------------------
3513. A NETWORK ANALYSIS OF COMMITTEES IN THE UNITED STATES HOUSE OF
REPRESENTATIVES
Mason A. Porter and Peter J. Mucha and M.E.J. Newman and and Casey
M. Warmbrand
Network theory provides a powerful tool for the representation and
analysis
of complex systems of interacting agents. Here we investigate the
United States
House of Representatives network of committees and subcommittees, with
committees connected according to ``interlocks'' or common
membership. Analysis
of this network reveals clearly the strong links between different
committees,
as well as the intrinsic hierarchical structure within the House as a
whole. We
show that network theory, combined with the analysis of roll call
votes using
singular value decomposition, successfully uncovers political and
organizational correlations between committees in the House without
the need to
incorporate other political information.
http://front.math.ucdavis.edu/nlin.AO/0505043
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3514. SOME RANDOM TIMES AND MARTINGALES ASSOCIATED WITH $BES_{0}
(\DELTA)$ PROCESSES $(0<\DELTA<2)$
Ashkan Nikeghbali
In this paper, we study Bessel processes of dimension $\delta\equiv2
(1-\mu)$,
with $0<\delta<2$, and some related martingales and random times. Our
approach
is based on martingale techniques and the general theory of stochastic
processes (unlike the usual approach based on excursion theory),
although for
$0<\delta<1$, these processes are even not semimartingales. The last
time
before 1 when a Bessel process hits 0, called $g_{\mu}$, plays a key
role in
our study: we characterize its conditional distribution and extend Paul
L\'{e}vy's arc sine law and a related result of Jeulin about the
standard
Brownian Motion. We also introduce some remarkable families of
martingales
related to the Bessel process, thus obtaining in some cases a one
parameter
extension of some results of Az\'{e}ma and Yor in the Brownian setting:
martingales which have the same set of zeros as the Bessel process
and which
satisfy the stopping theorem for $g_{\mu}$, a one parameter extension of
Az\'{e}ma's second martingale, etc. Throughout our study, the local
time of the
Bessel process also plays a central role and we shall establish some
of its
elementary properties.
http://front.math.ucdavis.edu/math.PR/0505423
---------------------------------------------------------------
3515. ON A FAST, ROBUST ESTIMATOR OF THE MODE: COMPARISONS TO OTHER
ROBUST ESTIMATORS WITH APPLICATIONS
David R. Bickel and Rudolf Fruehwirth
Advances in computing power enable more widespread use of the mode,
which is
a natural measure of central tendency since, as the most probable
value, it is
not influenced by the tails in the distribution. The properties of the
half-sample mode, which is a simple and fast estimator of the mode of a
continuous distribution, are studied. The half-sample mode is less
sensitive to
outliers than most other estimators of location, including many other
low-bias
estimators of the mode. Its breakdown point is one half, equal to
that of the
median. However, because of its finite rejection point, the half-
sample mode is
much less sensitive to outliers that are all either greater or less
than the
other values of the sample. This is confirmed by applying the mode
estimator
and the median to samples drawn from normal, lognormal, and Pareto
distributions contaminated by outliers. It is also shown that the
half-sample
mode, in combination with a robust scale estimator, is a highly
robust starting
point for iterative robust location estimators such as Huber's M-
estimator. The
half-sample mode can easily be generalized to modal intervals
containing more
or less than half of the sample. An application of such an estimator
to the
finding of collision points in high-energy proton-proton interactions is
presented.
http://front.math.ucdavis.edu/math.ST/0505419
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