[PAS] Probability Abstracts 102
Probability Abstract Service
pas at lists.imstat.org
Tue Mar 4 05:05:32 CST 2008
Probability Abstracts 102
This document contains abstracts 6511-6752 from
January-1-2008 to February-29-2008.
They have been mailed on March 4th, 2008.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_102.shtml
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6511. SOME EXAMPLES OF ABSOLUTE CONTINUITY OF MEASURES IN STOCHASTIC
FLUID DYNAMICS
B. Ferrario
A non linear Ito equation in a Hilbert space is studied by means of
Girsanov
theorem. We consider a non linearity of polynomial growth in suitable
norms,
including that of quadratic type which appears in the Kuramoto-
Sivashinsky
equation and in the Navier-Stokes equation. We prove that Girsanov
theorem
holds for the 1-dimensional stochastic Kuramoto-Sivashinsky equation
and for a
modification of the 2- and 3-dimensional stochastic Navier-Stokes
equation. In
this way, we prove existence and uniqueness of solutions for these
stochastic
equations. Moreover, the asymptotic behaviour for large time is
characterized.
http://arxiv.org/abs/0801.0496
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6512. MAXIMUM AND ENTROPIC REPULSION FOR A GAUSSIAN MEMBRANE MODEL IN
THE CRITICAL DIMENSION
Noemi Kurt
We consider the real-valued centered Gaussian field on the four-
dimensional
integer lattice, whose covariance matrix is given by the Green's
function of
the discrete Bilaplacian. This is interpreted as a model for a
semiflexible
membrane. $d=4$ is the critical dimension for this model. We discuss
the effect
of a hard wall on the membrane, via a multiscale analysis of the
maximum of the
field. We use analytic and probabilistic tools to describe the
correlation
structure of the field.
http://arxiv.org/abs/0801.0551
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6513. RANDOM TURN WALK ON A HALF LINE WITH CREATION OF PARTICLES AT
THE ORIGIN
J.W. van de Leur and A. Yu. Orlov
We consider a version of random motion of hard core particles on the
semi-lattice $ 1, 2, 3,...$, where in each time instant one of three
possible
events occurs, viz., (a) a randomly chosen particle hops to a free
neighboring
site, (b) a particle is created at the origin (namely, at site 1)
provided that
site 1 is free and (c) a particle is eliminated at the origin
(provided that
the site 1 is occupied). Relations to the BKP equation are explained.
Namely,
the tau functions of two different BKP hierarchies provide generating
functions
respectively (I) for transition weights between different particle
configurations and (II) for an important object: a normalization
function which
plays the role of the statistical sum for our non-equilibrium system.
As an
example we study a model where the hopping rate depends on two
parameters ($r$
and $\beta$). For time $\time\to\infty$ we obtain the asymptotic
configuration
of particles obtained from the initial empty state (the state without
particles) and find an analog of the first order transition at $
\beta=1$.
http://arxiv.org/abs/0801.0066
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6514. SOME REMARKS ON TANGENT MARTINGALE DIFFERENCE SEQUENCES IN $L^1$-
SPACES
Sonja Cox and Mark Veraar
Let X be a Banach space. Suppose that for all $p\in (1, \infty)$ a
constant
$C_{p,X}$ depending only on X and p exists such that for any two X-
valued
martingales f and g with tangent martingale difference sequences one has
\[\E\|f\|^p \leq C_{p,X} \E\|g\|^p (*).\] This property is equivalent
to the
UMD condition. In fact, it is still equivalent to the UMD condition if
in
addition one demands that either f or g satisfy the so-called (CI)
condition.
However, for some applications it suffices to assume that (*) holds
whenever g
satisfies the (CI) condition. We show that the class of Banach spaces
for which
(*) holds whenever only g satisfies the (CI) condition is more general
than the
class of UMD spaces, in particular it includes the space L^1. We state
several
problems related to (*) and other decoupling inequalities.
http://arxiv.org/abs/0801.0695
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6515. ARBITRAGE FREE MODELS IN MARKETS WITH TRANSACTION COSTS
Erhan Bayraktar
In \cite{Gua} the notion of stickiness for stochastic processes was
introduced. It was also shown that stickiness implies absense of
arbitrage in a
market with proportional transaction costs. In this paper, we
investigate the
notion of stickiness further. In particular, we show that stickiness is
invariant under composition with continuous functions. We also prove a
time
change result on stickiness. As an application we provide sufficient
conditions
for continuous semimartingales to be sticky (A counter example show
that not
all semi-martingales are sticky). As a result, our paper provides an
extended
class of stochastic processes that are consistent with the no arbitrage
property in a market with friction.
http://arxiv.org/abs/0801.0718
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6516. CONVERGENCE OF MULTI-DIMENSIONAL QUANTIZED $SDE$'S
Gilles Pag\`es (PMA) and Afef Sellami (PMA)
We quantize a multidimensional $SDE$ (in the Stratanovich sense) by
solving
the related $ODE$'s in which the Brownian motion has been replaced by
the
components of stationary quantizers. We make a connection with rough
path
theory to show that such quantizations converge toward the solution of
the
$SDE$. In some particular cases, we show that this procedure provide
some rate
optimal quantizations of the equation.
http://arxiv.org/abs/0801.0726
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6517. LOWER LARGE DEVIATIONS FOR MAXIMAL FLOWS THROUGH A BOX IN FIRST
PASSAGE PERCOLATION
Rapha\"el Rossignol and Marie Th\'eret
We consider the standard first passage percolation model in $
\mathbb{Z}^d$
for $d\geq 2$. We are interested in two quantities, the maximal flow $
\tau$
between the lower half and the upper half of the box, and the maximal
flow
$\phi$ between the top and the bottom of the box. A standard subadditive
argument yields the law of large numbers for $\tau$. Kesten and Zhang
have
proved the law of large numbers for $\phi$. The two variables grow
linearly
with the surface $s$ of the basis of the box, with the same
deterministic
speed. We study the probabilities that the rescaled variables $\tau /s
$ and
$\phi /s$ are abnormally small. Using a concentration inequality, we
show that
these probabilities decay exponentially fast with $s$, when $s$ grows to
infinity. Moreover, we prove an associated large deviation principle
of speed
$s$ for $\tau /s$, and for $\phi /s$. For $\phi$, we require either
that the
box is sufficiently flat, or that its sides are parallel to the
coordinates
hyperplanes.
http://arxiv.org/abs/0801.0967
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6518. LAWS OF LARGE NUMBERS FOR CONTINUOUS BELIEF MEASURES ON COMPACT
SPACES
Yann Rebille
We prove for outer continuous belief measures defined on compact spaces
strong and weak laws of large numbers as Kolmogorov's one for
measures. These
results contribute to M. Marinacci's (Journal of Economic Theory 84
(1999)
145-195) though with different methods.
http://arxiv.org/abs/0801.0976
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6519. IMPRECISE MARKOV CHAINS AND THEIR LIMIT BEHAVIOUR
Gert de Cooman and Filip Hermans and Erik Quaeghebeur
When the parameters of a finite Markov chain in discrete time, i.e., its
initial and transition probabilities, are not well known, we can and
should
perform a sensitivity analysis. This is done by considering as basic
uncertainty models the so-called credal sets that these probabilities
are known
or believed to belong to, and by allowing the probabilities to vary
over such
sets. This leads to the definition of an imprecise Markov chain. We
show that
the time evolution of such a system can be studied efficiently using
so-called
lower and upper expectations, which are equivalent mathematical
representations
of credal sets. We also study how the inferred credal set about the
state at
time n evolves as n goes to infinity, and we show that under quite
unrestrictive conditions, this credal set converges to a uniquely
invariant
credal set, regardless of the credal set given for the initial state
of the
system. We thus effectively prove a Perron-Frobenius Theorem for a
special
class of non-linear dynamical systems in discrete time.
http://arxiv.org/abs/0801.0980
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6520. LAW OF LARGE NUMBERS FOR NON-ADDITIVE MEASURES
Yann Rebille
Our aim is to give for some classes non-additive measures some limit
theorems. For balanced games we obtain a weak and strong law of large
numbers
for bounded random variables, a sharper conclusion is obtain with
exact games.
We provide an extension to upper enveloppe measures.
http://arxiv.org/abs/0801.0984
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6521. MODERATE DEVIATIONS FOR RANDOM FIELDS AND RANDOM COMPLEX ZEROES
Boris Tsirelson
Moderate deviations for random complex zeroes are deduced from a new
theorem
on moderate deviations for random fields.
http://arxiv.org/abs/0801.1050
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6522. ON THE ROBUSTNESS OF POWER-LAW RANDOM GRAPHS IN THE FINITE
MEAN, INFINITE VARIANCE REGION
I. Norros and H. Reittu
We consider a conditionally Poissonian random graph model where the mean
degrees, `capacities', follow a power-tailed distribution with finite
mean and
infinite variance. Such a graph of size $N$ has a giant component
which is
super-small in the sense that the typical distance between vertices is
of the
order of $\log\log N$. The shortest paths travel through a core
consisting of
nodes with high mean degrees. In this paper we derive upper bounds of
the
typical distance when an upper part of the core is removed, including
the case
that the whole core is removed.
http://arxiv.org/abs/0801.1079
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6523. FRACTIONAL BROWNIAN MOTION IN PRESENCE OF TWO FIXED ADSORBING
BOUNDARIES
G. Oshanin
We study the long-time asymptotics of the probability P_t that the
Riemann-Liouville fractional Brownian motion with Hurst index H does
not escape
from a fixed interval [-L,L] up to time t. We show that for any H
\in ]0,1],
for both subdiffusion and superdiffusion regimes, this probability obeys
\ln(P_t) \sim - t^{2 H}/L^2, i.e. may decay slower than exponential
(subdiffusion) or faster than exponential (superdiffusion). This
implies that
survival probability S_t of particles undergoing fractional Brownian
motion in
a one-dimensional system with randomly placed traps follows \ln(S_t)
\sim -
n^{2/3} t^{2H/3} as t \to \infty, where n is the mean density of traps.
http://arxiv.org/abs/0801.0676
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6524. ON THE EXTREMAL RAYS OF THE CONE OF POSITIVE, POSITIVE DEFINITE
FUNCTIONS
Philippe Jaming (MAPMO) and Mat\'e Matolcsi and Szilard Gy. R\'evesz
The aim of this paper is to investigate the cone of non-negative,
radial,
positive-definite functions in the set of continuous functions on $\R^d
$.
Elements of this cone admit a Choquet integral representation in terms
of the
extremals. The main feature of this article is to characterize some
large
classes of such extremals. In particular, we show that there many other
extremals than the gaussians, thus disproving a conjecture of G.
Choquet and
that no reasonable conjecture can be made on the full set of
extremals. The
last feature of this article is to show that many characterizations of
positive
definite functions available in the literature are actually particular
cases of
the Choquet integral representations we obtain.
http://arxiv.org/abs/0801.0941
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6525. IMPRECISE PROBABILITY TREES: BRIDGING TWO THEORIES OF IMPRECISE
PROBABILITY
Gert de Cooman and Filip Hermans
We give an overview of two approaches to probability theory where
lower and
upper probabilities, rather than probabilities, are used: Walley's
behavioural
theory of imprecise probabilities, and Shafer and Vovk's game-
theoretic account
of probability. We show that the two theories are more closely related
than
would be suspected at first sight, and we establish a correspondence
between
them that (i) has an interesting interpretation, and (ii) allows us to
freely
import results from one theory into the other. Our approach leads to
an account
of probability trees and random processes in the framework of Walley's
theory.
We indicate how our results can be used to reduce the computational
complexity
of dealing with imprecision in probability trees, and we prove an
interesting
and quite general version of the weak law of large numbers.
http://arxiv.org/abs/0801.1196
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6526. STOCHASTIC PROCESSES AND THEIR SPECTRAL REPRESENTATIONS OVER
NON-ARCHIMEDEAN FIELDS
S.V. Ludkovsky
The article is devoted to stochastic processes with values in finite-
and
infinite-dimensional vector spaces over infinite fields $\bf K$ of zero
characteristics with non-trivial non-archimedean norms. For different
types of
stochastic processes controlled by measures with values in $\bf K$ and
in
complete topological vector spaces over $\bf K$ stochastic integrals are
investigated. Vector valued measures and integrals in spaces over $\bf
K$ are
studied. Theorems about spectral decompositions of non-archimedean
stochastic
processes are proved.
http://arxiv.org/abs/0801.1209
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6527. OPTIMAL CO-ADAPTED COUPLING FOR THE SYMMETRIC RANDOM WALK ON
THE HYPERCUBE
Stephen B. Connor and Saul D. Jacka
Let X and Y be two simple symmetric continuous-time random walks on the
vertices of the n-dimensional hypercube. We consider the class of co-
adapted
couplings of these processes, and describe an intuitive coupling which
is shown
to be the fastest in this class.
http://arxiv.org/abs/0801.1220
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6528. ON THE SINGULARITY OF RANDOM MATRICES WITH INDEPENDENT ENTRIES
Laurent Bruneau and Francois Germinet
We consider n by n real matrices whose entries are non-degenerate random
variables that are independent but non necessarily identically
distributed, and
show that the probability that such a matrix is singular is O(1/
sqrt{n}). The
purpose of this note is to provide a short and elementary proof of
this fact
using a Bernoulli decomposition of arbitrary non degenerate random
variables.
http://arxiv.org/abs/0801.1221
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6529. BALANCED ROUTING OF RANDOM CALLS
Malwina J. Luczak and Colin McDiarmid
We consider an online routing problem in continuous time, where calls
have
Poisson arrivals and exponential durations. The first-fit dynamic
alternative
routing algorithm sequentially selects up to $d$ random two-link
routes between
the two endpoints of a call, via an intermediate node, and assigns the
call to
the first route with spare capacity on each link, if there is such a
route. The
balanced dynamic alternative routing algorithm simultaneously selects
$d$
random two-link routes; and the call is accepted on a route minimising
the
maximum of the loads on its two links, provided neither of these two
links is
saturated.
We determine the capacities needed for these algorithms to route
calls
successfully, and find that the balanced algorithm requires a much
smaller
capacity.
http://arxiv.org/abs/0801.1260
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6530. FINITELY ADDITIVE SUPERMARTINGALES
Gianluca Cassese
The concept of finitely additive supermartingales, originally due to
Bochner,
is revived and developed. We exploit it to study measure
decompositions over
filtered probability spaces and the properties of the associated
Dol\'{e}ans-Dade measure. We obtain versions of the Doob Meyer
decomposition
and, as an application, we establish a version of the Bichteler and
Dellacherie
theorem with no exogenous probability measure.
http://arxiv.org/abs/0801.1262
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6531. EXCHANGEABLE LOWER PREVISIONS
Gert de Cooman and Erik Quaeghebeur and Enrique Miranda
We extend de Finetti's (1937) notion of exchangeability to finite and
countable sequences of variables, when a subject's beliefs about them
are
modelled using coherent lower previsions rather than (linear)
previsions. We
prove representation theorems in both the finite and the countable
case, in
terms of sampling without and with replacement, respectively. We also
establish
a convergence result for sample means of exchangeable sequences.
Finally, we
study and solve the problem of exchangeable natural extension: how to
find the
most conservative (point-wise smallest) coherent and exchangeable lower
prevision that dominates a given lower prevision.
http://arxiv.org/abs/0801.1265
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6532. GAME-THEORETIC BROWNIAN MOTION
Vladimir Vovk
This paper suggests a perfect-information game, along the lines of
Levy's
characterization of Brownian motion, that formalizes the process of
Brownian
motion in game-theoretic probability. This is perhaps the simplest
situation
where probability emerges in a non-stochastic environment.
http://arxiv.org/abs/0801.1309
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6533. ON MAXIMA OF PERIODOGRAMS OF STATIONARY PROCESSES
Zhengyan Lin and Weidong Liu
We consider the limit distribution of maxima of periodograms for
stationary
processes. Our method is based on $m$-dependent approximation for
stationary
processes and a moderate deviation result.
http://arxiv.org/abs/0801.1357
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6534. CENTRAL AND $L^2$-CONCENTRATION OF 1-LIPSCHITZ MAPS INTO $
\MATHBB{R}$-TREES
Kei Funano
In this paper, we examine the L\'{e}vy-Milman concentration phenomenon
of
1-Lipschitz maps from mm-spaces to $\mathbb{R}$-trees. Our main
theorems assert
that the concentration to $\mathbb{R}$-trees follows from the
concentration to
the real line.
http://arxiv.org/abs/0801.1371
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6535. LARGE DEVIATIONS FOR STOCHASTIC EVOLUTION EQUATIONS WITH SMALL
MULTIPLICATIVE NOISE
Wei Liu
The Freidlin-Wentzell large deviation principle is established for the
distributions of stochastic evolution equations with general monotone
drift and
small multiplicative noise. Roughly speaking, the assumptions one need
for
large deviation principle are classical monotone condition on drift
part (as
for the existence and uniqueness of solution) and Lipschitz condition on
diffusion coefficient. As applications we can apply the main result to
different type examples of SPDEs (e.g. stochastic reaction-diffusion
equation,
stochastic porous media and fast diffusion equations, stochastic p-
Laplacian
equation) in Hilbert space. The weak convergence approach is employed
to verify
the Laplace principle, which is equivalent to large deviation
principle in our
framework.
http://arxiv.org/abs/0801.1443
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6536. LONGEST INCREASING SUBSEQUENCES, PLANCHEREL-TYPE MEASURE AND THE
HECKE INSERTION ALGORITHM
Hugh Thomas and Alexander Yong
We define and study the Plancherel-Hecke probability measure on Young
diagrams; the Hecke algorithm of [Buch-Kresch-Shimozono-Tamvakis-Yong
'06] is
interpreted as a polynomial-time exact sampling algorithm for this
measure.
Using the results of [Thomas-Yong '07] on jeu de taquin for increasing
tableaux, a symmetry property of the Hecke algorithm is proved, in
terms of
longest strictly increasing/decreasing subsequences of words. This
parallels
classical theorems of [Schensted '61] and of [Knuth '70],
respectively, on the
Schensted and Robinson-Schensted-Knuth algorithms. We investigate, and
conjecture about, the limit typical shape of the measure, in analogy
with work
of [Vershik-Kerov '77], [Logan-Shepp '77] and others on the ``longest
increasing subsequence problem'' for permutations. We also include a
related
extension of [Aldous-Diaconis '99] on patience sorting. Together,
these results
provide a new rationale for the study of increasing tableau
combinatorics,
distinct from the original algebraic-geometric ones concerning K-
theoretic
Schubert calculus.
http://arxiv.org/abs/0801.1319
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6537. ESTIMATION OF ORDINAL PATTERN PROBABILITIES IN FRACTIONAL
BROWNIAN MOTION
Mathieu Sinn and Karsten Keller
For equidistant discretizations of fractional Brownian motion (fBm), the
probabilities of ordinal patterns of order d=2 are monotonically
related to the
Hurst parameter H. By plugging the sample relative frequency of those
patterns
indicating changes between up and down into the monotonic relation to
H, one
obtains the Zero Crossing (ZC) estimator of the Hurst parameter which
has found
considerable attention in mathematical and applied research.
In this paper, we generally discuss the estimation of ordinal pattern
probabilities in fBm. As it turns out, according to the sufficiency
principle,
for ordinal patterns of order d=2 any reasonable estimator is an affine
functional of the sample relative frequency of changes. We establish
strong
consistency of the estimators and show them to be asymptotically
normal for
H<3/4. Further, we derive confidence intervals for the Hurst parameter.
Simulation studies show that the ZC estimator has larger variance but
less bias
than the HEAF estimator of the Hurst parameter.
http://arxiv.org/abs/0801.1598
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6538. RANDOM SUBGRAPHS OF THE 2D HAMMING GRAPH: THE SUPERCRITICAL PHASE
Remco van der Hofstad and Malwina J. Luczak
We study random subgraphs of the 2-dimensional Hamming graph H(2,n),
which is
the Cartesian product of two complete graphs on $n$ vertices. Let $p$
be the
edge probability, and write $p=\frac{1+\vep}{2(n-1)}$ for some $\vep
\in \R$. In
Borgs et al., Random subgraphs of finite graphs: I. The scaling window
under
the triangle condition, Rand. Struct. Alg. (2005), and in Borgs et
al., Random
subgraphs of finite graphs: II. The lace expansion and the triangle
condition,
Ann. Probab. (2005), the size of the largest connected component was
estimated
precisely for a large class of graphs including H(2,n) for $\vep\leq
\Lambda
V^{-1/3}$, where $\Lambda > 0$ is a constant and $V=n^2$ denotes the
number of
vertices in H(2,n). Until now, no matching lower bound on the size in
the
supercritical regime has been obtained.
In this paper we prove that, when $\vep\gg (\log{V})^{1/3}
V^{-1/3}$, then
the largest connected component has size close to $2\vep V$ with high
probability. We thus obtain a law of large numbers for the largest
connected
component size, and show that the corresponding values of $p$ are
supercritical. Barring the factor $(\log{\chs{V}})^{1/3}$, this
identifies the
size of the largest connected component all the way down to the
critical $p$
window.
http://arxiv.org/abs/0801.1607
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6539. THE SECOND LARGEST COMPONENT IN THE SUPERCRITICAL 2D HAMMING GRAPH
Malwina J. Luczak and Joel Spencer
The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices
$(i,j)$, $1\leq i,j\leq n$, two vertices being adjacent when they
share a
common coordinate. We examine random subgraphs of H(2,n) in
percolation with
edge probability $p$, so that the average degree $2(n-1)p=1+\epsilon$.
Previous
work by van der Hofstad and Luczak had shown that in the barely
supercritical
region $n^{-2/3}\ln^{1/3}n\ll \epsilon \ll 1$ the largest component
has size
$\sim 2\epsilon n$. Here we show that the second largest component has
size
close to $\epsilon^{-2}$, so that the dominant component has emerged.
http://arxiv.org/abs/0801.1608
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6540. COMMENTS ON "REVERSE AUCTION: THE LOWEST POSITIVE INTEGER GAME"
Adrian P. Flitney
In Zeng et al. [Fluct. Noise Lett. 7 (2007) L439--L447] the analysis
of the
lowest unique positive integer game is simplified by some reasonable
assumptions that make the problem tractable for arbitrary numbers of
players.
However, here we show that the solution obtained for rational players
is not a
Nash equilibrium and that a rational utility maximizer with full
computational
capability would arrive at a solution with a superior expected payoff.
An exact
solution is presented for the three- and four-player cases and an
approximate
solution for an arbitrary number of players.
http://arxiv.org/abs/0801.1535
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6541. A GEOMETRIC PREFERENTIAL ATTACHMENT MODEL WITH FITNESS
H. van den Esker
We study a random graph $G_n$, which combines aspects of geometric
random
graphs and preferential attachment. The resulting random graphs have
power-law
degree sequences with finite mean and possibly infinite variance. In
particular, the power-law exponent can be any value larger than 2.
The vertices of $G_n$ are $n$ sequentially generated vertices
chosen at
random in the unit sphere in $\mathbb R^3$. A newly added vertex has $m
$ edges
attached to it and the endpoints of these edges are connected to old
vertices
or to the added vertex itself. The vertices are chosen with probability
proportional to their current degree plus some initial attractiveness
and
multiplied by a function, depending on the geometry.
http://arxiv.org/abs/0801.1612
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6542. ON EXCHANGEABLE RANDOM VARIABLES AND THE STATISTICS OF LARGE
GRAPHS AND HYPERGRAPHS
Tim D. Austin (UC and Los Angeles)
De Finetti's classical result identifying the law of an exchangeable
family
of random variables as a mixture of i.i.d. laws was extended to
structure
theorems for more complex notions of exchangeability by Aldous, Hoover
and
Kallenberg. On the other hand, such exchangeable laws were first
related to
questions from combinatorics in an independent analysis by Fremlin and
Talagrand, and again more recently in work of Tao, where they appear
as a
natural proxy for the `leading order statistics' of colourings of
large graphs
or hypergraphs. Moreover, this relation appears implicitly in the
study of
various more bespoke formalisms for handling `limit objects' of
sequences of
dense graphs or hypergraphs in a number of recent works. However, the
connection between these works and the earlier probabilistic
structural results
seems to have gone largely unappreciated.
In this survey we recall the basic results of the theory of
exchangeable
laws, and then explain the probabilistic versions of various interesting
questions from graph and hypergraph theory that their connection
motivates
(particularly extremal questions on the testability of properties for
graphs
and hypergraphs).
We also locate the notions of exchangeability of interest to us in
the
context of other classes of probability measures subject to various
symmetries,
in particular contrasting the methods employed to analyze exchangeable
laws
with related structural results in ergodic theory, particular the
Furstenberg-Zimmer structure theorem for probability-preserving $\bbZ$-
systems,
which underpins Furstenberg's ergodic-theoretic proof of Szemer\'edi's
Theorem.
http://arxiv.org/abs/0801.1698
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6543. FREIDLIN-WENTZELL'S LARGE DEVIATIONS FOR STOCHASTIC EVOLUTION
EQUATIONS
Jiagang Ren and Xicheng Zhang
We prove a Freidlin-Wentzell large deviation principle for general
stochastic
evolution equations with small perturbation multiplicative noises. In
particular, our general result can be used to deal with a large class
of quasi
linear stochastic partial differential equations, such as stochastic
porous
medium equations and stochastic reaction diffusion equations with
polynomial
growth zero order term and $p$-Laplacian second order term.
http://arxiv.org/abs/0801.1830
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6544. HARMONIC MEASURE AND SLE
D. Beliaev and S. Smirnov
In this paper we rigorously compute the average multifractal spectrum of
harmonic measure on the boundary of SLE clusters.
http://arxiv.org/abs/0801.1792
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6545. THE LEXICOGRAPHIC FIRST OCCURRENCE OF A I-II-III PATTERN
Torey Burton and Anant P. Godbole and Brett M. Kindle
Consider a random permutation $\pi\in{\cal S}_n$. In this paper,
perhaps best
classified as a contribution to discrete probability distribution
theory, we
study the {\it first} occurrence $X=X_n$ of a I-II-III-pattern, where
"first"
is interpreted in the lexicographic order induced by the 3-subsets of
$[n]=\{1,2,...,n\}$. Of course if the permutation is I-II-III-avoiding
then the
first I-II-III-pattern never occurs, and thus $\e(X)=\infty$ for each
$n$; to
avoid this case, we also study the first occurrence of a I-II-III-
pattern given
a bijection $f:{\bf Z}^+\to{\bf Z}^+$.
http://arxiv.org/abs/0801.1876
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6546. EFFECTIVE RESISTANCE OF RANDOM TREES
Louigi Addario-Berry and Nicolas Broutin and Gabor Lugosi
We investigate the effective resistance R_n and conductance C_n
between the
root and leaves a binary tree of height n. In this electrical network,
the
resistance of each edge e at distance d from the root is defined by
r_e=2^d X_e
where the X_e are i.i.d. positive random variables bounded away from
zero and
infinity. It is shown that E(R_n) = n*E(X_e) - (V(X_e)/E(X_e))*ln n +
O(1) and
V(R_n)=O(1). Some of the results are extended to the case when the
underlying
tree is a supercritical Galton--Watson tree. (In this case the correct
scale
for r_e is b^dX_e where b is the branching number of the tree.)
http://arxiv.org/abs/0801.1909
---------------------------------------------------------------
6547. POSITIVELY AND NEGATIVELY EXCITED RANDOM WALKS ON INTEGERS,
WITH BRANCHING PROCESSES
Elena Kosygina and Martin P.W. Zerner
We consider excited random walks on the integers with a bounded number
of
i.i.d. cookies per site which may induce drifts both to the left and
to the
right. We extend the criteria for recurrence and transience by M.
Zerner and
for positivity of speed by A.-L. Basdevant and A. Singh to this case
and also
prove an annealed central limit theorem. The proofs are based on
results from
the literature concerning branching processes with migration and make
use of a
certain renewal structure.
http://arxiv.org/abs/0801.1924
---------------------------------------------------------------
6548. CONVEXITY AND SMOOTHNESS OF SCALE FUNCTIONS AND DE FINETTI'S
CONTROL PROBLEM
A. E. Kyprianou and V. Rivero and R. Song
Motivated by a classical control problem from actuarial mathematics,
we study
smoothness and convexity properties of $q$-scale functions for
spectrally
negative L\'evy processes. Continuing from the very recent work of
\cite{APP2007} and \cite{Loe} we strengthen their collective
conclusions by
showing, amongst other results, that whenever the L\'evy measure has a
non-decreasing density which is log convex then for $q>0$ the scale
function
$W^{(q)}$ is convex on some half line $(a^*,\infty)$ where $a^*$ is
the largest
value at which $W^{(q)\prime}$ attains its global minimum. As a
consequence we
deduce that de Finetti's classical actuarial control problem is solved
by a
barrier strategy where the barrier is positioned at height $a^*$.
http://arxiv.org/abs/0801.1951
---------------------------------------------------------------
6549. GEOMETRIC GAMMA MAX-INFINITELY DIVISIBLE MODELS
S. Satheesh and E. Sandhya
A transformation of gamma max-infinitely divisible laws viz. geometric
gamma
max-infinitely divisible laws is considered in this paper. Some of its
distributional and divisibility properties are discussed and a random
time
changed extremal process corresponding to this distribution is
presented. A new
kind of invariance (stability) under geometric maxima is proved and a
max-AR(1)
model corresponding to it is also discussed.
http://arxiv.org/abs/0801.2083
---------------------------------------------------------------
6550. THE LOCAL TIME OF THE CLASSICAL RISK PROCESS
F. Cortes and J.A. Le\'on and J. Villa
In this paper we give an explicit expression for the local time of the
classical risk process and associate it with the density of an
occupational
measure. To do so, we approximate the local time by a suitable
sequence of
absolutely continuous random fields. Also, as an application, we
analyze the
mean of the times $s \in [0,T]$ such that $0\leq X_{s} \leq X_{s+
\epsilon} $
for some given $\epsilon>0$.
http://arxiv.org/abs/0801.2106
---------------------------------------------------------------
6551. Q-INVARIANT FUNCTIONS FOR SOME GENERALIZATIONS OF THE ORNSTEIN-
UHLENBECK SEMIGROUP
P. Patie
We show that the multiplication operator associated to a fractional
power of
a Gamma random variable, with parameter q>0, maps the convex cone of the
1-invariant functions for a self-similar semigroup into the convex
cone of the
q-invariant functions for the associated Ornstein-Uhlenbeck (for short
OU)
semigroup. We also describe the harmonic functions for some other
generalizations of the OU semigroup. Among the various applications, we
characterize, through their Laplace transforms, the laws of first
passage times
above and overshoot for certain two-sided stable OU processes and also
for
spectrally negative semi-stable OU processes. These Laplace transforms
are
expressed in terms of a new family of power series which includes the
generalized Mittag-Leffler functions.
http://arxiv.org/abs/0801.2111
---------------------------------------------------------------
6552. BOUNDS ON THE POINCARE CONSTANT OF ULTRA LOG-CONCAVE RANDOM
VARIABLES
Oliver Johnson
We consider the discrete Poincar\'{e} constant, which relates the
variance of
a function to the expected square of its finite difference. We give an
explicit
bound on the Poincar\'{e} constant of ultra log-concave random
variables in
terms of their first two moments, and discuss how this bound relates to
calculations performed by other authors.
http://arxiv.org/abs/0801.2112
---------------------------------------------------------------
6553. A STUDY OF COUNTS OF BERNOULLI STRINGS VIA CONDITIONAL POISSON
PROCESSES
Fred W. Huffer and Jayaram Sethuraman and Sunder Sethuraman
We say that a string of length $d$ occurs, in a Bernoulli sequence, if a
success is followed by exactly $(d-1)$ failures before the next
success. The
counts of such $d$-strings are of interest, and in specific independent
Bernoulli sequences are known to correspond to asymptotic $d$-cycle
counts in
random permutations.
In this note, we give a new framework, in terms of conditional
Poisson
processes, which allows for a quick characterization of the joint
distribution
of the counts of all $d$-strings, in a general class of Bernoulli
sequences, as
certain mixtures of the product of Poisson measures. This general class
includes all Bernoulli sequences considered before, as well many new
sequences.
http://arxiv.org/abs/0801.2115
---------------------------------------------------------------
6554. EXPONENTIAL BOUNDS IN THE LAW OF ITERATED LOGARITHM FOR
MARTINGALES
E. Ostrovsky and L.Sirota
In this paper non-asymptotic exponential estimates are derived for
tail of
maximum martingale distribution by naturally norming in the spirit of
the
classical Law of Iterated Logarithm.
Key words: Martingales, exponential estimations, moment, Banach
spaces of
random variables, tail of distribution, conditional expectation.
http://arxiv.org/abs/0801.2125
---------------------------------------------------------------
6555. FAR FIELD ASYMPTOTICS OF SOLUTIONS TO CONVECTION EQUATION WITH
ANOMALOUS DIFFUSION
Lorenzo Brandolese (ICJ) and Grzegorz Karch
The initial value problem for the conservation law $\partial_t
u+(-\Delta)^{\alpha/2}u+\nabla \cdot f(u)=0$ is studied for $\alpha\in
(1,2)$
and under natural polynomial growth conditions imposed on the
nonlinearity. We
find the asymptotic expansion as $|x|\to \infty$ of solutions to this
equation
corresponding to initial conditions, decaying sufficiently fast at
infinity.
http://arxiv.org/abs/0801.1884
---------------------------------------------------------------
6556. THE ORIGIN OF INFINITELY DIVISIBLE DISTRIBUTIONS: FROM DE
FINETTI'S PROBLEM TO LEVY-KHINTCHINE FORMULA
Francesco Mainardi and Sergei Rogosin
The article provides an historical survey of the early contributions on
infinitely divisible distributions starting from the pioneering works
of de
Finetti in 1929 up to the canonical forms developed in the thirties by
Kolmogorov, Levy and Khintchine. Particular attention is paid to
single out the
personal contributions of the above authors that were published in
Italian,
French or Russian during the period 1929-1938. In Appendix we report the
translation from the Russian into English of a fundamental paper by
Khintchine
published in Moscow in 1937.
http://arxiv.org/abs/0801.1910
---------------------------------------------------------------
6557. N-MONOTONE EXACT FUNCTIONALS
Gert de Cooman and Matthias C. M. Troffaes and Enrique Miranda
We study n-monotone functionals, which constitute a generalisation of
n-monotone set functions. We investigate their relation to the
concepts of
exactness and natural extension, which generalise the notions of
coherence and
natural extension in the behavioural theory of imprecise
probabilities. We
improve upon a number of results in the literature, and prove among
other
things a representation result for exact n-monotone functionals in
terms of
Choquet integrals.
http://arxiv.org/abs/0801.1962
---------------------------------------------------------------
6558. A FUNCTIONAL CENTRAL LIMIT THEOREM FOR REGENERATIVE CHAINS
G. Maillard and S. Sch\"opfer
Using the regenerative scheme of Comets, Fern\'andez and Ferrari
(2002), we
establish a functional central limit theorem (FCLT) for discrete time
stochastic processes (chains) with summable memory decay. Furthermore,
under
stronger assumptions on the memory decay, we identify the limiting
variance in
terms of the process only. As applications, we define classes of binary
autoregressive processes and power-law Ising chains for which the FCLT
is
fulfilled.
http://arxiv.org/abs/0801.2263
---------------------------------------------------------------
6559. MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF
MULTIDIMENSIONAL DIFFUSIONS: TRUNCATED LOCAL LIMIT THEOREM
Alexey M. Kulik
For a difference approximations of multidimensional diffusion, the
truncated
local limit theorem is proved. Under very mild conditions on the
distribution
of the difference terms, this theorem provides that the transition
probabilities of these approximations, after truncation of some
asymptotically
negligible terms, possess a densities that converge uniformly to the
transition
probability density for the limiting diffusion and satisfy a uniform
diffusion-type estimates. The proof is based on the new version of the
Malliavin calculus for the product of finite family of measures, that
may
contain non-trivial singular components. An applications for uniform
estimates
for mixing and convergence rates for difference approximations to
SDE's and for
convergence of difference approximations for local times of
multidimensional
diffusions are given.
http://arxiv.org/abs/0801.2319
---------------------------------------------------------------
6560. ANALYSIS OF THE STOCHASTIC FITZHUGH-NAGUMO SYSTEM
Stefano Bonaccorsi and Elisa Mastrogiacomo
In this paper we study a system of stochastic differential equations
with
dissipative nonlinearity which arise in certain neurobiology models.
Besides
proving existence, uniqueness and continuous dependence on the initial
datum,
we shall be mainly concerned with the asymptotic behaviour of the
solution. We
prove the existence of an invariant ergodic measure $\nu$ associated
with the
transition semigroup $P_t$; further, we identify its infinitesimal
generator in
the space $L^2(H;\nu)$.
http://arxiv.org/abs/0801.2325
---------------------------------------------------------------
6561. THE EINSTEIN RELATION FOR RANDOM WALKS ON GRAPHS
Andras Telcs
This paper investigates the Einstein relation; the connection between
the
volume growth, the resistance growth and the expected time a random
walk needs
to leave a ball on a weighted graph. The Einstein relation is proved
under
different set of conditions. In the simplest case it is shown under
the volume
doubling and time comparison principles. This and the other set of
conditions
provide the basic framework for the study of (sub-) diffusive behavior
of the
random walks on weighted graphs.
http://arxiv.org/abs/0801.2336
---------------------------------------------------------------
6562. UPPER BOUNDS FOR TRANSITION PROBABILITIES ON GRAPHS AND
ISOPERIMETRIC INEQUALITIES
Andras Telcs
In this paper necessary and sufficient conditions are presented for heat
kernel upper bounds for random walks on weighted graphs. Several
equivalent
conditions are given in the form of isoperimetric inequalities.
http://arxiv.org/abs/0801.2341
---------------------------------------------------------------
6563. RANDOM WALKS ON GRAPHS WITH VOLUME AND TIME DOUBLING
Andras Telcs
This paper studies the on- and off-diagonal upper estimate and the two-
sided
transition probability estimate of random walks on weighted graphs.
http://arxiv.org/abs/0801.2351
---------------------------------------------------------------
6564. ON WASSERSTEIN GEOMETRY OF THE SPACE OF GAUSSIAN MEASURES
Asuka Takatsu
The space which consists of measures having finite second moment is an
infinite dimensional metric space endowed with Wasserstein distance,
while the
space of Gaussian measures on Euclidean space is parameterized by mean
and
covariance matrices, hence a finite dimensional manifold. By
restricting to the
space of Gaussian measures inside the space of probability measures,
we manege
to provide detailed descriptions of the Wasserstein geometry from a
Riemannian
geometric viewpoint. In particular, using the results from the Monge-
Kantrovich
transport theory, an explicit expression of geodesics interpolating two
Gaussian measures. It follows that the space of Gaussian measures is
geodesically convex in the space of probability measures. Also, a
Riemannian
metric which induces the Wasserstein distance is specified. Using the
Riemannian metric, a formula for the sectional curvatures of the space
of
Gaussian measures on the plane is written out in terms of the
eigenvalues of
the covariance matrix.
http://arxiv.org/abs/0801.2250
---------------------------------------------------------------
6565. THE VOLUME AND TIME COMPARISON PRINCIPLE AND TRANSITION
PROBABILITY ESTIMATES FOR RANDOM WALKS
Andras Telcs
This paper presents necessary and sufficient conditions for on- and
off-diagonal transition probability estimates for random walks on
weighted
graphs. On the integer lattice and on may fractal type graphs both the
volume
of a ball and the mean exit time from a ball is independent of the
centre,
uniform in space. Here the upper estimate is given without such
restriction and
two-sided estimate is given if uniformity in the space assumed only
for the
mean exit time.
http://arxiv.org/abs/0801.2393
---------------------------------------------------------------
6566. STOCHASTIC POROUS MEDIA EQUATION AND SELF-ORGANIZED CRITICALITY
Viorel Barbu (Institute of Mathematics "Octav Mayer" and Iasi and
Romania) and Giuseppe Da Prato (Scuola Normale Superiore di Pisa,
Italy) and Michael
R\"ockner (Faculty of Mathematics, Bielefeld, Germany and
Departments of
Mathematics and Statistics, Purdue University, USA)
The existence and uniqueness of nonnegative strong solutions for
stochastic
porous media equations with noncoercive monotone diffusivity function
and
Wiener forcing term is proven. The finite time extinction of solutions
with
high probability is also proven in 1-D. The results are relevant for
self-organized critical behaviour of stochastic nonlinear diffusion
equations
with critical states.
http://arxiv.org/abs/0801.2478
---------------------------------------------------------------
6567. BAXTER'S INEQUALITY FOR FRACTIONAL BROWNIAN MOTION-TYPE
PROCESSES WITH HURST INDEX LESS THAN 1/2
Akihiko Inoue and Yukio Kasahara and Punam Phartyal
The aim of this paper is to prove an analogue of Baxter's inequality for
fractional Brownian motion-type processes with Hurst index less than
1/2. This
inequality is concerned with the norm estimate of the difference between
finite- and infinite-past predictor coefficients.
http://arxiv.org/abs/0801.2509
---------------------------------------------------------------
6568. THERMODYNAMIC LIMIT FOR THE INVARIANT MEASURES IN SUPERCRITICAL
ZERO RANGE PROCESSES
In\'es Armend\'ariz and Michail Loulakis
We prove a strong form of the equivalence of ensembles for the invariant
measures of zero range processes conditioned to a supercritical
density of
particles. It is known that in this case there is a single site that
accomodates a macroscopically large number of the particles in the
system. We
show that in the thermodynamic limit the rest of the sites have joint
distribution equal to the grand canonical measure at the critical
density. This
improves the result of Gro\ss kinsky, Sch\"{u}tz and Spohn, where
convergence
is obtained for the finite dimensional marginals. We obtain as
corollaries
limit theorems for the order statistics of the components and for the
fluctuations of the bulk.
http://arxiv.org/abs/0801.2511
---------------------------------------------------------------
6569. THE VARIANCE OF THE SHOCK IN THE HAD PROCESS
Cristian F. Coletti and Pablo A. Ferrari and Leandro P.R. Pimentel
We consider the Hammersley-Aldous-Diaconis (HAD) process with sinks and
sources such that there is a microscopic shock at every time $t$;
denote $Z(t)$
its position. We show that the mean and variance of $Z(t)$ are linear
functions
of $t$ and compute explicitely the respective constants in function of
the left
and right densities. Furthermore, we describe the dependence of $Z(t)$
on the
initial configuration in the scale $\sqrt t$ and, as a corollary,
prove a
central limit theorem.
http://arxiv.org/abs/0801.2526
---------------------------------------------------------------
6570. RECURRENCE TIMES AND LARGE DEVIATIONS
Yong Moo Chung
We give a criterion to determine the large deviation rate functions for
abstract dynamical systems on towers. As an application of this
criterion we
show the level 2 large deviation principle for some class of smooth
interval
maps with nonuniform hyperbolicity.
http://arxiv.org/abs/0801.2409
---------------------------------------------------------------
6571. UNIFORMLY SPREAD MEASURES AND VECTOR FIELDS
Mikhail Sodin and Boris Tsirelson
We show that two different ideas of uniform spreading of locally finite
measures in the d-dimensional Euclidean space are equivalent. The
first idea is
formulated in terms of finite distance transportations to the Lebesgue
measure,
while the second idea is formulated in terms of vector fields
connecting a
given measure with the Lebesgue measure.
http://arxiv.org/abs/0801.2505
---------------------------------------------------------------
6572. EDGEWORTH EXPANSION OF THE LARGEST EIGENVALUE DISTRIBUTION
FUNCTION OF GOE
Leonard N. Choup
In this paper we focus on the large n probability distribution
function of
the largest eigenvalue in the Gaussian Orthogonal Ensemble of n by n
matrices
(GOEn). We prove an Edgeworth type Theorem for the largest eigenvalue
probability distribution function of GOEn. The correction terms to the
limiting
probability distribution are expressed in terms of the same Painleve II
functions appearing in the Tracy-Widom distribution. We conclude with
a brief
discussion of the GSEn case.
http://arxiv.org/abs/0801.2620
---------------------------------------------------------------
6573. TOTAL-VARIATION CUTOFF IN BIRTH-AND-DEATH CHAINS
Jian Ding and Eyal Lubetzky and Yuval Peres
The cutoff phenomenon describes a case where a Markov chain exhibits a
sharp
transition in its convergence to stationarity. In 1996, Diaconis
surveyed this
phenomenon, and asked how one could recognize its occurrence in
families of
finite ergodic Markov chains. In 2004, the third author noted that a
necessary
condition for cutoff in a family of reversible chains is that the
product of
the mixing-time and spectral-gap tends to infinity, and conjectured
that in
many settings, this condition should also be sufficient. Diaconis and
Saloff-Coste (2006) verified this conjecture for birth-and-death
chains with
the convergence measured in separation. It is natural to ask whether the
conjecture holds for these chains in the more widely used total-
variation
distance.
In this work, we confirm the above conjecture for all continuous-
time or lazy
discrete-time birth-and-death chains, with convergence measured via
total-variation distance. Namely, if the product of the mixing-time and
spectral-gap tends to infinity, the chains exhibit cutoff at the maximal
hitting time of the stationary distribution median, with a window of
at most
the geometric mean between the relaxation-time and mixing-time.
http://arxiv.org/abs/0801.2625
---------------------------------------------------------------
6574. HARNACK INEQUALITY AND APPLICATIONS FOR STOCHASTIC DIFFERENTIAL
EQUATIONS WITH JUMPS
Feng-Yu Wang and Chenggui Yuan
Gradient estimates and a Harnack inequality are established for the
semigroup
associated to stochastic differential equations driven by Poisson
processes. As
applications, estimates of the transition probability density, the
compactness
and ultraboundedness of the semigroup are studied in terms of the
corresponding
invariant measure.
http://arxiv.org/abs/0801.2668
---------------------------------------------------------------
6575. CONSTRUCTION OF AN EDWARDS' PROBABILITY MEASURE ON $\MATHCAL{C}
(\MATHBB{R}_+, \MATHBB{R})$
Joseph Najnudel
In this article, we prove that the measures $\mathbb{Q}_T$ associated
to the
one-dimensional Edwards' model on the interval $[0,T]$ converge to a
limit
measure $\mathbb{Q}$ when $T$ goes to infinity, in the following
sense : for
all $s \geq 0$ and for all events $\Lambda_s$ depending on the canonical
process only up to time $s$, $\mathbb{Q}_T (\Lambda_s) \to \mathbb{Q}
(\Lambda_s)$.
Moreover, we prove that, if $\mathbb{P}$ is Wiener measure,
there exists a martingale $(D_s)_{s \in \mathbb{R}_+}$ such that $
\mathbb{Q}
(\Lambda_s) = \mathbb{E}_{\mathbb{P}} (\mathds{1}_{\Lambda_s} D_s)$,
and we
give an explicit expression for this martingale.
http://arxiv.org/abs/0801.2751
---------------------------------------------------------------
6576. DISCRETE APPROXIMATION OF A STABLE SELF-SIMILAR STATIONARY
INCREMENTS PROCESS
Cl\'ement Dombry (LMA) and Nadine Guillotin-Plantard (ICJ)
The aim of this paper is to present a result of discrete approximation
of
some class of stable self-similar stationary increments processes. The
properties of such processes were intensively investigated, but little
is known
on the context in which such processes can arise. To our knowledge,
discretisation and convergence theorems are available only in the case
of
stable L\'evy motions and fractional Brownian motions. This paper
yields new
results in this direction. Our main result is the convergence of the
random
rewards schema, which was firstly introduced by Cohen and
Samorodnitsky, and
that we consider in a more general setting. Strong relationships with
Kesten
and Spitzer's random walk in random sceneries are evidenced. Finally,
we study
some path properties of the limit process.
http://arxiv.org/abs/0801.2753
---------------------------------------------------------------
6577. INTEGRABILITY OF EXIT TIMES AND BALLISTICITY FOR RANDOM WALKS
IN DIRICHLET ENVIRONMENT
Laurent Tournier (ICJ)
We consider random walks in Dirichlet environment, introduced by
Enriquez and
Sabot in 2006. As this distribution on environments is not uniformly
elliptic,
the annealed integrability of exit times out of a given finite subset
is a
non-trivial property. We provide here an explicit equivalent condition
for this
integrability to happen, on general directed graphs. Such integrability
problems arise for instance from the definition of Kalikow auxiliary
random
walk. Using our condition, we prove a refined version of the
ballisticity
criterion given by Enriquez and Sabot.
http://arxiv.org/abs/0801.2875
---------------------------------------------------------------
6578. ON BESOV REGULARITY OF BROWNIAN MOTIONS IN INFINITE DIMENSIONS
Tuomas Hytonen and Mark Veraar
We extend to the vector-valued situation some earlier work of
Ciesielski and
Roynette on the Besov regularity of the paths of the classical
Brownian motion.
We also consider a Brownian motion as a Besov space valued random
variable. It
turns out that a Brownian motion, in this interpretation, is a
Gaussian random
variable with some pathological properties. We prove estimates for the
first
moment of the Besov norm of a Brownian motion. To obtain such results we
estimate expressions of the form $\E \sup_{n\geq 1}\|\xi_n\|$, where the
$\xi_n$ are independent centered Gaussian random variables with values
in a
Banach space. Using isoperimetric inequalities we obtain two-sided
inequalities
in terms of the first moments and the weak variances of $\xi_n$.
http://arxiv.org/abs/0801.2959
---------------------------------------------------------------
6579. EVERY MINOR-CLOSED PROPERTY OF SPARSE GRAPHS IS TESTABLE
Itai Benjamini and Oded Schramm and Asaf Shapira
Testing a property $P$ of graphs in the bounded degree model deals
with the
following problem: given a graph $G$ of bounded degree $d$ we should
distinguish (with probability 0.9, say) between the case that $G$
satisfies $P$
and the case that one should add/remove at least $\epsilon d n$ edges
of $G$ to
make it satisfy $P$. In sharp contrast to property testing of dense
graphs,
which is relatively well understood, very few properties are known to be
testable in bounded degree graphs with a constant number of queries.
In this paper we identify for the first time a large (and natural)
family of
properties that can be efficiently tested in bounded degree graphs, by
showing
that every minor-closed graph property can be tested with a constant
number of
queries. As a special case, we infer that many well studied graph
properties,
like being planar, outer-planar, series-parallel, bounded genus, bounded
tree-width and several others, are testable with a constant number of
queries.
None of these properties was previously known to be testable even with
$o(n)$
queries. The proof combines results from the theory of graph minors with
results on convergent sequences of sparse graphs, which rely on
martingale
arguments.
http://arxiv.org/abs/0801.2797
---------------------------------------------------------------
6580. MALLIAVIN CALCULUS AND DECOUPLING INEQUALITIES IN BANACH SPACES
Jan Maas
We develop a theory of Malliavin calculus for Banach space valued random
variables. Using radonifying operators instead of symmetric tensor
products we
extend the Wiener-Ito isometry to Banach spaces. In the white noise
case we
obtain two sided L^p-estimates for multiple stochastic integrals in
arbitrary
Banach spaces. It is shown that the Malliavin derivative is bounded on
vector-valued Wiener-Ito chaoses. Our main tools are decoupling
inequalities
for vector-valued random variables. In the opposite direction we use
Meyer's
inequalities to give a new proof of a decoupling result for Gaussian
chaoses in
UMD Banach spaces.
http://arxiv.org/abs/0801.2899
---------------------------------------------------------------
6581. $C^1$-GENERIC SYMPLECTIC DIFFEOMORPHISMS: PARTIAL HYPERBOLICITY
AND LYAPUNOV EXPONENTS
Jairo Bochi
It is proven that for a $C^1$-generic symplectic diffeomorphism $f$ of
any
closed manifold, the Oseledets splitting along almost every orbit is
either
trivial or partially hyperbolic. In addition, if $f$ is not Anosov
then all the
exponents in the center bundle vanish. This establishes in full a result
announced by Ma\~n\'e in the ICM 1983. The main technical novelty is a
probabilistic method for the construction of perturbations (using random
walks).
http://arxiv.org/abs/0801.2960
---------------------------------------------------------------
6582. A NEW CONCEPT OF STRONG CONTROLLABILITY VIA THE SCHUR COMPLEMENT
IN ADAPTIVE TRACKING
Bernard Bercu and Victor Vazquez
We propose a new concept of strong controllability associated with the
Schur
complement of a suitable limiting matrix. This concept allows us to
extend the
previous results associated with multidimensional ARX models. On the
one hand,
we carry out a sharp analysis of the almost sure convergence for both
least
squares and weighted least squares algorithms. On the other hand, we
also
provide a central limit theorem and a law of iterated logarithm for
these two
stochastic algorithms. Our asymptotic results are illustrated by
numerical
simulations.
http://arxiv.org/abs/0801.2991
---------------------------------------------------------------
6583. ESTIMATION OF QUADRATIC VARIATION FOR TWO-PARAMETER DIFFUSIONS
Anthony R\'eveillac
In this paper we give a central limit theorem for the weighted quadratic
variations process of a two-parameter Brownian motion. As an
application, we
show that the discretized quadratic variations $\sum_{i=1}^{[n s]}
\sum_{j=1}^{[n t]} | \Delta_{i,j} Y |^2$ of a two-parameter diffusion
$Y=(Y_{(s,t)})_{(s,t)\in[0,1]^2}$ observed on a regular grid $G_n$ is an
asymptotically normal estimator of the quadratic variation of $Y$ as $n
$ goes
to infinity.
http://arxiv.org/abs/0801.3027
---------------------------------------------------------------
6584. A LOWER BOUND FOR THE CHUNG-DIACONIS-GRAHAM RANDOM PROCESS
Martin Hildebrand
Chung, Diaconis, and Graham considered random processes of the form
X_{n+1}=a_n X_n+b_n (mod p) where p is odd, X_0=0, a_n=2 always, and
b_n are
i.i.d. for n=0,1,2,... . In this paper, we show that if
P(b_n=-1)=P{b_n=0)=P(b_n=1)=1/3, then there exists a constant c>1 such
that c
log_2 p steps are not enough to make X_n get close to uniformly
distributed on
the integers mod p.
http://arxiv.org/abs/0801.3094
---------------------------------------------------------------
6585. APPROXIMATE WORD MATCHES BETWEEN TWO RANDOM SEQUENCES
Conrad J. Burden and Miriam R. Kantorovitz and Susan R. Wilson
Given two sequences over a finite alphabet $\mathcal{L}$, the $D_2$
statistic
is the number of $m$-letter word matches between the two sequences. This
statistic is used in bioinformatics for expressed sequence tag database
searches. Here we study a generalization of the $D_2$ statistic in the
context
of DNA sequences, under the assumption of strand symmetric Bernoulli
text. For
$k<m$, we look at the count of $m$-letter word matches with up to $k$
mismatches. For this statistic, we compute the expectation, give upper
and
lower bounds for the variance and prove its distribution is
asymptotically
normal.
http://arxiv.org/abs/0801.3145
---------------------------------------------------------------
6586. THE HEAVY TRAFFIC LIMIT OF AN UNBALANCED GENERALIZED PROCESSOR
SHARING MODEL
Kavita Ramanan and Martin I. Reiman
This work considers a server that processes $J$ classes using the
generalized
processor sharing discipline with base weight vector $\alpha=(\alpha
_1,...,\alpha_J)$ and redistribution weight vector
$\beta=(\beta_1,...,\beta_J)$. The invariant manifold $\mathcal{M}$ of
the
so-called fluid limit associated with this model is shown to have the
form
$\mathcal{M}=\{x\in\mathbb{R}_+^J:x_j=0 for j\in\mathcal{S}\}$, where
$\mathcal{S}$ is the set of strictly subcritical classes, which is
identified
explicitly in terms of the vectors $\alpha$ and $\beta$ and the long-run
average work arrival rates $\gamma_j$ of each class $j$. In addition,
under
general assumptions, it is shown that when the heavy traffic condition
$\sum_{j=1}^J\gamma_j=\sum_{j=1}^J\alpha_j$ holds, the functional
central limit
of the scaled unfinished work process is a reflected diffusion process
that
lies in $\mathcal{M}$. The reflected diffusion limit is characterized
by the
so-called extended Skorokhod map and may fail to be a semimartingale.
This
generalizes earlier results obtained for the simpler, balanced case
where
$\gamma_j=\alpha_j$ for $j=1,...,J$, in which case $\mathcal{M}=
\mathbb{R}_+^J$
and there is no state-space collapse. Standard techniques for obtaining
diffusion approximations cannot be applied in the unbalanced case due
to the
particular structure of the GPS model. Along the way, this work also
establishes a comparison principle for solutions to the extended
Skorokhod map
associated with this model, which may be of independent interest.
http://arxiv.org/abs/0801.3174
---------------------------------------------------------------
6587. THE EXPECTED DURATION OF RANDOM SEQUENTIAL ADSORPTION
Aidan Sudbury
When gas molecules bind to a surface they may do so in such a way that
the
adsorption of one molecule inhibits the arrival of others. We consider
random
sequential adsorption in which the empty sites of a graph are
irreversibly
occupied in random order by a variety of types of ``particles.'' In a
finite
region the process terminates when no more particles can arrive. A
universal
asymptotic formula for the mean duration is given.
http://arxiv.org/abs/0801.3184
---------------------------------------------------------------
6588. INTENSITY PROCESS AND COMPENSATOR: A NEW FILTRATION EXPANSION
APPROACH AND THE JEULIN--YOR THEOREM
Xin Guo and Yan Zeng
Let $(X_t)_{t\ge0}$ be a continuous-time, time-homogeneous strong Markov
process with possible jumps and let $\tau$ be its first hitting time
of a Borel
subset of the state space. Suppose $X$ is sampled at random times and
suppose
also that $X$ has not hit the Borel set by time $t$. What is the
intensity
process of $\tau$ based on this information? This question from credit
risk
encompasses basic mathematical problems concerning the existence of an
intensity process and filtration expansions, as well as some
conceptual issues
for credit risk. By revisiting and extending the famous Jeulin--Yor
[Lecture
Notes in Math. 649 (1978) 78--97] result regarding compensators under
a general
filtration expansion framework, a novel computation methodology for the
intensity process of a stopping time is proposed. En route, an analogous
characterization result for martingales of Jacod and Skorohod [Lecture
Notes in
Math. 1583 (1994) 21--35] under local jumping filtration is derived.
http://arxiv.org/abs/0801.3191
---------------------------------------------------------------
6589. TIME DISCRETIZATION AND MARKOVIAN ITERATION FOR COUPLED FBSDES
Christian Bender and Jianfeng Zhang
In this paper we lay the foundation for a numerical algorithm to
simulate
high-dimensional coupled FBSDEs under weak coupling or monotonicity
conditions.
In particular, we prove convergence of a time discretization and a
Markovian
iteration. The iteration differs from standard Picard iterations for
FBSDEs in
that the dimension of the underlying Markovian process does not
increase with
the number of iterations. This feature seems to be indispensable for an
efficient iterative scheme from a numerical point of view. We finally
suggest a
fully explicit numerical algorithm and present some numerical examples
with up
to 10-dimensional state space.
http://arxiv.org/abs/0801.3203
---------------------------------------------------------------
6590. DEGENERATE STOCHASTIC DIFFERENTIAL EQUATIONS ARISING FROM
CATALYTIC BRANCHING NETWORKS
Richard F. Bass and Edwin A. Perkins
We establish existence and uniqueness for the martingale problem
associated
with a system of degenerate SDE's representing a catalytic branching
network.
For example, in the hypercyclic case:
$$dX_{t}^{(i)}=b_i(X_t)dt+\sqrt{2\gamma_{i}(X_{t})
X_{t}^{(i+1)}X_{t}^{(i)}}dB_{t}^{i}, X_t^{(i)}\ge 0, i=1,..., d,$$ where
$X^{(d+1)}\equiv X^{(1)}$, existence and uniqueness is proved when $
\gamma$ and
$b$ are continuous on the positive orthant, $\gamma$ is strictly
positive, and
$b_i>0$ on $\{x_i=0\}$. The special case $d=2$, $b_i=\theta_i-x_i$ is
required
in work of Dawson-Greven-den Hollander-Sun-Swart on mean fields limits
of block
averages for 2-type branching models on a hierarchical group. The
proofs make
use of some new methods, including Cotlar's lemma to establish
asymptotic
orthogonality of the derivatives of an associated semigroup at different
times,and a refined integration by parts technique from Dawson-
Perkins]. As a
by-product of the proof we obtain the strong Feller property of the
associated
resolvent.
http://arxiv.org/abs/0801.3257
---------------------------------------------------------------
6591. A NOTE ABOUT CONDITIONAL ORNSTEIN-UHLENBECK PROCESSES
Amel Bentata (PMA)
In this short note, the identity in law, which was obtained by P.
Salminen,
between on one hand, the Ornstein-Uhlenbeck process with parameter
gamma,
killed when it reaches 0, and on the other hand, the 3-dimensional
radial
Ornstein-Uhlenbeck process killed exponentially at rate gamma and
conditioned
to hit 0, is derived from a simple absolute continuity relationship.
http://arxiv.org/abs/0801.3261
---------------------------------------------------------------
6592. THE EXECUTION GAME
Ciamac C. Moallemi and Beomsoo Park and Benjamin Van Roy
We consider a trader who aims to liquidate a large position in the
presence
of an arbitrageur who hopes to profit from the trader's activity. The
arbitrageur is uncertain about the trader's position and learns from
observed
market activity. This is a dynamic game with asymmetric information.
We present
an algorithm for computing perfect Bayesian equilibrium behavior and
conduct
numerical experiments. Our results demonstrate that the trader's
strategy
differs in important ways from one that would be optimal in the
absence of an
arbitrageur. In particular, the trader's actions depend on and
influence the
arbitrageur's beliefs. Accounting for the presence of a strategic
adversary can
greatly reduce transaction costs.
http://arxiv.org/abs/0801.3001
---------------------------------------------------------------
6593. POISSON SUSPENSIONS AND ENTROPY FOR INFINITE TRANSFORMATIONS
Elise Janvresse and Tom Meyerovitch and Emmanuel Roy and Thierry De
La Rue
The Poisson entropy of an infinite-measure-preserving transformation is
defined as the Kolmogorov entropy of its Poisson suspension. In this
article,
we relate Poisson entropy with other definitions of entropy for infinite
transformations: For quasi-finite transformations we prove that
Poisson entropy
coincides with Krengel's and Parry's entropy. In particular, this
implies that
for null-recurrent Markov chains, the usual formula for the entropy $-
\sum q_i
p_{i,j}\log p_{i,j}$ holds in any of the definitions for entropy.
Poisson
entropy dominates Parry's entropy in any conservative transformation.
We also
prove that relative entropy (in the sense of Danilenko and Rudolph)
coincides
with the relative Poisson entropy. Thus, for any factor of a
conservative
transformation, difference of the Krengel's entropy is equal to the
difference
of the Poisson entropies. Finally, we prove the existence of a maximal
(Pinsker) factor with zero (Poisson, Krengel, Parry) entropy for quasi-
finite
transformations. This answers affirmatively the question about
existence of a
Pinsker factor in the sense of Krengel for quasi-finite
transformations, a
question raised in arXiv:0705.2148v3.
http://arxiv.org/abs/0801.3155
---------------------------------------------------------------
6594. OCCUPATION DENSITIES FOR CERTAIN PROCESSES RELATED TO
FRACTIONAL BROWNIAN MOTION
Khalifa Es-Sebaiy and David Nualart and Youssef Ouknine and Ciprian
Tudor (CES and SAMOS)
In this paper we establish the existence of a square integrable
occupation
density for two classes of stochastic processes. First we consider a
Gaussian
process with an absolutely continuous random drift, and secondly we
handle the
case of a (Skorohod) integral with respect to the fractional Brownian
motion
with Hurst parameter $H>\frac 12$. The proof of these results uses a
general
criterion for the existence of a square integrable local time, which
is based
on the techniques of Malliavin calculus.
http://arxiv.org/abs/0801.3314
---------------------------------------------------------------
6595. THE LINEAGE PROCESS IN GALTON--WATSON TREES AND GLOBALLY
CENTERED DISCRETE SNAKES
Jean-Fran\c{c}ois Marckert
We consider branching random walks built on Galton--Watson trees with
offspring distribution having a bounded support, conditioned to have $n
$ nodes,
and their rescaled convergences to the Brownian snake. We exhibit a
notion of
``globally centered discrete snake'' that extends the usual settings
in which
the displacements are supposed centered. We show that under some
additional
moment conditions, when $n$ goes to $+\infty$, ``globally centered
discrete
snakes'' converge to the Brownian snake. The proof relies on a precise
study of
the lineage of the nodes in a Galton--Watson tree conditioned by the
size, and
their links with a multinomial process [the lineage of a node $u$ is
the vector
indexed by $(k,j)$ giving the number of ancestors of $u$ having $k$
children
and for which $u$ is a descendant of the $j$th one]. Some consequences
concerning Galton--Watson trees conditioned by the size are also
derived.
http://arxiv.org/abs/0801.3330
---------------------------------------------------------------
6596. CONVEXITY, TRANSLATION INVARIANCE AND SUBADDITIVITY FOR $G$-
EXPECTATIONS AND RELATED RISK MEASURES
Long Jiang
Under the continuous assumption on the generator $g$, Briand et al.
[Electron. Comm. Probab. 5 (2000) 101--117] showed some connections
between $g$
and the conditional $g$-expectation
$({\mathcal{E}}_g[\cdot|{\mathcal{F}}_t])_{t\in[0,T]}$ and Rosazza
Gianin
[Insurance: Math. Econ. 39 (2006) 19--34] showed some connections
between $g$
and the corresponding dynamic risk measure $(\rho^g_t)_{t\in[0,T]}$.
In this
paper we prove that, without the additional continuous assumption on $g
$, a
$g$-expectation ${\mathcal{E}}_g$ satisfies translation invariance if
and only
if $g$ is independent of $y$, and ${\mathcal{E}}_g$ satisfies
convexity (resp.
subadditivity) if and only if $g$ is independent of $y$ and $g$ is
convex
(resp. subadditive) with respect to $z$. By these conclusions we
deduce that
the static risk measure $\rho^g$ induced by a $g$-expectation $
{\mathcal{E}}_g$
is a convex (resp. coherent) risk measure if and only if $g$ is
independent of
$y$ and $g$ is convex (resp. sublinear) with respect to $z$. Our
results extend
the results in Briand et al. [Electron. Comm. Probab. 5 (2000)
101--117] and
Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19--34] on these
subjects.
http://arxiv.org/abs/0801.3340
---------------------------------------------------------------
6597. EVOLUTIONARILY STABLE STRATEGIES OF RANDOM GAMES, AND THE
VERTICES OF RANDOM POLYGONS
Sergiu Hart and Yosef Rinott and Benjamin Weiss
An evolutionarily stable strategy (ESS) is an equilibrium strategy
that is
immune to invasions by rare alternative (``mutant'') strategies.
Unlike Nash
equilibria, ESS do not always exist in finite games. In this paper we
address
the question of what happens when the size of the game increases: does
an ESS
exist for ``almost every large'' game? Letting the entries in the $n
\times n$
game matrix be independently randomly chosen according to a
distribution $F$,
we study the number of ESS with support of size $2.$ In particular, we
show
that, as $n\to \infty$, the probability of having such an ESS: (i)
converges to
1 for distributions $F$ with ``exponential and faster decreasing
tails'' (e.g.,
uniform, normal, exponential); and (ii) converges to $1-1/\sqrt{e}$ for
distributions $F$ with ``slower than exponential decreasing
tails'' (e.g.,
lognormal, Pareto, Cauchy). Our results also imply that the expected
number of
vertices of the convex hull of $n$ random points in the plane
converges to
infinity for the distributions in (i), and to 4 for the distributions
in (ii).
http://arxiv.org/abs/0801.3353
---------------------------------------------------------------
6598. ONE-DIMENSIONAL STEPPING STONE MODELS, SARDINE GENETICS AND
BROWNIAN LOCAL TIME
Richard Durrett and Mateo Restrepo
Consider a one-dimensional stepping stone model with colonies of size
$M$ and
per-generation migration probability $\nu$, or a voter model on $
\mathbb{Z}$ in
which interactions occur over a distance of order $K$. Sample one
individual at
the origin and one at $L$. We show that if $M\nu/L$ and $L/K^2$
converge to
positive finite limits, then the genealogy of the sample converges to
a pair of
Brownian motions that coalesce after the local time of their
difference exceeds
an independent exponentially distributed random variable. The
computation of
the distribution of the coalescence time leads to a one-dimensional
parabolic
differential equation with an interesting boundary condition at 0.
http://arxiv.org/abs/0801.3370
---------------------------------------------------------------
6599. CONVERGENCE OF FINITE-DIMENSIONAL LAWS OF THE WEIGHTED
QUADRATIC VARIATIONS PROCESS FOR SOME FRACTIONAL BROWNIAN SHEETS
Anthony Reveillac
In this paper we state and prove a central limit theorem for the
finite-dimensional laws of the quadratic variations process of certain
fractional Brownian sheets. The main tool of this article is a method
developed
by Nourdin and Nualart based on the Malliavin calculus.
http://arxiv.org/abs/0801.3416
---------------------------------------------------------------
6600. QUENCHED CONVERGENCE OF A SEQUENCE OF SUPERPROCESSES IN R^D
AMONG POISSONIAN OBSTACLES
Amandine Veber
We prove a convergence theorem for a sequence of super-Brownian motions
moving among hard Poissonian obstacles, when the intensity of the
obstacles
grows to infinity but their diameters shrink to zero in an appropriate
manner.
The superprocesses are shown to converge in probability for the law
$\mathbf{P}$ of the obstacles, and $\mathbf{P}$-almost surely for a
subsequence, towards a superprocess with underlying spatial motion
given by
Brownian motion and (inhomogeneous) branching mechanism $\psi(u,x)$ of
the form
$\psi(u,x)= u^2+ \kappa(x)u$, where $\kappa(x)$ depends on the density
of the
obstacles. This work draws on similar questions for a single Brownian
motion.
In the course of the proof, we establish precise estimates for
integrals of
functions over the Wiener sausage, which are of independent interest.
http://arxiv.org/abs/0801.3444
---------------------------------------------------------------
6601. ON THE CONDENSED DENSITY OF THE GENERALIZED EIGENVALUES OF
PENCILS OF HANKEL GAUSSIAN RANDOM MATRICES AND APPLICATIONS
Piero Barone
Pencils of Hankel matrices whose elements have a joint Gaussian
distribution
with nonzero mean and not identical covariance are considered. An
approximation
to the distribution of the squared modulus of their determinant is
computed
which allows to get a closed form approximation of the condensed
density of the
generalized eigenvalues of the pencils. Implications of this result
for solving
several moments problems are discussed and some numerical examples are
provided.
http://arxiv.org/abs/0801.3352
---------------------------------------------------------------
6602. STATISTICAL ARBITRAGE AND OPTIMAL TRADING WITH TRANSACTION COSTS
IN FUTURES MARKETS
Theodoros Tsagaris
We consider the Brownian market model and the problem of expected
utility
maximization of terminal wealth. We, specifically, examine the problem
of
maximizing the utility of terminal wealth under the presence of
transaction
costs of a fund/agent investing in futures markets. We offer some
preliminary
remarks about statistical arbitrage strategies and we set the
framework for
futures markets, and introduce concepts such as margin, gearing and
slippage.
The setting is of discrete time, and the price evolution of the
futures prices
is modelled as discrete random sequence involving Ito's sums. We
assume the
drift and the Brownian motion driving the return process are non-
observable and
the transaction costs are represented by the bid-ask spread. We provide
explicit solution to the optimal portfolio process, and we offer an
example
using logarithmic utility.
http://arxiv.org/abs/0801.3348
---------------------------------------------------------------
6603. HARMONIC ANALYSIS OF STOCHASTIC EQUATIONS AND BACKWARD
STOCHASTIC DIFFERENTIAL EQUATIONS
Freddy Delbaen and Shanjian Tang
The BMO martingale theory is extensively used to study nonlinear
multi-dimensional stochastic equations (SEs) in $\cR^p$ ($p\in [1,
\infty)$)
and backward stochastic differential equations (BSDEs) in $\cR^p\times
\cH^p$
($p\in (1, \infty)$) and in $\cR^\infty\times \bar{\cH^\infty}^{BMO}$,
with the
coefficients being allowed to be unbounded. In particular, the
probabilistic
version of Fefferman's inequality plays a crucial role in the
development of
our theory, which seems to be new. Several new results are consequently
obtained. The particular multi-dimensional linear case for SDEs and
BSDEs are
separately investigated, and the existence and uniqueness of a
solution is
connected to the property that the elementary solutions-matrix for the
associated homogeneous SDE satisfies the reverse H\"older inequality
for some
suitable exponent $p\ge 1$. Finally, we establish some relations between
Kazamaki's quadratic critical exponent $b(M)$ of a BMO martingale $M$
and the
spectral radius of the solution operator for the $M$-driven SDE, which
lead to
a characterization of Kazamaki's quadratic critical exponent of BMO
martingales
being infinite.
http://arxiv.org/abs/0801.3505
---------------------------------------------------------------
6604. MAJORIZING MEASURES AND PROPORTIONAL SUBSETS OF BOUNDED
ORTHONORMAL SYSTEMS
Olivier Guedon and Shahar Mendelson and Alain Pajor and Nicole
Tomczak-Jaegermann
In this article we prove that for any orthonormal system $
(\vphi_j)_{j=1}^n
\subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k <n$,
there exists
a subset $I$ of cardinality greater than $n-k$ such that on $\spa\
{\vphi_i\}_{i
\in I}$, the $L_1$ norm and the $L_2$ norm are equivalent up to a
factor $\mu
(\log \mu)^{5/2}$, where $\mu = \sqrt{n/k} \sqrt{\log k}$. The proof
is based
on a new estimate of the supremum of an empirical process on the unit
ball of a
Banach space with a good modulus of convexity, via the use of majorizing
measures.
http://arxiv.org/abs/0801.3556
---------------------------------------------------------------
6605. A CONCENTRATION INEQUALITY FOR INTERVAL MAPS WITH AN INDIFFERENT
FIXED POINT
J.-R. Chazottes and P. Collet and F. Redig and E. Verbitskiy
For a map of the unit interval with an indifferent fixed point, we
prove an
upper bound for the variance of all observables of $n$ variables
$K:[0,1]^n\to\R$ which are componentwise Lipschitz. The proof is based
on
coupling and decay of correlation properties of the map. We then give
various
applications of this inequality to the almost-sure central limit
theorem, the
kernel density estimation, the empirical measure and the periodogram.
http://arxiv.org/abs/0801.3567
---------------------------------------------------------------
6606. A UNIFYING FORMULATION OF THE FOKKER-PLANCK-KOLMOGOROV EQUATION
FOR GENERAL STOCHASTIC HYBRID SYSTEMS
Julien Bect
A general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for
stochastic hybrid systems is presented, within the framework of
Generalized
Stochastic Hybrid Systems (GSHS). The FPK equation describes the time
evolution
of the probability law of the hybrid state. Our derivation is based on
the
concept of mean jump intensity, which is related to both the usual
stochastic
intensity (in the case of spontaneous jumps) and the notion of
probability
current (in the case of forced jumps). This work unifies all
previously known
instances of the FPK equation for stochastic hybrid systems, and
provides GSHS
practitioners with a tool to derive the correct evolution equation for
the
probability law of the state in any given example.
http://arxiv.org/abs/0801.3725
---------------------------------------------------------------
6607. STABLE LAWS AND PRODUCTS OF POSITIVE RANDOM MATRICES
Hubert Hennion (Universit\'e de Rennes I) and Loic Herv\'e (Institut
National des Sciences Appliqu\'ees de Rennes)
Let $S$ be the multiplicative semigroup of $q\times q$ matrices with
positive
entries such that every row and every column contains a strictly
positive
element. Denote by $(X_n)_{n\geq1}$ a sequence of independent
identically
distributed random variables in $S$ and by $X^{(n)} = X_n ... X_1$, $ n
\geq 1$,
the associated left random walk on $S$. We assume that $(X_n)_{n\geq1}$
verifies the contraction property
$\P(\bigcup_{n\geq1}[X^{(n)} \in S^\circ])>0$,
where $S^\circ $ is the subset of all matrices which have strictly
positive
entries. We state conditions on the distribution of the random matrix
$X_1$
which ensure that the logarithms of the entries, of the norm, and of the
spectral radius of the products $X^{(n)}$, $n\ge 1$, are in the domain
of
attraction of a stable law.
http://arxiv.org/abs/0801.3780
---------------------------------------------------------------
6608. NOVEL BOUNDS ON MARGINAL PROBABILITIES
Joris M. Mooij and Hilbert J. Kappen
We derive two related novel bounds on single-variable marginal
probability
distributions in factor graphs with discrete variables. The first method
propagates bounds over a subtree of the factor graph rooted in the
variable,
and the second method propagates bounds over the self-avoiding walk tree
starting at the variable. By construction, both methods not only bound
the
exact marginal probability distribution of a variable, but also its
approximate
Belief Propagation marginal (``belief''). Thus, apart from providing a
practical means to calculate bounds on marginals, our contribution
also lies in
an increased understanding of the error made by Belief Propagation.
Empirically, we show that our bounds often outperform existing bounds
in terms
of accuracy and/or computation time. We also show that our bounds can
yield
nontrivial results for medical diagnosis inference problems.
http://arxiv.org/abs/0801.3797
---------------------------------------------------------------
6609. CONSTRUCTION AND UNIQUENESS FOR REFLECTED BSDE UNDER LINEAR
INCREASING CONDITION
G. Jia and Mingyu Xu
In this paper, we study the uniqueness of the solution of reflected
BSDE with
one or two barriers, under continuous and linear increasing condition of
generator $g$. Before that we study the construction of solution of of
reflected BSDE with one or two barriers.
http://arxiv.org/abs/0801.3718
---------------------------------------------------------------
6610. ALGORITHM FOR SOLVING OPTIMIZATION PROBLEMS WITH INTERVAL
VALUED PROBABILITY MEASURE
Phantipa Thipwiwatpotjana and Weldon A. Lodwick
We are concerned with three types of uncertainties: probabilistic,
possibilitistic and interval. By using possibility and necessity
measures as an
Interval Valued Probability Measure (IVPM), we present IVPM's interval
expected
values whose possibility distributions are in the form of polynomials.
By
working with interval expected values of independent uncertainty
coefficients
in a linear optimization problem together with operations suggested in
Lodwick
and Jamison (2007), the problem after applying these operations
becomes a
linear programming problem with constant coefficients. This is
achieved by the
application of two functions. The first is applied to the interval
coefficients, v: I -> R^k, where I= {[a,b] | a <= b}. The second is u:
R^k ->
R, applied to the product we got from a previous function. Similar
concepts
hold for any types of optimization problems with linear constraints.
Moreover,
it implied that optimization problems containing all three types of
uncertainties in one problem can be solved as ordinary optimization
problems.
http://arxiv.org/abs/0801.3816
---------------------------------------------------------------
6611. SMOOTH SOLUTIONS OF NON-LINEAR STOCHASTIC PARTIAL DIFFERENTIAL
EQUATIONS
Xicheng Zhang
In this paper, we study the regularities of solutions of nonlinear
stochastic
partial differential equations in the framework of Hilbert scales.
Then we
apply our general result to several typical nonlinear SPDEs such as
stochastic
Burgers and Ginzburg-Landau's equations on the real line, stochastic 2D
Navier-Stokes equations in the whole space and a stochastic tamed 3D
Navier-Stokes equation in the whole space, and obtain the existence of
their
respectively smooth solutions.
http://arxiv.org/abs/0801.3883
---------------------------------------------------------------
6612. AN LQ PROBLEM FOR THE HEAT EQUATION ON THE HALFLINE WITH
DIRICHLET BOUNDARY CONTROL AND NOISE
G. Fabbri and B. Goldys
A linear quadratic problem for a system governed by a heat equation
with a
Dirichlet boundary control and a Dirichlet boundary noise on halfline is
studied. To this end the problem is reformulated as a stochastic
evolution
equation in a certain weighted L2 space. An appropriate choice of
weight allows
us to prove a stronger regularity for the boundary terms appearing in
the
infinite dimensional state equation. The direct solution of the Riccati
equation related to the associated non-stochastic problem is used to
find the
solution of the problem in feedback form and to write the value
function of the
problem.
http://arxiv.org/abs/0801.3888
---------------------------------------------------------------
6613. EXIT PROBLEMS RELATED TO THE PERSISTENCE OF SOLITONS FOR THE
KORTEWEG-DE VRIES EQUATION WITH SMALL NOISE
Anne De Bouard (CMAP) and Eric Gautier (CREST)
We consider two exit problems for the Korteweg-de Vries equation
perturbed by
an additive white in time and colored in space noise of amplitude a. The
initial datum gives rise to a soliton when a=0. It has been proved
recently
that the solution remains in a neighborhood of a randomly modulated
soliton for
times at least of the order of a^{-2}. We prove exponential upper and
lower
bounds for the small noise limit of the probability that the exit time
from a
neighborhood of this randomly modulated soliton is less than T, of the
same
order in a and T. We obtain that the time scale is exactly the right
one. We
also study the similar probability for the exit from a neighborhood of
the
deterministic soliton solution. We are able to quantify the gain of
eliminating
the secular modes to better describe the persistence of the soliton.
http://arxiv.org/abs/0801.3894
---------------------------------------------------------------
6614. ON LARGE INTERSECTION AND SELF-INTERSECTION LOCAL TIMES IN
DIMENSION FIVE OR MORE
Amine Asselah
We show a remarkable similarity between strategies to realize a large
intersection or self-intersection local times in dimension five or
more. This
leads to the same rate functional for large deviation principles for
the two
objects obtained respectively by Chen and Morters, and by the present
author.
We also present a new estimate for the distribution of high level sets
for a
random walk, with application to the geometry of the intersection set
of two
high level sets of the local times of two independent random walks.
http://arxiv.org/abs/0801.3918
---------------------------------------------------------------
6615. ON THE INFIMUM CONVOLUTION INEQUALITY
Rafa{\l} Lata{\l}a and Jakub Onufry Wojtaszczyk
In the paper we study the infimum convolution inequalites. Such an
inequality
was first introduced by B. Maurey to give the optimal concentration of
measure
behaviour for the product exponential measure. We show how IC-
inequalities are
tied to concentration and study the optimal cost functions for an
arbitrary
probability measure. In particular, we show the optimal IC-inequality
for
product log-concave measures and for uniform measures on the l_p^n
balls. Such
an optimal inequality implies, for a given measure, in particular the
Central
Limit Theorem of Klartag and the tail estimates of Paouris.
http://arxiv.org/abs/0801.4036
---------------------------------------------------------------
6616. CARRY PROPAGATION IN MULTIPLICATION BY CONSTANTS
Alexander Izsak and Nicholas Pippenger
Suppose that a random n-bit number V is multiplied by an odd constant M,
greater than or equal to 3, by adding shifted versions of the number V
corresponding to the 1s in the binary representation of the constant
M. Suppose
further that the additions are performed by carry-save adders until
the number
of summands is reduced to two, at which time the final addition is
performed by
a carry-propagate adder. We show that in this situation the
distribution of the
length of the longest carry-propagation chain in the final addition is
the same
(up to terms tending to 0 as n tends to infinity) as when two
independent n-bit
numbers are added, and in particular the mean and variance are the
same (again
up to terms tending to 0). This result applies to all possible orders of
performing the carry-save additions.
http://arxiv.org/abs/0801.4040
---------------------------------------------------------------
6617. NO ARBITRAGE CONDITIONS FOR SIMPLE TRADING STRATEGIES
Erhan Bayraktar and Hasanjan Sayit
Strict local martingales may admit arbitrage opportunities with
respect to
the class of simple trading strategies. (Since there is no possibility
of using
doubling strategies in this framework, the losses are not assumed to
be bounded
from below.) We show that for a class of non-negative strict local
martingales,
the strong Markov property implies the no arbitrage property with
respect to
the class of simple trading strategies. This result can be seen as a
generalization of a similar result on three dimensional Bessel process
in [3].
We also pro- vide no arbitrage conditions for stochastic processes
within the
class of simple trading strategies with shortsale restriction.
http://arxiv.org/abs/0801.4047
---------------------------------------------------------------
6618. CRITICAL PERCOLATION ON CAYLEY GRAPHS OF GROUPS ACTING ON TREES
Iva Kozakova
This article presents a method for finding the critical probability
$p_c$ for
the Bernoulli bond percolation on graphs with the so called tree-like
structure. Such graphs can be decomposed into a tree of pieces which
have
finitely many isomorphism classes. This class of graphs includes the
Cayley
graphs of amalgamated products, HNN extensions or general groups
acting on
trees. It also includes all transitive graphs with more than one end.
The idea of the method is to find a multi-type Galton-Watson
branching
process (with a parameter $p$) which has finite expected population
size if and
only if the expected percolation cluster size is finite. This provides a
sufficient information about $p_c$. In particular if the pairwise
intersections
of pieces are finite, then $p_c$ is the smallest positive $p$ for which
$\det(M-1)=0$, where $M$ is the first-moment matrix of the branching
process.
If the pieces of the tree-like structure are finite, then $p_c$ is an
algebraic
number, and we give an algorithm computing $p_c$ as a root of some
algebraic
function.
We show that any Cayley graph of a group acting on a tree with
finite vertex
stabilizers with respect to any finite generating set has a tree-like
structure
with finite pieces. In particular we show how to compute $p_c$ of the
Cayley
graph of a free group with respect to any finite generating set.
http://arxiv.org/abs/0801.4153
---------------------------------------------------------------
6619. FROM COMBINATORICS TO LARGE DEVIATIONS FOR THE INVARIANT
MEASURES OF SOME MULTICLASS PARTICLE SYSTEMS
Davide Gabrielli
We prove large deviation principles (LDP) for the invariant measures
of the
multiclass totally asymmetric simple exclusion process (TASEP) and the
multiclass Hammersely-Aldous-Diaconis (HAD) process on a torus. The
proof is
based on a combinatorial representation of the measures in terms of a
\emph{collapsing procedure} introduced in \cite{A} for the 2-class
TASEP and
then generalized in \cite{FM1}, \cite{FM2} and \cite{FM3} to the
multiclass
TASEP and the multiclass HAD process. The rate functionals are written
in terms
of variational problems that we solve in the cases of 2-class processes.
http://arxiv.org/abs/0801.4156
---------------------------------------------------------------
6620. FORECASTING VOLATILITY WITH THE MULTIFRACTAL RANDOM WALK MODEL
Jean Duchon (IF) and Raoul Robert (IF) and Vincent Vargas (CEREMADE)
We study the problem of forecasting volatility for the multifractal
random
walk model. In order to avoid the ill posed problem of estimating the
correlation length T of the model, we introduce a limiting object
defined in a
quotient space; formally, this object is an infinite range
logvolatility. For
this object and the non limiting object, we obtain precise prediction
formulas
and we apply them to the problem of forecasting volatility and pricing
options
with the MRW model in the absence of a reliable estimate of the average
volatility and T.
http://arxiv.org/abs/0801.4220
---------------------------------------------------------------
6621. A PERMUTATION MODEL FOR FREE RANDOM VARIABLES AND ITS CLASSICAL
ANALOGUE
Florent Benaych-Georges (PMA) and Ion Nechita (ICJ)
In this paper, we generalize a permutation model for free random
variables
which was first proposed by Biane in \cite{biane}. We also construct its
classical probability analogue, by replacing the group of permutations
with the
group of subsets of a finite set endowed with the symmetric difference
operation. These models provide explicit examples of non random
matrices which
are asymptotically free or independent. The moments and the free (resp.
classical) cumulants of the limiting distributions are expressed in
terms of a
special subset of (noncrossing) pairings. At the end of the paper we
present
some combinatorial applications of our results.
http://arxiv.org/abs/0801.4229
---------------------------------------------------------------
6622. LOWER BOUNDS FOR TRANSITION PROBABILITIES ON GRAPHS
Andras Telcs
The paper presents two results. The first one provides separate
conditions
for the upper and lower estimate of the distribution of the exit time
from
balls of a random walk on a weighted graph. The main result of the
paper is
that the lower estimate follows from the elliptic Harnack inequality.
The
second result is an off-diagonal lower bound for the transition
probability of
the random walk.
http://arxiv.org/abs/0801.4260
---------------------------------------------------------------
6623. NECESSARY AND SUFFICIENT OPTIMALITY CONDITIONS FOR RELAXED AND
STRICT CONTROL PROBLEMS
Seid Bahlali
We consider a stochastic control problem where the set of strict
(classical)
controls is not necessarily convex, and the system is governed by a
nonlinear
stochastic differential equation, in which the control enters both the
drift
and the diffusion coefficients. By introducing a new approach, we
establish
necessary as well as sufficient conditions of optimality for two
models. The
first concerns the relaxed controls, who are a measure-valued
processes in
which an optimal solution exists. The second is a particular case of
the first
and relates to strict control problems. These results are given in the
form of
global stochastic maximum principle by using only the first order
expansion and
the associated adjoint equation. This improves all the previous works
on the
subject.
http://arxiv.org/abs/0801.4285
---------------------------------------------------------------
6624. NECESSARY AND SUFFICIENT OPTIMALITY CONDITIONS FOR RELAXED AND
STRICT CONTROL PROBLEMS OF FORWARD-BACKWARD SYSTEMS
Seid Bahlali
We consider a stochastic control problem of nonlinear forward-backward
systems, where the set of strict (classical) controls need not be
convex and
the coefficients depend explicitly on the variable control. By
introducing a
new approach, we establish necessary as well as sufficient conditions of
optimality, in the form of global stochastic maximum principle, for
two models.
The first concerns the relaxed controls, who are a measure-valued
processes.
The second is a restriction of the first to strict control problems
http://arxiv.org/abs/0801.4326
---------------------------------------------------------------
6625. THE STABILITY OF CONDITIONAL MARKOV PROCESSES AND MARKOV CHAINS
IN RANDOM ENVIRONMENTS
Ramon van Handel
We consider a discrete time hidden Markov model where the signal is a
stationary Markov chain. When conditioned on the observations, the
signal is a
Markov chain with stationary transition probabilities under the
conditional
measure. It is shown that this conditional signal is weakly ergodic
when the
signal is weakly ergodic and the observations are nondegenerate. This
permits a
delicate exchange of the intersection and supremum of sigma-fields,
which has
direct implications for the stability of nonlinear filters. The proof
relies on
an extension of results on the weak ergodicity of Markov chains in
random
environments to general state spaces. Finally it is shown that the
main results
can be lifted to the continuous time setting. The results partially
resolve a
long-standing gap in the proof of a result of H. Kunita (1971).
http://arxiv.org/abs/0801.4366
---------------------------------------------------------------
6626. ASYMPTOTICS FOR THE SURVIVAL PROBABILITY OF A ROUSE CHAIN MONOMER
G.Oshanin (LPTMC and University of Paris 6 and Paris and France)
We study the long-time asymptotical behavior of the survival
probability P_t
of a tagged monomer of an infinitely long Rouse chain in presence of
two fixed
absorbing boundaries, placed at x = \pm L. Mean-square displacement of
a tagged
monomer obeys \bar{X^2(t)} \sim t^{1/2} at all times, which signifies
that its
dynamics is an anomalous diffusion process.
Constructing lower and upper bounds on P_t, which have the same
time-dependence but slightly differ by numerical factors in the
definition of
the characteristic relaxation time, we show that P_t is a stretched-
exponential
function of time, \ln(P_t) \sim - t^{1/2}/L^2. This implies that the
distribution function of the first exit time from a fixed interval [-
L,L] for
such an anomalous diffusion has all moments.
http://arxiv.org/abs/0801.2914
---------------------------------------------------------------
6627. FIRST-EXIT-TIME PROBABILITY DENSITY TAILS FOR A LOCAL HEIGHT OF
A NON-EQUILIBRIUM GAUSSIAN INTERFACE
G.Oshanin (LPTMC and University of Paris 6 and France)
We study the long-time behavior of the probability density Q_t of the
first
exit time from a bounded interval [-L,L] for a stochastic non-
Markovian process
h(t) describing fluctuations at a given point of a two-dimensional,
infinite in
both directions Gaussian interface. We show that Q_t decays when t \to
\infty
as a power-law $^{-1 - \alpha}, where \alpha is non-universal and
proportional
to the ratio of the thermal energy and the elastic energy of a
fluctuation of
size L. The fact that \alpha appears to be dependent on L, which is
rather
unusual, implies that the number of existing moments of Q_t depends on
the size
of the window [-L,L]. A moment of an arbitrary order n, as a function
of L,
exists for sufficiently small L, diverges when L approaches a certain
threshold
value L_n, and does not exist for L > L_n. For L > L_1, the
probability density
Q_t is normalizable but does not have moments.
http://arxiv.org/abs/0801.3975
---------------------------------------------------------------
6628. CANONICAL MOMENTS AND RANDOM SPECTRAL MEASURES
Fabrice Gamboa Alain Rouault
We study some connections between the random moment problem and the
random
matrix theory. A uniform pick in a space of moments can be lifted into
the
spectral probability measure of the pair (A;e) where A is a random
matrix from
a classical ensemble and e is a fixed unit vec- tor. This random
measure is a
weighted sampling among the eigenvalues of A. We also study the large
deviations properties of this random measure when the dimension of the
matrix
grows. The rate function for these large deviations involves the
reversed
Kullback information.
http://arxiv.org/abs/0801.4400
---------------------------------------------------------------
6629. ON THE BIRTH-AND-ASSASSINATION PROCESS, WITH AN APPLICATION TO
SCOTCHING A RUMOR IN A NETWORK
Charles Bordenave
We give new formulas on the total number of born particles in the stable
birth-and-assassination process, and prove that it has an heavy-tailed
distribution. We also establish that this process is a scaling limit
of a
process of rumor scotching in a network, and is related to a predator-
prey
dynamics.
http://arxiv.org/abs/0801.4499
---------------------------------------------------------------
6630. SUBORDINATED DISCRETE SEMIGROUPS OF OPERATORS
Nick Dungey
Given a power-bounded linear operator T in a Banach space and a
probability F
on the non-negative integers, one can form a `subordinated' operator S
= \sum_k
F(k) T^k. We obtain asymptotic properties of the subordinated discrete
semigroup (S^n: n=1,2,...) under certain conditions on F. In
particular, we
study probabilities F with the property that S satisfies the Ritt
resolvent
condition whenever T is power-bounded. Examples and counterexamples of
this
property are discussed. The hypothesis of power-boundedness of T can
sometimes
be replaced by the weaker Kreiss resolvent condition.
http://arxiv.org/abs/0801.4557
---------------------------------------------------------------
6631. CONVEX ORDERING FOR RANDOM VECTORS USING PREDICTABLE
REPRESENTATION
Marc Arnaudon (LMA) and Jean-Christophe Breton (LMCA) and Nicolas
Privault
We prove convex ordering results for random vectors admitting a
predictable
representation in terms of a Brownian motion and a non-necessarily
independent
jump component. Our method uses forward-backward stochastic calculus and
extends previous results in the one-dimensional case. We also study a
geometric
interpretation of convex ordering for discrete measures in connection
with the
conditions set on the jump heights and intensities of the considered
processes.
http://arxiv.org/abs/0801.4621
---------------------------------------------------------------
6632. REFRACTED LEVY PROCESSES AND RUIN
Andreas E. Kyprianou and Ronnie Loeffen
Motivated by classical considerations from the theory of risk theory we
investigate the problem of ruin for a so-called refracted L\'evy
process. The
latter is a L\'evy processes whose dynamics change by subtracting off
a fixed
linear drift (of suitable size) whenever the aggregate process is
above a
pre-specified level. More formally, whenever it exists, a refracted L
\'evy
process is described by the unique weak solution to the stochastic
differential
equation \[ \D U_t = - \delta \mathbf{1}_{(U_t >b)}\D t + \D X_t \]
where
$X=\{X_t :t\geq 0\}$ is a L\'evy process with law $\mathbb{P}$ and $b,
\delta\in \mathbb{R}$ such that the resulting process $U$ may visit
the half
line $(b,\infty)$ with positive probability. In the light of
connection with a
certain dividend payment strategy on risk processes, we are particularly
interested in the case that $X$ is spectrally negative, $b>0$ and $0<
\delta<\mathbb{E}(X_1)$. For that case we provide some new identities
for
certain functionals of the path of the refracted process which are of
relevance
to the ruin problem.
http://arxiv.org/abs/0801.4655
---------------------------------------------------------------
6633. STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL PROBLEM OF
BACKWARD SYSTEMS WITH TERMINAL CONDITION IN L1
Seid Bahlali
We consider a stochastic control problem, where the control domain is
convex
and the system is governed by a nonlinear backward stochastic
differential
equation. With a L1 terminal data, we derive necessary optimality
conditions in
the form of stochastic maximum principle.
http://arxiv.org/abs/0801.4666
---------------------------------------------------------------
6634. THE STRICT AND RELAXED STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL
CONTROL PROBLEM OF BACKWARD SYSTEMS
Seid Bahlali
We consider a stochastic control problem where the set of controls is
not
necessarily convex and the system is governed by a nonlinear backward
stochastic differential equation. We establish necessary as well as
sufficient
conditions of optimality for two models. The first concerns the strict
(classical) controls. The second is an extension of the first to relaxed
controls, who are a measure valued processes.
http://arxiv.org/abs/0801.4668
---------------------------------------------------------------
6635. A GENERAL STOCHASTIC MAXIMUM PRINCIPLE FOR MIXED RELAXED-
SINGULAR CONTROL PROBLEMS
Seid Bahlali
We consider in this paper, mixed relaxed-singular stochastic control
problems, where the control variable has two components, the first being
measure-valued and the second singular. The control domain is not
necessarily
convex and the system is governed by a nonlinear stochastic differential
equation, in which the measure-valued part of the control enters both
the drift
and the diffusion coefficients. We establish necessary optimality
conditions,
of the Pontryagin maximum principle type, satisfied by an optimal
relaxed-singular control, which exist under general conditions on the
coefficients. The proof is based on the strict singular stochastic
maximum
principle established by Bahlali-Mezerdi, Ekeland's variational
principle and
some stability properties of the trajectories and adjoint processes with
respect to the control variable.
http://arxiv.org/abs/0801.4669
---------------------------------------------------------------
6636. GRADIENT ESTIMATE AND HARNACK INEQUALITY ON NON-COMPACT
RIEMANNIAN MANIFOLDS
Marc Arnaudon (LMA) and Anton Thalmaier and Feng-Yu Wang
A new type of gradient estimate is established for diffusion
semigroups on
non-compact complete Riemannian manifolds. As applications, a global
Harnack
inequality with power and a heat kernel estimate are derived for
diffusion
semigroups on arbitrary complete Riemannian manifolds.
http://arxiv.org/abs/0801.4708
---------------------------------------------------------------
6637. THE BERNOULLI SIEVE REVISITED
Alexander Gnedin and Alex Iksanov and Pavlo Negadajlov and Uwe Roesler
We consider an occupancy scheme in which `balls' are identified with $n$
points sampled from the standard exponential distribution, while the
role of
`boxes' is played by the spacings induced by an independent random
walk with
positive and non-lattice steps. We discuss the asymptotic behaviour of
five
quantities: the index $K_n^*$ of the last occupied box, the number $K_n
$ of
occupied boxes, the number $K_{n,0}$ of empty boxes whose index is at
most
$K_n^*$, the index $W_n$ of the first empty box and the number of
balls $Z_n$
in the last occupied box. It is shown that the limiting distribution of
properly scaled and centered $K_n^*$ coincides with that of the number
of
renewals not exceeding $\log n$. A similar result is shown for $K_n$
and $W_n$
under a side condition that prevents occurrence of very small boxes. The
condition also ensures that $K_{n,0}$ converges in distribution.
Limiting
results for
$Z_n$ are established under an assumption of regular variation.
http://arxiv.org/abs/0801.4725
---------------------------------------------------------------
6638. STOCHASTIC EXTREMA AS STATIONARY PHASES OF CHARACTERISTIC
FUNCTIONS
S. Nikitin
The paper is dealing with semi-classical asymptotics of a characteristic
function for a stochastic process. The main technical tool is provided
by the
stationary phase method. The extremal range for a stochastic process
is defined
by limit values of the complex logarithm of the characteristic
function. The
paper also outlines a numerical method for calculating stochastic
extrema.
http://arxiv.org/abs/0801.4726
---------------------------------------------------------------
6639. HEDGING STRATEGIES AND MINIMAL VARIANCE PORTFOLIOS FOR EUROPEAN
AND EXOTIC OPTIONS IN A LEVY MARKET
Wing Yan Yip and Sofia Olhede and David Stephens
This paper presents hedging strategies for European and exotic options
in a
Levy market. By applying Taylor's Theorem, dynamic hedging portfolios
are con-
structed under different market assumptions, such as the existence of
power
jump assets or moment swaps. In the case of European options or
baskets of
European options, static hedging is implemented. It is shown that
perfect
hedging can be achieved. Delta and gamma hedging strategies are
extended to
higher moment hedging by investing in other traded derivatives
depending on the
same underlying asset. This development is of practical importance as
such
other derivatives might be readily available. Moment swaps or power
jump assets
are not typically liquidly traded. It is shown how minimal variance
portfolios
can be used to hedge the higher order terms in a Taylor expansion of the
pricing function, investing only in a risk-free bank account, the
underlying
asset and potentially variance swaps. The numerical algorithms and
performance
of the hedging strategies are presented, showing the practical utility
of the
derived results.
http://arxiv.org/abs/0801.4941
---------------------------------------------------------------
6640. STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN
MOTION AND STANDARD BROWNIAN MOTION
Jo\~ao Guerra and David Nualart
We prove an existence and uniqueness theorem for solutions of
multidimensional, time dependent, stochastic differential equations
driven
simultaneously by a multidimensional fractional Brownian motion with
Hurst
parameter H>1/2 and a multidimensional standard Brownian motion. The
proof
relies on some a priori estimates, which are obtained using the
methods of
fractional integration, and the classical Ito stochastic calculus. The
existence result is based on the Yamada-Watanabe theorem.
http://arxiv.org/abs/0801.4963
---------------------------------------------------------------
6641. DEGENERATE STOCHASTIC DIFFERENTIAL EQUATIONS FOR CATALYTIC
BRANCHING NETWORKS
Sandra M. Kliem
Uniqueness of the martingale problem corresponding to a degenerate SDE
which
models catalytic branching networks is proven. This work is an
extension of a
paper by Dawson and Perkins to arbitrary catalytic branching networks.
As part
of the proof estimates on the corresponding semigroup are found in
terms of
weighted Holder norms for arbitrary networks, which are proven to be
equivalent
to the semigroup norm for this generalized setting.
-----
On prouve l'unicite d'un probleme de martingale correspondant a une
EDS
degeneree, qui apparait comme un modele de reseaux avec branchement
catalytique. Ce travail est une extension des resultats de Dawson et
Perkins au
cas de reseaux generaux. On obtient en particulier des estimees pour le
semi-groupe des reseaux generaux, sous forme de normes de Holder
ponderees; et
on etablit l'equivalence de ces normes avec des normes de semi-groupe
dans ce
contexte general.
http://arxiv.org/abs/0802.0035
---------------------------------------------------------------
6642. ON THE SUPREMUM OF RANDOM DIRICHLET POLYNOMIALS WITH
MULTIPLICATIVE COEFFICIENTS
Mikhail Lifshits and Michel Weber
We study the supremum of some random Dirichlet polynomials with
independent
coefficients and obtain sharp upper and lower bounds for supremum
expectation
thus extending the results from our previous work (see
http://arXiv.org/abs/math/0703691). Our approach in proving these
results is
entirely based on methods of stochastic processes, in particular the
metric
entropy method.
http://arxiv.org/abs/0802.0071
---------------------------------------------------------------
6643. LIMIT THEOREMS FOR LARGE DIMENSIONAL SAMPLE MEANS, SAMPLE
COVARIANCE MATRICES AND HOTELLING'S T^2 STATISTICS
Guangming Pan and Wang Zhou
In this paper, we prove the central limit theorem for Hotelling's $T^2$
statistics when the dimension of the random vectors is proportional to
the
sample size via investigating asymptotic independence and random
quadratic
forms involving sample means and sample covariance matrices.
http://arxiv.org/abs/0802.0082
---------------------------------------------------------------
6644. OCCUPATION TIME FLUCTUATION LIMITS OF INFINITE VARIANCE
EQUILIBRIUM BRANCHING SYSTEMS
Piotr Milos
We establish limit theorems for the fluctuations of the rescaled
occupation
time of a $(d,\alpha,\beta)$-branching particle system. It consists of
particles moving according to a symmetric $\alpha$-stable motion in
$\mathbb{R}^d$. The branching law is in the domain of attraction of a
(1+$\beta$)-stable law and the initial condition is an equilibrium
random
measure for the system (defined below). In the paper we treat
separately the
cases of intermediate $\alpha/\beta<d<(1+\beta)\alpha/\beta$, critical
$d=(1+\beta)\alpha/\beta$ and large $d>(1+\beta)\alpha/\beta $
dimensions. In
the most interesting case of intermediate dimensions we obtain a
version of a
fractional stable motion. The long-range dependence structure of this
process
is also studied. Contrary to this case, limit processes in critical
and large
dimensions have independent increments.
http://arxiv.org/abs/0802.0187
---------------------------------------------------------------
6645. V-VARIABLE FRACTALS: FRACTALS WITH PARTIAL SELF SIMILARITY
Michael Barnsley and John E. Hutchinson and \"Orjan Stenflo
We establish properties of a new type of fractal which has partial self
similarity at all scales. For any collection of iterated functions
systems with
an associated probability distribution and any positive integer V
there is a
corresponding class of V-variable fractal sets or measures. These V-
variable
fractals can be obtained from the points on the attractor of a single
deterministic iterated function system. Existence, uniqueness and
approximation
results are established under average contractive assumptions. We also
obtain
extensions of some basic results concerning iterated function systems.
http://arxiv.org/abs/0802.0064
---------------------------------------------------------------
6646. HARNACK INEQUALITY AND APPLICATIONS FOR STOCHASTIC EVOLUTION
EQUATIONS WITH MONOTONE DRIFT
Wei Liu
In this paper, the dimension-free Harnack inequality is proved for
transition
semigroups of solutions to a large class of stochastic evolution
equations with
monotone drift. As a conseqence, the strong Feller property, ergodic
property
and hyper-(or ultra-)contractivity are established for corresponding
semigroups. The main results can be applied to many concrete stochastic
evolution equations such as stochastic reaction-diffusion equation,
stochastic
p-Laplacian equation in Hilbert space.
http://arxiv.org/abs/0802.0289
---------------------------------------------------------------
6647. NONDIFFERENTIABLE FUNCTIONS OF ONE DIMENSIONAL SEMIMARTINGALES
George Lowther
In this paper we consider decompositions of processes of the form
Y=f(t,X)
where X is a one dimensional semimartingale, but f is not required to be
differentiable so Ito's formula does not apply.
First, in the case where f(t,x) is independent of t, we show that
requiring
it to be locally Lipschitz continuous in x is enough for an Ito style
decomposition to apply. This decomposes Y into a stochastic integral
term and a
term whose quadratic variation is well defined and has zero continuous
part.
For the time dependent case we show that the same decomposition
still holds
under the additional conditions that the left and right derivatives of
f(t,x)
in x exist, it is right-continuous in t, and that locally its
variation with
respect to t is integrable in x. In particular, in the continuous case
this
shows that Y is a Dirichlet process. We furthermore prove that such
processes
satisfy a decomposition into continuous martingale and purely
discontinuous
terms, and a Doob-Meyer style decomposition.
http://arxiv.org/abs/0802.0331
---------------------------------------------------------------
6648. IMPROVED MIXING TIME BOUNDS FOR THE THORP SHUFFLE AND L-REVERSAL
CHAIN
Ben Morris
We prove a theorem that reduces bounding the mixing time of a card
shuffle to
verifying a condition that involves only pairs of cards, then we use
it to
obtain improved bounds for two previously studied models.
E. Thorp introduced the following card shuffling model in 1973.
Suppose the
number of cards n is even. Cut the deck into two equal piles. Drop the
first
card from the left pile or from the right pile according to the
outcome of a
fair coin flip. Then drop from the other pile. Continue this way until
both
piles are empty. We obtain a mixing time bound of O(log^4 n).
Previously, the
best known bound was O(log^{29} n) and previous proofs were only valid
for n a
power of 2.
We also analyze the following model, called the L-reversal chain,
introduced
by Durrett. There are n cards arrayed in a circle. Each step, an
interval of
cards of length at most L is chosen uniformly at random and its order is
reversed. Durrett has conjectured that the mixing time is O(max(n, n^3/
L^3) log
n). We obtain a bound that is within a factor O(log^2 n) of this,the
first
bound within a poly log factor of the conjecture.
http://arxiv.org/abs/0802.0339
---------------------------------------------------------------
6649. EXACT EXPONENTIAL BOUNDS FOR THE RANDOM FIELD MAXIMUM
DISTRIBUTION VIA THE MAJORING MEASURES (GENERIC CHAINING)
E. Ostrovsky and E. Rogover
In this paper non-asymptotic exact exponential estimates are derived
for the
tail of maximum distribution of random field in the terms of majoring
measures
or, equally, generic chaining.
http://arxiv.org/abs/0802.0349
---------------------------------------------------------------
6650. JENSEN'S INEQUALITY FOR G-CONVEX FUNCTION UNDER G-EXPECTATION
Guangyan Jia and Shige Peng
A real valued function defined on}$\mathbb{R}$ {\small is called}$g$
{\small
--convex if it satisfies the following \textquotedblleft generalized
Jensen's
inequality\textquotedblright under a given}$g${\small -expectation,
i.e.,
}$h(\mathbb{E}^{g}[X])\leq \mathbb{E}% ^{g}[h(X)]${\small, for all
random
variables}$X$ {\small such that both sides of the inequality are
meaningful. In
this paper we will give a necessary and sufficient conditions for a
}$C^{2}${\small -function being}$% g ${\small -convex. We also studied
some
more general situations. We also studied}$g${\small -concave and}$g$
{\small
-affine functions.
http://arxiv.org/abs/0802.0373
---------------------------------------------------------------
6651. ABSOLUTE CONTINUITY AND SINGULARITY OF TWO PROBABILITY MEASURES
ON A FILTERED SPACE
S.S. Gabriyelyan
Let $\mu$ and $\nu$ be fixed probability measures on a filtered space
$(\Omega, ({\cal F}_t)_{t\in {\bf R}^{+}}, {\cal F})$. Denote by $
\mu_T $ and
$\nu_T $ (respectively, $\mu_{T-} $ and $\nu_{T-} $) the restrictions of
measures $\mu$ and $\nu$ on ${\cal F}_T $ (respectively, on ${\cal
F}_{T-} $)
for a stopping time $T$. We can find a Hahn-decomposition of $\mu_T $
and
$\nu_T $ using a Hahn-decomposition of measures $\mu$, $\nu$, and a
Hellinger
process $h_t$ in the strict sense of order 1/2. The norm of the
absolutely
continuity component of $\mu_{T-} $ relative to $\nu_{T-} $ in terms
of density
processes and Hellinger integrals is computed.
http://arxiv.org/abs/0802.0385
---------------------------------------------------------------
6652. A QUADRATIC REGRESSION PROBLEM FOR TWO-STATE ALGEBRAS WITH
APPLICATION TO THE CENTRAL LIMIT THEOREM
Marek Bozejko and Wlodzimierz Bryc
We extend a free version of the Laha-Lukacs theorem to probability
spaces
with two-states. We then use this result to generalize a
noncommutative CLT of
Kargin to the two-state setting.
http://arxiv.org/abs/0802.0266
---------------------------------------------------------------
6653. JACK POLYNOMIALS AND FREE CUMULANTS
Michel Lassalle (CNRS and Marne la Vallee and France)
We study the coefficients in the expansion of Jack polynomials in
terms of
power sums. We express them as polynomials in the free cumulants of the
transition measure of an anisotropic Young diagram. We conjecture that
such
polynomials have nonnegative integer coefficients. This extends recent
results
about normalized characters of the symmetric group.
http://arxiv.org/abs/0802.0448
---------------------------------------------------------------
6654. LARGE DEVIATIONS FOR THE STOCHASTIC SHELL MODEL OF TURBULENCE
U. Manna and S.S. Sritharan and and P. Sundar
In this work we first prove the existence and uniqueness of a strong
solution
to stochastic GOY model of turbulence with a small multiplicative
noise. Then
using the weak convergence approach, Laplace principle for so- lutions
of the
stochastic GOY model is established in certain Polish space. Thus a
Wentzell-Freidlin type large deviation principle is established
utilizing
certain results by Varadhan and Bryc.
http://arxiv.org/abs/0802.0585
---------------------------------------------------------------
6655. A UNIQUENESS THEOREM FOR SOLUTION OF BSDES
Guangyan Jia
In this note, we prove that if $g$ is uniformly continuous in $z$,
uniformly
with respect to $(\oo,t)$ and independent of $y$, the solution to the
backward
stochastic differential equation (BSDE) with generator $g$ is unique.
http://arxiv.org/abs/0802.0616
---------------------------------------------------------------
6656. MULTIFRACTIONAL, MULTISTABLE, AND OTHER PROCESSES WITH
PRESCRIBED LOCAL FORM
K.J. Falconer and J. Levy Vehel
We present a general method for constructing stochastic processes with
prescribed local form. Such processes include variable amplitude
multifractional Brownian motion, multifractional $\alpha$-stable
processes, and
multistable processes, that is processes that are locally $\alpha(t)$-
stable
but where the stability index $\alpha(t)$ varies with $t$. In
particular we
construct multifractional multistable processes where both the local
self-similarity and stability indices vary.
http://arxiv.org/abs/0802.0645
---------------------------------------------------------------
6657. STABILIZATION AND LIMIT THEOREMS FOR GEOMETRIC FUNCTIONALS OF
GIBBS POINT PROCESSES
T. Schreiber and J. E. Yukich
Given a Gibbs point process $\P^{\Psi}$ on $\R^d$ having a weak enough
potential $\Psi$, we consider the random measures $\mu_\la := \sum_{x
\in
\P^{\Psi} \cap Q_\la} \xi(x, \P^{\Psi} \cap Q_\la) \delta_{x/\la^{1/d}}
$, where
$Q_{\la} := [-\la^{1/d}/2,\la^{1/d}/2]^d$ is the volume $\la$ cube and
where
$\xi(\cdot,\cdot)$ is a translation invariant stabilizing functional.
Subject
to $\Psi$ satisfying a localization property and translation
invariance, we
establish weak laws of large numbers for $\la^{-1} \mu_\la(f)$, $f$ a
bounded
test function on $\R^d$, and weak convergence of $\la^{-1/2} \mu_
\la(f),$
suitably centered, to a Gaussian field acting on bounded test
functions. The
result yields limit laws for geometric functionals on Gibbs point
processes
including the Strauss and area interaction point processes as well as
more
general point processes defined by the Widom-Rowlinson and hard-core
model. We
provide applications to random sequential packing on Gibbsian input, to
functionals of Euclidean graphs, networks, and percolation models on
Gibbsian
input, and to quantization via Gibbsian input.
http://arxiv.org/abs/0802.0647
---------------------------------------------------------------
6658. FRACTIONAL CAUCHY PROBLEMS ON BOUNDED DOMAINS
Mark M. Meerschaert and Erkan Nane and Palaniappan Vellaisamy
Fractional Cauchy problems replace the usual first order time
derivative by a
fractional derivative. This paper develops classical solutions and
stochastic
analogues for fractional Cauchy problems in a bounded domain $D\subset
\rd$
with Dirichlet boundary conditions. Stochastic solutions are
constructed via an
inverse stable subordinator whose scaling index corresponds to the
order of the
fractional time derivative. Dirichlet problems corresponding to iterated
Brownian motion in a bounded domain are then solved by establishing a
correspondence with the case of a half-derivative in time.
http://arxiv.org/abs/0802.0673
---------------------------------------------------------------
6659. ON THE ASYMPTOTIC NORMALITY OF THE CONDITIONAL MAXIMUM
LIKELIHOOD ESTIMATORS FOR THE TRUNCATED REGRESSION MODEL AND THE
TOBIT MODEL
Chunlin Wang
In this paper, we study the asymptotic normality of the conditional
maximum
likelihood (ML) estimators for the truncated regression model and the
Tobit
model. We show that under the general setting assumed in his book, the
conjectures made by Hayashi (2000) \footnote{see page 516, and page
520 of
Hayashi (2000).} about the asymptotic normality of the conditional ML
estimators for both models are true, namely, a sufficient condition is
the
nonsingularity of $\mathbf{x_tx'_t}$.
http://arxiv.org/abs/0802.0536
---------------------------------------------------------------
6660. ON THE DISTRIBUTION OF THE DOMINATION NUMBER OF A NEW FAMILY OF
PARAMETRIZED RANDOM DIGRAPHS
E. Ceyhan and C. E. Priebe
We derive the asymptotic distribution of the domination number of a new
family of random digraph called proximity catch digraph (PCD), which has
application to statistical testing of spatial point patterns and to
pattern
recognition. The PCD we use is a parametrized digraph based on two
sets of
points on the plane, where sample size and locations of the elements
of one is
held fixed, while the sample size of the other whose elements are
randomly
distributed over a region of interest goes to infinity. PCDs are
constructed
based on the relative allocation of the random set of points with
respect to
the Delaunay triangulation of the other set whose size and locations
are fixed.
We introduce various auxiliary tools and concepts for the derivation
of the
asymptotic distribution. We investigate these concepts in one Delaunay
triangle
on the plane, and then extend them to the multiple triangle case. The
methods
are illustrated for planar data, but are applicable in higher
dimensions also.
http://arxiv.org/abs/0802.0617
---------------------------------------------------------------
6661. TIME--SPACE WHITE NOISE ELIMINATES GLOBAL SOLUTIONS IN REACTION
DIFFUSION EQUATIONS
Juli\'an Fern\'andez Bonder and Pablo Groisman
We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} +
(u_+)^p$ ($p>1$), with time--space white noise produces that solutions
explodes
with probability one for every initial datum, opposite to the
deterministic
model where a positive stationary solution exists.
http://arxiv.org/abs/0802.0633
---------------------------------------------------------------
6662. SEMICLASSICAL ANALYSIS OF A RANDOM WALK ON A MANIFOLD
G. Lebeau and L. Michel
We prove sharp rate of convergence to stationarity for a natural
random walk
on a compact Riemannian manifold (M,g). The proof includes a detailed
study of
the spectral theory of the associated operator.
http://arxiv.org/abs/0802.0644
---------------------------------------------------------------
6663. ON THE LOCAL TIME OF THE ASYMMETRIC BERNOULLI WALK
Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz
We study some properties of the local time of the asymmetric Bernoulli
walk
on the line. These properties are very similar to the corresponding
ones of the
simple symmetric random walks in higher ($d\geq3$) dimension, which we
established in the recent years. The goal of this paper is to
highlight these
similarities.
http://arxiv.org/abs/0802.0765
---------------------------------------------------------------
6664. TRANSIENT NEAREST NEIGHBOR RANDOM WALK AND BESSEL PROCESS
Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz
We prove strong invariance principle between a transient Bessel
process and a
certain nearest neighbor (NN) random walk that is constructed from the
former
by using stopping times. It is also shown that their local times are
close
enough to share the same strong limit theorems. It is shown
furthermore, that
if the difference between the distributions of two NN random walks are
small,
then the walks themselves can be constructed so that they are close
enough.
Finally, some consequences concerning strong limit theorems are
discussed.
http://arxiv.org/abs/0802.0778
---------------------------------------------------------------
6665. ON THE LAMPERTI STABLE PROCESSES
M.E. Caballero and J.C. Pardo and J.L. P\'erez
We consider a new family of $\R^d$-valued L\'{e}vy processes that we
call
Lamperti stable. One of the advantages of this class is that the law
of many
related functionals can be computed explicitely (see for instance
\cite{cc},
\cite{ckp}, \cite{kp} and \cite{pp}). This family of processes shares
many
properties with the tempered stable and the layered stable processes,
defined
in Rosi\'nski \cite{ro} and Houdr\'e and Kawai \cite{hok}
respectively, for
instance their short and long time behaviour. Additionally, in the
real valued
case we find a series representation which is used for sample paths
simulation.
In this work we find general properties of this class and we also
provide many
examples, some of which appear in recent literature.
http://arxiv.org/abs/0802.0851
---------------------------------------------------------------
6666. CONTINUOUS LOCAL TIME OF A PURELY ATOMIC IMMIGRATION
SUPERPROCESS WITH DEPENDENT SPATIAL MOTION
Zenghu Li and Jie Xiong
A purely atomic immigration superprocess with dependent spatial motion
in the
space of tempered measures is constructed as the unique strong
solution of a
stochastic integral equation driven by Poisson processes based on the
excursion
law of a Feller branching diffusion, which generalizes the work of
Dawson and
Li (2003). As an application of the stochastic equation, it is proved
that the
superprocess possesses a local time which is Holder continuous of order
$\alpha$ for every $\alpha< 1/2$. We establish two scaling limit
theorems for
the immigration superprocess, from which we derive scaling limits for
the
corresponding local time.
http://arxiv.org/abs/0802.0926
---------------------------------------------------------------
6667. STOCHASTIC EQUATIONS OF NON-NEGATIVE PROCESSES WITH JUMPS
Zongfei Fu and Zenghu Li
We study stochastic equations of non-negative processes with jumps. The
existence and uniqueness of strong solutions are established under
Lipschitz
and non-Lipschitz conditions. The comparison property of two solutions
are
proved under suitable conditions. The results are applied to stochastic
equations driven by one-sided Levy processes and those of continuous
state
branching processes with immigration.
http://arxiv.org/abs/0802.0933
---------------------------------------------------------------
6668. EXISTENCE OF NON-TRIVIAL HARMONIC FUNCTIONS ON CARTAN-HADAMARD
MANIFOLDS OF UNBOUNDED CURVATURE
Marc Arnaudon (LMA) and Anton Thalmaier and Stefanie Ulsamer
The Liouville property of a complete Riemannian manifold (i.e., the
question
whether there exist non-trivial bounded harmonic functions) attracted
a lot of
attention. For Cartan-Hadamard manifolds the role of lower curvature
bounds is
still an open problem. We discuss examples of Cartan-Hadamard
manifolds of
unbounded curvature where the limiting angle of Brownian motion
degenerates to
a single point on the sphere at infinity, but where nevertheless the
space of
bounded harmonic functions is as rich as in the non-degenerate case.
To see the
full boundary the point at infinity has to be blown up in a non-
trivial way.
Such examples indicate that the situation concerning the famous
conjecture of
Greene and Wu about existence of non-trivial bounded harmonic
functions on
Cartan-Hadamard manifolds is much more complicated than one might have
expected.
http://arxiv.org/abs/0802.0966
---------------------------------------------------------------
6669. FINITE SIZE SCALING FOR HOMOGENEOUS PINNING MODELS
Julien Sohier (PMA)
Pinning models are built from discrete renewal sequences by rewarding
(or
penalizing) the trajectories according to their number of renewal
epochs up to
time $N$, and $N$ is then sent to infinity. They are statistical
mechanics
models to which a lot of attention has been paid both because they are
very
relevant for applications and because of their {\sl exactly solvable
character}, while displaying a non-trivial phase transition (in fact, a
localization transition). The order of the transition depends on the
tail of
the inter-arrival law of the underlying renewal and the transition is
continuous when such a tail is sufficiently heavy: this is the case on
which we
will focus. The main purpose of this work is to give a mathematical
treatment
of the {\sl finite size scaling limit} of pinning models, namely
studying the
limit (in law) of the process close to criticality when the system
size is
proportional to the correlation length.
http://arxiv.org/abs/0802.1040
---------------------------------------------------------------
6670. REPRESENTATION OF THE PENALTY TERM OF DYNAMIC CONCAVE UTILITIES
Freddy Delbaen and Shige Peng and Emanuela Rosazza Gianin
In this paper we will provide a representation of the penalty term of
general
dynamic concave utilities (hence of dynamic convex risk measures) by
applying
the theory of g-expectations.
http://arxiv.org/abs/0802.1121
---------------------------------------------------------------
6671. HIDING THE DRIFT
Miklos Rasonyi and Walter Schachermayer and Richard Warnung
In this article we consider a Brownian motion with drift, denoted by
$S =
(S_t)_{t\ge0}$, of the form $dS_t = \mu_t dt + dB_t \qquad \text{for}
t \ge 0,$
with a specific non-trivial drift predictable with respect to $
\mathbb{F}^B$,
the natural filtration of the Brownian motion $B = (B_t)_{t\ge0}$. We
construct
a process $H = (H_t)_{t\ge0}$ also predictable with respect to $
\mathbb{F}^B$
such that $((H \cdot S)_t)_{t\ge 0}$ is a Brownian motion in its own
filtration. Furthermore, for any $\delta>0$, we refine this
construction such
that the drift $(\mu_t)_{t\ge0}$ only takes values in $]\mu-\delta,\mu+
\delta[$
for fixed $\mu>0$.
http://arxiv.org/abs/0802.1152
---------------------------------------------------------------
6672. LIMIT THEOREMS FOR HYBRIDIZATION REACTIONS ON OLIGONUCLEOTIDE
MICROARRAYS
Grzegorz A. Rempala and Iwona Pawlikowska
We derive herein the limiting laws for certain stationary
distributions of
birth-and-death processes related to the classical model of chemical
adsorption-desorption reactions due to Langmuir. The model has been
recently
considered in the context of a hybridization reaction on an
oligonucleotide DNA
microarray. Our results imply that the truncated gamma- and beta- type
distributions can be used as approximations to the observed
distributions of
the fluorescence readings of the oligo-probes on a microarray. These
findings
might be useful in developing new model-based, probe-specific methods of
extracting target concentrations from array fluorescence readings.
http://arxiv.org/abs/0802.1192
---------------------------------------------------------------
6673. LARGE DEVIATIONS OF LATTICE HAMILTONIAN DYNAMICS COUPLED TO
STOCHASTIC THERMOSTATS
T. Bodineau and R. Lefevere
We discuss the Donsker-Varadhan theory of large deviations in the
framework
of Hamiltonian systems thermostated by a Gaussian stochastic coupling.
We
derive a general formula for the Donsker-Varadhan large deviation
functional
for dynamics which satisfy natural properties under time reversal.
Next, we
discuss the characterization of the stationary state as the solution
of a
variational principle and its relation to the minimum entropy production
principle. Finally, we compute the large deviation functional of the
current in
the case of a harmonic chain thermostated by a Gaussian stochastic
coupling.
http://arxiv.org/abs/0802.1104
---------------------------------------------------------------
6674. FUNCTION SPACES AND CAPACITY RELATED TO A SUBLINEAR
EXPECTATION: APPLICATION TO G-BROWNIAN MOTION PATHES
Laurent Denis and Mingshang Hu and Shige Peng
In this paper we give some basic and important properties of several
typical
Banach spaces of functions of $G$-Brownian motion pathes induced by a
sublinear
expectation--G-expectation. Many results can be also applied to more
general
situations. A generalized version of Kolmogorov's criterion for
continuous
modification of a stochastic process is also obtained.
http://arxiv.org/abs/0802.1240
---------------------------------------------------------------
6675. FRACTIONAL TERM STRUCTURE MODELS: NO-ARBITRAGE AND CONSISTENCY
Alberto Ohashi
In this work we introduce Heath-Jarrow-Morton (HJM) interest rate models
driven by fractional Brownian motions. By using support arguments we
prove that
the resulting model is arbitrage-free under proportional transaction
costs in
the same spirit of Guasoni et al (2006, 2007). In particular, we
obtain a drift
condition which is similar in nature to the classical HJM no-arbitrage
drift
restriction.
The second part of this paper deals with consistency problems
related to the
fractional HJM dynamics. We give a fairly complete characterization of
finite-dimensional invariant manifolds for HJM models with fractional
Brownian
motion by means of Nagumo-type conditions. As an application, we
investigate
consistency of Nelson-Siegel family with respect to Ho-Lee and Hull-
White
models. It turns out that similar to the Brownian case such family
does not go
well with the fractional HJM dynamics with deterministic volatility.
In fact,
there is no nontrivial fractional interest rate model consistent with
the
Nelson-Siegel family.
http://arxiv.org/abs/0802.1288
---------------------------------------------------------------
6676. A GENERALIZATION OF DOOB'S MAXIMAL IDENTITY
Ashkan Nikeghbali
In this paper, using martingale techniques, we prove a generalization of
Doob's maximal identity in the setting of continuous nonnegative local
submartingales $(X_{t})$ of the form: $X_{t}=N_{t}+A_{t}$, where the
measure
$(dA_{t})$ is carried by the set $\left\{t: X_{t}=0\right\}$. In
particular, we
give a multiplicative decomposition for the Az\'ema supermartingale
associated
with some last passage times related to such processes and we prove
that these
non-stopping times contain very useful information. As a consequence,
we obtain
the law of the maximum of a continuous nonnegative local martingale $
(M_t)$
which satisfies $M_\infty=\psi(\sup_{t\geq0}M_t)$ for some measurable
function
$\psi$ as well as the law of the last time this maximum is reached.
http://arxiv.org/abs/0802.1317
---------------------------------------------------------------
6677. LARGE DEVIATIONS FOR THE BOUSSINESQ EQUATIONS UNDER RANDOM
INFLUENCES
Jinqiao Duan (IIT) and Annie Millet (CES and Samos and Pma)
A Boussinesq model for the Benard convection under random influences is
considered as a system of stochastic partial differential equations.
This is a
coupled system of stochastic Navier-Stokes equations and the transport
equation
for temperature. Large deviations are proved, using a weak convergence
approach
based on a variational representation of functionals of infinite
dimensional
Brownian motion.
http://arxiv.org/abs/0802.1335
---------------------------------------------------------------
6678. ASYMPTOTICS OF THE SPECTRAL GAP FOR THE INTERCHANGE PROCESS ON
LARGE HYPERCUBES
Shannon Starr and Matt Conomos
We consider the interchange process (IP) on the $d$-dimensional,
discrete
hypercube of side-length $n$. Specifically, we compare the spectral
gap of the
IP to the spectral gap of the random walk (RW) on the same graph. We
prove that
the two spectral gaps are asymptotically equivalent, in the limit $n \to
\infty$. This result gives further supporting evidence for a
conjecture of
Aldous, that the spectral gap of the IP equals the spectral gap of the
RW on
all finite graphs. Our proof is based on an argument invented by
Handjani and
Jungreis, who proved Aldous's conjecture for all trees.
http://arxiv.org/abs/0802.1368
---------------------------------------------------------------
6679. EXPLICIT COMPUTATIONS FOR A FILTERING PROBLEM WITH POINT
PROCESS OBSERVATIONS WITH APPLICATIONS TO CREDIT RISK
Vincent Leijdekker and Peter Spreij
We consider the intensity-based approach for the modeling of default
times of
one or more companies. In this approach the default times are defined
as the
jump times of a Cox process, which is a Poisson process conditional on
the
realization of its intensity. We assume that the intensity follows the
Cox-Ingersoll-Ross model. This model allows one to calculate survival
probabilities and prices of defaultable bonds explicitly. In this
paper we
assume that the Brownian motion, that drives the intensity, is not
observed.
Using filtering theory for point process observations, we are able to
derive
dynamics for the intensity and its moment generating function, given the
observations of the Cox process. A transformation of the dynamics of the
conditional moment generating function allows us to solve the filtering
problem, between the jumps of the Cox process, as well as at the jumps.
Assuming that the initial distribution of the intensity is of the
Gamma type,
we obtain an explicit solution to the filtering problem for all t>0. We
conclude the paper with the observation that the resulting conditional
moment
generating function at time t corresponds to a mixture of Gamma
distributions.
http://arxiv.org/abs/0802.1407
---------------------------------------------------------------
6680. LAWS OF LARGE NUMBERS FOR EPIDEMIC MODELS WITH COUNTABLY MANY
TYPES
A.D. Barbour and M.J. Luczak
In modelling parasitic diseases, it is natural to distinguish hosts
according
to the number of parasites that they carry, leading to a countably
infinite
type space. Proving the analogue of the deterministic equations, used
in models
with finitely many types as a `law of large numbers' approximation to
the
underlying stochastic model, has previously either been done case by
case,
using some special structure, or else not attempted. In this paper, we
prove a
general theorem of this sort, and complement it with a rate of
convergence in
the $\ell_1$-norm.
http://arxiv.org/abs/0802.1478
---------------------------------------------------------------
6681. LEARNING NONSINGULAR PHYLOGENIES AND HIDDEN MARKOV MODELS
Elchanan Mossel and S\'{e}bastien Roch
In this paper we study the problem of learning phylogenies and hidden
Markov
models. We call a Markov model nonsingular if all transition matrices
have
determinants bounded away from 0 (and 1). We highlight the role of the
nonsingularity condition for the learning problem. Learning hidden
Markov
models without the nonsingularity condition is at least as hard as
learning
parity with noise, a well-known learning problem conjectured to be
computationally hard. On the other hand, we give a polynomial-time
algorithm
for learning nonsingular phylogenies and hidden Markov models.
http://arxiv.org/abs/cs/0502076
---------------------------------------------------------------
6682. ON LEARNING THRESHOLDS OF PARITIES AND UNIONS OF RECTANGLES IN
RANDOM WALK MODELS
S. Roch
In a recent breakthrough, [Bshouty et al., 2005] obtained the first
passive-learning algorithm for DNFs under the uniform distribution.
They showed
that DNFs are learnable in the Random Walk and Noise Sensitivity
models. We
extend their results in several directions. We first show that
thresholds of
parities, a natural class encompassing DNFs, cannot be learned
efficiently in
the Noise Sensitivity model using only statistical queries. In
contrast, we
show that a cyclic version of the Random Walk model allows to learn
efficiently
polynomially weighted thresholds of parities. We also extend the
algorithm of
Bshouty et al. to the case of Unions of Rectangles, a natural
generalization of
DNFs to $\{0,...,b-1\}^n$.
http://arxiv.org/abs/cs/0605048
---------------------------------------------------------------
6683. INCOMPLETE LINEAGE SORTING: CONSISTENT PHYLOGENY ESTIMATION
FROM MULTIPLE LOCI
Elchanan Mossel and Sebastien Roch
We introduce a simple algorithm for reconstructing phylogenies from
multiple
gene trees in the presence of incomplete lineage sorting, that is,
when the
topology of the gene trees may differ from that of the species tree.
We show
that our technique is statistically consistent under standard stochastic
assumptions, that is, it returns the correct tree given sufficiently
many
unlinked loci. We also show that it can tolerate moderate estimation
errors.
http://arxiv.org/abs/0710.0262
---------------------------------------------------------------
6684. PHYLOGENIES WITHOUT BRANCH BOUNDS: CONTRACTING THE SHORT,
PRUNING THE DEEP
Constantinos Daskalakis and Elchanan Mossel and Sebastien Roch
We introduce a new phylogenetic reconstruction algorithm which, unlike
most
previous rigorous inference techniques, does not rely on assumptions
regarding
the branch lengths or the depth of the tree. The algorithm returns a
forest
which is guaranteed to contain all edges that are: 1) sufficiently
long and 2)
sufficiently close to the leaves. How much of the true tree is recovered
depends on the sequence length provided. The algorithm is distance-
based and
runs in polynomial time.
http://arxiv.org/abs/0801.4190
---------------------------------------------------------------
6685. SHRINKAGE EFFECT IN ANCESTRAL MAXIMUM LIKELIHOOD
Elchanan Mossel and Sebastien Roch and Mike Steel
Ancestral maximum likelihood (AML) is a method that simultaneously
reconstructs a phylogenetic tree and ancestral sequences from extant
data
(sequences at the leaves). The tree and ancestral sequences maximize the
probability of observing the given data under a Markov model of sequence
evolution, in which branch lengths are also optimized but constrained
to take
the same value on any edge across all sequence sites. AML differs from
the more
usual form of maximum likelihood (ML) in phylogenetics because ML
averages over
all possible ancestral sequences. ML has long been known to be
statistically
consistent -- that is, it converges on the correct tree with probability
approaching 1 as the sequence length grows. However, the statistical
consistency of AML has not been formally determined, despite informal
remarks
in a literature that dates back 20 years. In this short note we prove
a general
result that implies that AML is statistically inconsistent. In
particular we
show that AML can `shrink' short edges in a tree, resulting in a tree
that has
no internal resolution as the sequence length grows. Our results apply
to any
number of taxa.
http://arxiv.org/abs/0802.0914
---------------------------------------------------------------
6686. PROPERTIES OF THE DENSITY FOR A THREE DIMENSIONAL STOCHASTIC
WAVE EQUATION
Marta Sanz-Sol\'e
We consider a stochastic wave equation in space dimension three driven
by a
noise white in time and with an absolutely continuous correlation
measure given
by the product of a smooth function and a Riesz kernel. Let $p_{t,x}(y)
$ be the
density of the law of the solution $u(t,x)$ of such an equation at
points
$(t,x)\in]0,T]\times \IR^3$. We prove that the mapping $(t,x)\mapsto
p_{t,x}(y)$ owns the same regularity as the sample paths of the process
$\{u(t,x), (t,x)\in]0,T]\times \mathbbR^3\}$ established Dalang and
Sanz-Sol\'e
[Memoirs of the AMS, to appear]. The proof relies on Malliavin
calculus and
more explicitely, Watanabe's integration by parts formula and
estimates derived
form it.
http://arxiv.org/abs/0802.1607
---------------------------------------------------------------
6687. ASYMPTOTIC EQUIVALENCE AND CONTIGUITY OF SOME RANDOM GRAPHS
Svante Janson
We show that asymptotic equivalence, in a strong form, holds between two
random graph models with slightly differing edge probabilities under
substantially weaker conditions than what might naively be expected.
One application is a simple proof of a recent result by van den
Esker, van
der Hofstad and Hooghiemstra on the equivalence between graph
distances for
some random graph models.
http://arxiv.org/abs/0802.1637
---------------------------------------------------------------
6688. ASCENDING RUNS IN DEPENDENT UNIFORMLY DISTRIBUTED RANDOM
VARIABLES: APPLICATION TO WIRELESS NETWORKS
Nathalie Mitton (INRIA Futurs) and Katy Paroux (LM-Besan\c{c}on) and
Bruno Sericola (IRISA), S\'ebastien Tixeuil (INRIA Futurs)
We analyze in this paper the longest increasing contiguous sequence or
maximal ascending run of random variables with common uniform
distribution but
not independent. Their dependence is characterized by the fact that two
successive random variables cannot take the same value. Using a Markov
chain
approach, we study the distribution of the maximal ascending run and
we develop
an algorithm to compute it. This problem comes from the analysis of
several
self-organizing protocols designed for large-scale wireless sensor
networks,
and we show how our results apply to this domain.
http://arxiv.org/abs/0802.1387
---------------------------------------------------------------
6689. CONVERGENCE OF SOME LEADER ELECTION ALGORITHMS
Svante Janson and Christian Lavault and Guy Louchard
We start with a set of n players. With some probability P(n,k), we
kill n-k
players; the other ones stay alive, and we repeat with them. What is the
distribution of the number X_n of phases (or rounds) before getting
only one
player? We present a probabilistic analysis of this algorithm under some
conditions on the probability distributions P(n,k), including stochastic
monotonicity and the assumption that roughly a fixed proportion alpha
of the
players survive in each round.
We prove a kind of convergence in distribution for X_n-log_a n,
where the
basis a=1/alpha; as in many other similar problems there are
oscillations and
no true limit distribution, but suitable subsequences converge, and
there is an
absolutely continuous random variable Z such that the distribution of
X_n can
be approximated by Z+log_a n rounded to the nearest larger integer.
Applications of the general result include the leader election
algorithm
where players are eliminated by independent coin tosses and a
variation of the
leader election algorithm proposed by W.R. Franklin. We study the latter
algorithm further, including numerical results.
http://arxiv.org/abs/0802.1389
---------------------------------------------------------------
6690. A TRANSFERENCE METHOD IN QUANTUM PROBABILITY
Marius Junge and Javier Parcet
Working with a rather general notion of independence, we provide a
transference method which allows to compare the p-norm of sums of
independent
copies with the p-norm of sums of free copies. Our main technique is to
construct explicit operator space Lp embeddings preserving
independence to
reduce the problem to L1, where some recent results by the first-named
author
can be used. We find applications for noncommutative Khincthine/
Rosenthal type
inequalities and for noncommutative Lp embedding theory.
http://arxiv.org/abs/0802.1593
---------------------------------------------------------------
6691. CONDITIONS FOR STABILITY AND INSTABILITY OF RETRIAL QUEUEING
SYSTEMS WITH GENERAL RETRIAL TIMES
Tewfik Kernane (USTHB)
We study the stability of single server retrial queues under general
distribution for retrial times and stationary ergodic service times,
for three
main retrial policies studied in the literature: classical linear,
constant and
control policies. The approach used is the renovating events approach
to obtain
sufficient stability conditions by strong coupling convergence of the
process
modeling the dynamics of the system to a unique stationary ergodic
regime. We
also obtain instability conditions by convergence in distribution to
improper
limiting sequences.
http://arxiv.org/abs/0802.1812
---------------------------------------------------------------
6692. MOMENT EXPLOSIONS AND LONG-TERM BEHAVIOR OF AFFINE STOCHASTIC
VOLATILITY MODELS
Martin Keller-Ressel
We consider a class of asset pricing models, where the risk-neutral
joint
process of log-price and its stochastic variance is an affine process
in the
sense of Duffie, Filipovic and Schachermayer [2003]. First we obtain
conditions
for the price process to be conservative and a martingale. Then we
present some
results on the long-term behavior of the model, including an
expression for the
invariant distribution of the stochastic variance process. We study
moment
explosions of the price process, and provide explicit expressions for
the time
at which a moment of given order becomes infinite. We discuss
applications of
these results, in particular to the asymptotics of the implied
volatility
smile, and conclude with some calculations for the Heston model, a
model of
Bates and the Barndorff-Nielsen-Shephard model.
http://arxiv.org/abs/0802.1823
---------------------------------------------------------------
6693. LEVY-SHEFFER SYSTEMS AND THE LONGSTAFF-SCHWARTZ ALGORITHM FOR
AMERICAN OPTION PRICING
Stefan Gerhold
Glasserman and Yu (Ann. Appl. Probab. 14, 2004, p. 2090) have
investigated
the mean square error in the Longstaff-Schwartz algorithm for American
option
pricing, assuming that the underlying process is (geometric) Brownian
motion.
In this note we provide similar convergence results for the standard
Poisson,
Gamma, Pascal, and Meixner processes, pointing out the connection of the
problem to the L\'evy-Sheffer systems introduced by Schoutens.
http://arxiv.org/abs/0802.1831
---------------------------------------------------------------
6694. ASYMPTOTIC BEHAVIOUR OF RANDOMLY REFLECTING BILLIARDS IN
UNBOUNDED TUBULAR DOMAINS
Mikhail V. Menshikov and Marina Vachkovskaia and Andrew R. Wade
We study stochastic billiards in infinite planar domains with
curvilinear
boundaries: that is, piecewise deterministic motion with randomness
introduced
via random reflections at the domain boundary. Physical motivation for
the
process originates with ideal gas models in the Knudsen regime, with
particles
reflecting off microscopically rough surfaces. We classify the process
into
recurrent and transient cases. We also give almost-sure results on the
long-term behaviour of the location of the particle, including a
super-diffusive rate of escape in the transient case. A key step in
obtaining
our results is to relate our process to an instance of a one-dimensional
stochastic process with asymptotically zero drift, for which we prove
some new
almost-sure bounds of independent interest. We obtain some of these
bounds via
an application of general semimartingale criteria, which are again of
some
independent interest.
http://arxiv.org/abs/0802.1865
---------------------------------------------------------------
6695. MARKOVIAN EMBEDDINGS OF GENERAL RANDOM STRINGS
Manuel Lladser
Let A be a finite set and X a sequence of A-valued random variables.
We do
not assume any particular correlation structure between these random
variables;
in particular, X may be a non-Markovian sequence. An adapted embedding
of X is
a sequence of the form R(X_1), R(X_1,X_2), R(X_1,X_2,X_3), etc where R
is a
transformation defined over finite length sequences. In this extended
abstract
we characterize a wide class of adapted embeddings of X that result in a
first-order homogeneous Markov chain. We show that any transformation
R has a
unique coarsest refinement R' in this class such that R'(X_1),
R'(X_1,X_2),
R'(X_1,X_2,X_3), etc is Markovian. (By refinement we mean that
R'(u)=R'(v)
implies R(u)=R(v), and by coarsest refinement we mean that R' is a
deterministic function of any other refinement of R in our class of
transformations.) We propose a specific embedding that we denote as
R^X which
is particularly amenable for analyzing the occurrence of patterns
described by
regular expressions in X. A toy example of a non-Markovian sequence of
0's and
1's is analyzed thoroughly: discrete asymptotic distributions are
established
for the number of occurrences of a certain regular pattern in
X_1,...,X_n, as n
tends to infinity, whereas a Gaussian asymptotic distribution is shown
to apply
for another regular pattern.
http://arxiv.org/abs/0802.1896
---------------------------------------------------------------
6696. ON A THEOREM OF V. BERNIK IN THE METRICAL THEORY OF DIOPHANTINE
APPROXIMATION
Victor Beresnevich
This paper goes back to a famous problem of Mahler in metrical
Diophantine
approximation. The problem has been settled by Sprindzuk and
subsequently
improved by Alan Baker and Vasili Bernik. In particular, Bernik's result
establishes a convergence Khintchine type theorem for Diophantine
approximation
by polynomials, that is it allows arbitrary monotonic error of
approximation.
In the present paper the monotonicity assumption is completely removed.
http://arxiv.org/abs/0802.1910
---------------------------------------------------------------
6697. DIFFEOMORPHISMS OF THE CIRCLE AND BROWNIAN MOTIONS ON AN
INFINITE-DIMENSIONAL SYMPLECTIC GROUP
Maria Gordina and Mang Wu
An embedding of the group $\Diff(S^{1})$ of orientation preserving
diffeomorphims of the unit circle $S^1$ into an infinite-dimensional
symplectic
group, $\Sp(\infty)$, is studied. The authors prove that this
embedding is not
surjective. A Brownian motion is constructed on $\Sp(\infty)$. This
study is
motivated by recent work of H. Airault, S. Fang and P. Malliavin.
http://arxiv.org/abs/0802.1955
---------------------------------------------------------------
6698. ON THE KERT\'ESZ LINE: SOME RIGOROUS BOUNDS
Jean Ruiz (CPT) and Marc Wouts (MODAL'x)
We study the Kert\'esz line of the $q$--state Potts model at (inverse)
temperature $\beta$, in presence of an external magnetic field $h$.
This line
separates two regions of the phase diagram according to the existence
or not of
an infinite cluster in the Fortuin-Kasteleyn representation of the
model. It is
known that the Kert\'esz line $h_K (\beta)$ coincides with the line of
first
order phase transition for small fields when $q$ is large enough. Here
we prove
that the first order phase transition implies a jump in the density of
the
infinite cluster, hence the Kert\'esz line remains below the line of
first
order phase transition. We also analyze the region of large fields and
prove,
using techniques of stochastic comparisons, that $h_K (\beta)$ equals $
\log (q
- 1) - \log (\beta - \beta_p)$ to the leading order, as $\beta$ goes to
$\beta_p = - \log (1 - p_c)$ where $p_c$ is the threshold for bond
percolation.
http://arxiv.org/abs/0802.1826
---------------------------------------------------------------
6699. NON-HOMOGENEOUS POLYGONAL MARKOV FIELDS IN THE PLANE: GRAPHICAL
REPRESENTATIONS AND GEOMETRY OF HIGHER ORDER CORRELATIONS
Tomasz Schreiber
We consider polygonal Markov fields originally introduced by Arak and
Surgailis (1989). Our attention is focused on fields with nodes of
order two,
which can be regarded as continuum ensembles of non-intersecting
contours in
the plane, sharing a number of features with the two-dimensional Ising
model.
We introduce non-homogeneous version of polygonal fields in anisotropic
enviroment. For these fields we provide a class of new graphical
constructions
and random dynamics. These include a generalised dynamic representation,
generalised and defective disagreement loop dynamics as well as a
generalised
contour birth and death dynamics. Next, we use these constructions as
tools to
obtain new exact results on the geometry of higher order correlations of
polygonal Markov fields in their consistent regime.
http://arxiv.org/abs/0802.2115
---------------------------------------------------------------
6700. META-STABILITY AND CONDENSED ZERO-RANGE PROCESSES ON FINITE SETS
J. Beltran and C. Landim
We propose a definition o meta-stability and obtain sufficient
conditions for
a sequence of Markov processes on finite state spaces to be meta-
stable. In the
reversible case, these conditions reduce to estimates of the capacity
and the
measure of certain meta-stable sets. We prove that a class of condensed
zero-range processes with asymptotically decreasing jump rates is meta-
stable.
http://arxiv.org/abs/0802.2171
---------------------------------------------------------------
6701. EFFICIENT HEDGING AND RISK MINIMIZATION
Marie-Amelie Morlais
In that paper, we solve dynamically a partial hedging problem for an
American
contingent claim: assuming superhedging is not feasible, we explain in
this
context the notion of efficient hedging by introducing a risk
minimization
criterion: we consider here the problem of minimizing the conditional
expected
loss for a given convex and non decreasing loss function. To solve this
problem, we provide a connection between the dynamic convex risk
functional
introduced and the solution of a quadratic RBSDE (Reflected Backward
Stochastic
Differential Equations): this is achieved by studying the properties of
specific non linear expectations.
http://arxiv.org/abs/0802.2172
---------------------------------------------------------------
6702. EXPLICIT PARAMETRIX AND LOCAL LIMIT THEOREMS FOR SOME
DEGENERATE DIFFUSION PROCESSES
Valentin Konakov (CMI RAS) and Stephane Menozzi (PMA) and Stanislav
Molchanov
For a class of degenerate diffusion processes of rank 2, i.e. when only
Poisson brackets of order one are needed to span the whole space, we
obtain a
parametrix representation of the density from which we derive some
explicit
Gaussian controls that characterize the additional singularity induced
by the
degeneracy. We then give a local limit theorem with the usual
convergence rate
for an associated Markov chain approximation. The key point is that
the "weak"
degeneracy allows to exploit the techniques first introduced by
Konakov and
Molchanov and then developed by Konakov and Mammen that rely on Gaussian
approximations.
http://arxiv.org/abs/0802.2229
---------------------------------------------------------------
6703. CYLINDRICAL WIENER PROCESSES
Markus Riedle
In this work cylindrical Wiener processes on Banach spaces are defined
by
means of cylindrical stochastic processes, which are a well considered
mathematical object. This approach allows a definition which is a simple
straightforward extension of the real-valued situation. We apply this
definition to introduce a stochastic integral with respect to
cylindrical
Wiener processes. Again, this definition is a straightforward
extension of the
real-valued situation which results now in simple conditions on the
integrand.
In particular, we do not have to put any geometric constraints on the
Banach
space under consideration. Finally, we relate this integral to well-
known
stochastic integrals in literature.
http://arxiv.org/abs/0802.2261
---------------------------------------------------------------
6704. RATIONAL FUNCTIONS ASSOCIATED WITH THE WHITE NOISE SPACE AND
RELATED TOPICS
Daniel Alpay and David Levanony
Motivated by the hyper-holomorphic case we introduce and study rational
functions in the setting of Hida's white noise space. The Fueter
polynomials
are replaced by a basis computed in terms of the Hermite functions,
and the
Cauchy-Kovalevskaya product is replaced by the Wick product.
http://arxiv.org/abs/0802.2373
---------------------------------------------------------------
6705. MARKOV LOOPS AND RENORMALIZATION
Yves Le Jan (LM-Orsay)
We study Poissonnian ensembles of Markov loops and the associated
renormalized self-intersection local times.
http://arxiv.org/abs/0802.2478
---------------------------------------------------------------
6706. A GENERAL BALLOT THEOREM
L. Addario-Berry and B.A. Reed
We prove an analogue of the classical ballot theorem that holds for any
random walk in the range of attraction of the normal distribution. Our
result
is best possible: we exhibit examples demonstrating that if any of our
hypotheses are removed, our conclusions may no longer hold.
http://arxiv.org/abs/0802.2491
---------------------------------------------------------------
6707. CONSTRUCTION OF A STATIONARY FIFO QUEUE WITH IMPATIENT CUSTOMERS
Pascal Moyal
In this paper, we study the stability of queues with impatient
customers.
Under general stationary ergodic assumptions, we first provide some
conditions
for such a queue to be regenerative (i.e. to empty a.s. an infinite
number of
times). In the particular case of a single server operating in First
in, First
out, we prove the existence (in some cases, on an enlarged probability
space)
of a stationary workload. This is done by studying stochastic
recursions under
the Palm settings, and by stochastic comparison of stochastic
recursions.
http://arxiv.org/abs/0802.2495
---------------------------------------------------------------
6708. NON COMMUTATIVE CONDITIONAL EXPECTATIONS, PREDICTION AND A NEW
LOOK AT SOME QUANTUM PARADOXES
Henryk Gzyl
When the result of an observation is taken into account by means of a
non-commutative conditional expectation, exactly as in classical
prediction
theory, some of the usual paradoxes cease to be so. The moral of this
note is
that the mystery of the probabilistic interpretation of quantum
mechanics lies
in the superposition principle
http://arxiv.org/abs/0802.2297
---------------------------------------------------------------
6709. ON THE SHUFFLING ALGORITHM FOR DOMINO TILINGS
Eric Nordenstam
We study the dynamics of a certain discrete model of interacting
particles
that comes from the so called shuffling algorithm for sampling a
random tiling
of an Aztec diamond. It turns out that the transition probabilities
have a
particularly convenient determinantal form. An analogous formula in a
continuous setting has recently been obtained by Jon Warren studying
certain
model of interlacing Brownian motions which can be used to construct
Dyson's
non-intersecting Brownian motion.
We conjecture that Warren's model can be recovered as a scaling
limit of our
discrete model and prove some partial results in this direction. As an
application to one of these results we use it to rederive the known
result that
random tilings of an Aztec diamond, suitably rescaled near a turning
point,
converge to the GUE minor process.
http://arxiv.org/abs/0802.2592
---------------------------------------------------------------
6710. COUPLING-CUTOFFS FOR RANDOM WALKS ON THE HYPERCUBE
Stephen Connor
We consider a simple independence coupling for two continuous-time
random
walks on the hypercube, and investigate when the tail probability of the
coupling time exhibits `cutoff behaviour'. We not only provide a
necessary and
sufficient condition for this so-called `coupling-cutoff' to occur,
but also
prove a general bound on the window size of the cutoff, making use of
the
Lambert W-function. The results may be generalised to n-tuples of
independent
Markov processes for which each component may be coupled at an
exponential
rate.
http://arxiv.org/abs/0802.2641
---------------------------------------------------------------
6711. COMPLETE MOMENT AND INTEGRAL CONVERGENCE FOR SUMS OF NEGATIVELY
ASSOCIATED RANDOM VARIABLES
Han-Ying Liang and Deli Li and Andrew Rosalsky
For a sequence of identically distributed negatively associated random
variables $\{X_n; n\geq 1\}$ with partial sums $S_n=\sum_{i=1}^nX_i, n
\geq 1$,
refinements are presented of the classical Baum-Katz and Lai complete
convergence theorems. More specifically, necessary and sufficient moment
conditions are provided for complete moment convergence of the form $$
\sum_{n
\ge n_0} n^{r -2 -\frac{1}{pq}} a_n E(\max_{1 \le k \le n}|S_k|
^{\frac{1}{q}} -
\epsilon b_n^{\frac{1}{pq}})^+ < \infty $$ to hold where $r>1, q>0$
and either
$n_0=1, 0<p<2, a_n=1, b_n=n$ or $n_0=3, p=2, a_n=(\log n)^{-\frac{1}
{2q}},
b_n=n\log n$. These results extend results of Chow (1988) and Li and
Sp\u{a}taru (2005) from the independent and identically distributed
case to the
identically distributed negatively associated setting. The complete
moment
convergence is also shown to be equivalent to a form of complete
integral
convergence.
http://arxiv.org/abs/0802.2645
---------------------------------------------------------------
6712. PROOF(S) OF THE LAMPERTI REPRESENTATION OF CONTINUOUS-STATE
BRANCHING PROCESSES
Maria-Emilia Caballero and Amaury Lambert (CMAP and FESE) and
Geronimo Uribe Bravo
The representation of continuous-state branching processes (CSBPs) as
time-changed L\'evy processes with no negative jumps was discovered by
John
Lamperti in 1967 but was never proved. The goal of this paper is to
provide a
proof, and we actually provide two. The first one relies on studying the
time-change, using martingales and the L\'evy-It\^o representation of L
\'evy
processes. It gives insight into a stochastic differential equation
satisfied
by CSBPs and on its relevance to the branching property. The other
method
studies the time-change in a discrete model, where an analogous Lamperti
representation is evident, and provides functional approximations to
Lamperti
transforms by introducing a new topology on Skorohod space. Some
classical
arguments used to study CSBPs are reconsidered and simplified.
http://arxiv.org/abs/0802.2693
---------------------------------------------------------------
6713. ORDER-OPTIMAL CONSENSUS THROUGH RANDOMIZED PATH AVERAGING
F. Benezit and A.G. Dimakis and P. Thiran and M. Vetterli
Gossip algorithms have recently received significant attention, mainly
because they constitute simple and robust message-passing schemes for
distributed information processing over networks. However for many
topologies
that are realistic for wireless ad-hoc and sensor networks (like grids
and
random geometric graphs), the standard nearest-neighbor gossip
converges as
slowly as flooding ($O(n^2)$ messages).
A recently proposed algorithm called geographic gossip improves
gossip
efficiency by a $\sqrt{n}$ factor, by exploiting geographic
information to
enable multi-hop long distance communications. In this paper we prove
that a
variation of geographic gossip that averages along routed paths,
improves
efficiency by an additional $\sqrt{n}$ factor and is order optimal
($O(n)$
messages) for grids and random geometric graphs.
We develop a general technique (travel agency method) based on
Markov chain
mixing time inequalities, which can give bounds on the performance of
randomized message-passing algorithms operating over various graph
topologies.
http://arxiv.org/abs/0802.2587
---------------------------------------------------------------
6714. MEASURES AND THEIR RANDOM REALS
Jan Reimann and Theodore A. Slaman
We study the randomness properties of reals with respect to arbitrary
probability measures on Cantor space. We show that every non-recursive
real is
non-trivially random with respect to some measure. The probability
measures
constructed in the proof may have atoms. If one rules out the
existence of
atoms, i.e. considers only continuous measures, it turns out that every
non-hyperarithmetical real is random for a continuous measure. On the
other
hand, examples of reals not random for a continuous measure can be found
throughout the hyperarithmetical Turing degrees.
http://arxiv.org/abs/0802.2705
---------------------------------------------------------------
6715. STOCHASTIC 2-D NAVIER-STOKES EQUATION WITH ARTIFICIAL
COMPRESSIBILITY
Utpal Manna and Jose-Luis Menaldi and and Sivaguru S. Sritharan
In this paper we study the stochastic Navier-Stokes equation with
artificial
compressibility. The main results of this work are the existence and
uniqueness
theorem for strong solutions and the limit to incompressible flow. These
results are obtained by utilizing a local monotonicity property of the
sum of
the Stokes operator and the nonlinearity.
http://arxiv.org/abs/0802.2901
---------------------------------------------------------------
6716. ON THE TIME TO REACH MAXIMUM FOR A VARIETY OF CONSTRAINED
BROWNIAN MOTIONS
Satya. N. Majumdar (LPTMS) and Julien Randon-Furling (LPTMS) and
Michael J. Kearney, Marc Yor (PMA)
We derive P(M,t_m), the joint probability density of the maximum M and
the
time t_m at which this maximum is achieved for a class of constrained
Brownian
motions. In particular, we provide explicit results for excursions,
meanders
and reflected bridges associated with Brownian motion. By subsequently
integrating over M, the marginal density P(t_m) is obtained in each
case in the
form of a doubly infinite series. For the excursion and meander, we
analyse the
moments and asymptotic limits of P(t_m) in some detail and show that the
theoretical results are in excellent accord with numerical
simulations. Our
primary method of derivation is based on a path integral technique;
however, an
alternative approach is also outlined which is founded on certain
"agreement
formulae" that are encountered more generally in probabilistic studies
of
Brownian motion processes.
http://arxiv.org/abs/0802.2619
---------------------------------------------------------------
6717. SPECTRAL REPRESENTATION OF SOME NON STATIONARY ALPHA-STABLE
PROCESSES
Nourddine Azzaoui (IMB)
In this paper, we give a new covariation spectral representation of
some non
stationary symmetric $\alpha$-stable processes (S$\alpha$S). This
representation is based on a weaker covariation pseudo additivity
condition
which is more general than the condition of independence. This work
can be seen
as a generalization of the covariation spectral representation of
processes
expressed as stochastic integrals with respect to independent increments
S$\alpha$S processes (see Cambanis (1983)) or with respect to the
general
concept of independently scattered S$\alpha$S measures (Samorodnitsky
and Taqqu
1994). Relying on this result we investigate the non stationarity
structure of
some harmonisable S$\alpha$S processes especially those having
periodic or
almost-periodic covariation functions.
http://arxiv.org/abs/0802.2998
---------------------------------------------------------------
6718. MULTIPLE STRATONOVICH INTEGRAL AND HU--MEYER FORMULA FOR L\'EVY
PROCESSES
Merc\`e Farr\'e and Maria Jolis and Frederic Utzet
A multiple stochastic integral of Stratonovich type with respect to a L
\'evy
process is constructed, and its relationship with the multiple Ito
integral is
shown through a Hu-Meyer formula
http://arxiv.org/abs/0802.3112
---------------------------------------------------------------
6719. THE VIRGIN ISLAND MODEL
Martin Hutzenthaler
A continuous mass population model with local competition is constructed
where every emigrant colonizes an unpopulated island. The population
founded by
an emigrant is modeled as excursion from zero of an one-dimensional
diffusion.
With this excursion measure, we construct a process which we call
Virgin Island
Model. Furthermore, a necessary and sufficient condition for
extinction of the
total population is obtained for finite initial total mass.
http://arxiv.org/abs/0802.3145
---------------------------------------------------------------
6720. SCALING LIMITS OF (1+1)-DIMENSIONAL PINNING MODELS WITH
LAPLACIAN INTERACTION
Francesco Caravenna and Jean-Dominique Deuschel
We consider a random field \phi: {1, ..., N} -> R with Laplacian
interaction
of the form \sum_i V(\Delta \phi_i), where \Delta is the discrete
Laplacian and
the potential V(.) is symmetric and uniformly strictly convex. The
pinning
model is defined by giving the field a reward \epsilon \ge 0 each time
it
touches the x-axis, that plays the role of a defect line. It is known
that this
model exhibits a phase transition between a delocalized regime
(\epsilon <
\epsilon_c) and a localized one (\epsilon > \epsilon_c), where 0 <
\epsilon_c <
\infty.
In this paper we give a precise pathwise description of the
transition,
extracting the full scaling limits of the model. We show in particular
that in
the delocalized regime the field wanders away from the defect line at
a typical
distance N^{3/2}, while in the localized regime the distance is just
O((log
N)^2). A subtle scenario shows up in the critical regime (\epsilon =
\epsilon_c), where the field, suitably rescaled, converges in
distribution
toward the derivative of a symmetric stable Levy process of index 2/5.
Our
approach is based on Markov renewal theory.
http://arxiv.org/abs/0802.3154
---------------------------------------------------------------
6721. HARMONIC MEASURE AND WINDING OF RANDOM CONFORMAL PATHS: A
COULOMB GAS PERSPECTIVE
Bertrand Duplantier and Ilia Binder
We consider random conformally invariant paths in the complex plane
(SLEs).
Using the Coulomb gas method in conformal field theory, we rederive
the mixed
multifractal exponents associated with both the harmonic measure and
winding
(rotation or monodromy) near such critical curves, previously obtained
by
quantum gravity methods. The results also extend to the general cases of
harmonic measure moments and winding of multiple paths in a star
configuration.
http://arxiv.org/abs/0802.2280
---------------------------------------------------------------
6722. RELATIONSHIP BETWEEN STOCHASTIC FLOWS AND CONNECTION FORMS
M. Neklyudov
In this article I will prove new representation for the Levi-Civita
connection in terms of the stochastic flow corresponding to Brownian
motion on
manifold.
http://arxiv.org/abs/0802.3255
---------------------------------------------------------------
6723. A MODEL OF CONTINUOUS TIME POLYMER ON THE LATTICE
David Marquez-Carreras and Carles Rovira and Samy Tindel
In this article, we try to give a rather complete picture of the
behavior of
the free energy for a model of directed polymer in a random
environment, in
which the polymer is a simple symmetric random walk on the lattice $
\Z^d$, and
the environment is a collection $\{W(t,x);t\ge 0, x\in \Z^d\}$ of i.i.d.
Brownian motions.
http://arxiv.org/abs/0802.3296
---------------------------------------------------------------
6724. ASYMPTOTIC BEHAVIOR OF WEIGHTED QUADRATIC VARIATIONS OF
FRACTIONAL BROWNIAN MOTION: THE CRITICAL CASE H=1/4
Ivan Nourdin (PMA) and Anthony R\'eveillac (LMA)
We derive the asymptotic behavior of weighted quadratic variations of
fractional Brownian motion $B$ with Hurst index H=1/4. This completes
the only
missing case in a very recent work by I. Nourdin, D. Nualart and C.A.
Tudor.
Moreover, as an application, we solve a recent conjecture of K. Burdzy
and J.
Swanson on the asymptotic behavior of the Riemann sums with
alternating signs
associated to B.
http://arxiv.org/abs/0802.3307
---------------------------------------------------------------
6725. VARIATIONS OF THE SOLUTION TO A STOCHASTIC HEAT EQUATION II
Krzysztof Burdzy and Jason Swanson
We consider the solution u(x,t) to a stochastic heat equation. For
fixed x,
the process F(t) = u(x,t) has a nontrivial quartic variation. It
follows that F
is not a semimartingale, so a stochastic integral with respect to F
cannot be
defined in the classical Ito sense. We show that for sufficiently
differentiable functions g, a stochastic integral \int g(F) dF exists
as a
limit of discrete, midpoint style Riemann sums, where the limit is
taken in
distribution in the Skorohod space of cadlag functions. Moreover, we
show that
this integral satisfies a change of variables formulas with a
correction term
that is an ordinary Ito integral with respect to a Brownian motion
that is
independent of F.
http://arxiv.org/abs/0802.3356
---------------------------------------------------------------
6726. THE SUBELLIPTIC HEAT KERNEL ON SU(2): REPRESENTATIONS,
ASYMPTOTICS AND GRADIENT BOUNDS
Fabrice Baudoin and Michel Bonnefont
The Lie group SU(2) endowed with its canonical subriemannian structure
appears as a three-dimensional model of a positively curved}
subelliptic space.
The goal of this work is to study the subelliptic heat kernel on it
and some
related functional inequalities.
http://arxiv.org/abs/0802.3320
---------------------------------------------------------------
6727. MOMENT PROBLEMS AND BOUNDARIES OF NUMBER TRIANGLES
Alexander Gnedin and Jim Pitman
The boundary problem for graphs like Pascal's but with general
multiplicities
of edges is related to a `backward' problem of moments of the
Hausdorff type.
http://arxiv.org/abs/0802.3410
---------------------------------------------------------------
6728. ON A NONHIERARCHICAL VERSION OF THE GENERALIZED RANDOM ENERGY
MODEL. II. ULTRAMETRICITY
Erwin Bolthausen and Nicola Kistler
We study the Gibbs measure of the nonhierarchical versions of the
Generalized
Random Energy Models introduced in previous work, [2]. We prove that the
ultrametricity holds only provided some nondegeneracy conditions on the
hamiltonian are met.
http://arxiv.org/abs/0802.3436
---------------------------------------------------------------
6729. A TRUNCATION APPROACH FOR FAST COMPUTATION OF DISTRIBUTION
FUNCTIONS
Xinjia Chen
In this paper, we propose a general approach for improving the
efficiency of
computing distribution functions. The idea is to truncate the domain of
summation or integration.
http://arxiv.org/abs/0802.3455
---------------------------------------------------------------
6730. THE AIZENMAN-SIMS-STARR AND GUERRA'S SCHEMES FOR THE SK MODEL
WITH MULTIDIMENSIONAL SPINS
Anton Bovier and Anton Klimovsky
We prove upper and lower bounds on the free energy in the
Sherrington-Kirkpatrick model with multidimensional (e.g., Heisenberg)
spins in
terms of the variational inequalities based on the corresponding Parisi
functional. We employ the comparison scheme of Aizenman, Sims and
Starr and the
one of Guerra involving the generalised random energy model-inspired
processes
and Ruelle's probability cascades. For this purpose an abstract
quenched large
deviations principle of the Gaertner-Ellis type is obtained. Using the
properties of Ruelle's probability cascades and the Bolthausen-Sznitman
coalescent, we derive Talagrand's representation of the Guerra
remainder term
for our model. We study the properties of the multidimensional Parisi
functional by establishing a link with a certain class of the non-linear
partial differential equations. Solving a problem posed by Talagrand,
we show
the strict convexity of the local Parisi functional. We prove the Parisi
formula for the local free energy in the case of the multidimensional
Gaussian
a priori distribution of spins using Talagrand's methodology of the a
priori
estimates.
http://arxiv.org/abs/0802.3467
---------------------------------------------------------------
6731. THE WAITING TIME FOR M MUTATIONS
Jason Schweinsberg
We consider a model of a population of fixed size N in which each
individual
gets replaced at rate one and each individual experiences a mutation
at rate
\mu. We calculate the asymptotic distribution of the time that it
takes before
there is an individual in the population with m mutations. Several
different
behaviors are possible, depending on how \mu changes with N. These
results have
applications to the problem of determining the waiting time for
regulatory
sequences to appear and to models of cancer development.
http://arxiv.org/abs/0802.3485
---------------------------------------------------------------
6732. RECONSTRUCTION OF RANDOM COLOURINGS
Allan Sly
Reconstruction problems have been studied in a number of contexts
including
biology, information theory and and statistical physics. We consider the
reconstruction problem for random $k$-colourings on the $\Delta$-ary
tree for
large $k$. Bhatnagar et. al. showed non-reconstruction when $\Delta \leq
\frac12 k\log k - o(k\log k)$ and reconstruction when $\Delta \geq k
\log k +
o(k\log k)$. We tighten this result and show non-reconstruction when $
\Delta
\leq k[\log k + \log \log k + 1 - \ln 2 -o(1)]$ and reconstruction
when $\Delta
\geq k[\log k + \log \log k + 1+o(1)]$.
http://arxiv.org/abs/0802.3487
---------------------------------------------------------------
6733. THE TIME CONSTANT VANISHES ONLY ON THE PERCOLATION CONE IN
DIRECTED FIRST PASSAGE PERCOLATION
Yu Zhang
We consider the directed first passage percolation model on ${\bf
Z}^2$. In
this model, we assign independently to each edge $e$ a passage time
$t(e)$ with
a common distribution $F$. We denote by $\vec{T}({\bf 0}, (r,\theta))$
the
passage time from the origin to $(r, \theta)$ by a northeast path for $
(r,
\theta)\in {\bf R}^+\times [0,\pi/2]$. It is known that $\vec{T}({\bf
0}, (r,
\theta))/r$ converges to a time constant $\vec{\mu}_F (\theta)$. Let
$\vec{p}_c$ denote the critical probability for oriented percolation.
In this
paper, we show that the time constant has a phase transition divided by
$\vec{p}_c$, as follows:
(1) If $F(0) < \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ for all
$0\leq
\theta\leq \pi/2$.
(2) If $F(0) = \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ if and
only if
$\theta\neq \pi/4$.
(3) If $F(0)=p > \vec{p}_c$, then there exists a percolation cone
between
$\theta_p^-$ and $\theta_p^+$ for
$0\leq \theta^-_p< \theta^+_p \leq \pi/2$ such that $\vec{\mu}
(\theta) >0$
if and only if $\theta\not\in [\theta_p^-, \theta^+_p]$. Furthermore,
all the
moments of $\vec{T}({\bf 0}, (r, \theta))$ converge whenever $\theta\in
[\theta_p^-, \theta^+_p]$. As applications, we describe the shape of the
directed growth model on the distribution of $F$. We give a phase
transition
for the shape divided by $\vec{p}_c$.
http://arxiv.org/abs/0802.3519
---------------------------------------------------------------
6734. ON EQUILIBRIUM PRICES IN CONTINUOUS TIME
V. Filipe Martins-da-Rocha and Frank Riedel
We combine general equilibrium theory and theorie generale of stochastic
processes to derive structural results about equilibrium state prices.
http://arxiv.org/abs/0802.3585
---------------------------------------------------------------
6735. VERAVERBEKE'S THEOREM AT LARGE - ON THE MAXIMUM OF SOME
PROCESSES WITH NEGATIVE DRIFT AND HEAVY TAIL INNOVATIONS
Philippe Barbe (CNRS) and Bill McCormick (UGA)
Veraverbeke's (1977) theorem relates the tail of the distribution of the
supremum of a random walk with negative drift to the tail of the
distribution
of its increments, or equivalently, the probability that a centered
random walk
with heavy-tail increments hits a moving linear boundary. We study
similar
problems for more general processes. In particular, we derive an
analogue of
Veraverbeke's theorem for fractional integrated ARMA models without
prehistoric
influence, when the innovations have regularly varying tails.
Furthermore, we
prove some limit theorems for the trajectory of the process,
conditionally on a
large maximum. Those results are obtained by using a general scheme of
proof
which we present in some detail and should be of value in other related
problems.
http://arxiv.org/abs/0802.3638
---------------------------------------------------------------
6736. RANDOM WALK ON A DISCRETE TORUS AND RANDOM INTERLACEMENTS
David Windisch
We investigate the relation between the local picture left by the
trajectory
of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d
time steps,
u > 0, and the model of random interlacements recently introduced by
Sznitman.
In particular, we show that for large N, the joint distribution of the
local
pictures in the neighborhoods of finitely many distant points left by
the walk
up to time u N^d converges to independent copies of the random
interlacement at
level u.
http://arxiv.org/abs/0802.3654
---------------------------------------------------------------
6737. A SIMPLE FORMULA FOR CONSTRUCTING CONFIDENCE INTERVAL FOR THE
MEAN OF BOUNDED RANDOM VARIABLES
Xinjia Chen
In this article, we derive an explicit formula for computing confidence
interval for the mean of bounded random variables. In additional to its
simplicity, the formula is very tight in comparison with existing
results in
literature.
http://arxiv.org/abs/0802.3458
---------------------------------------------------------------
6738. INTERVAL ESTIMATION OF BOUNDED VARIABLE MEANS VIA INVERSE SAMPLING
Xinjia Chen
In this paper, we develop interval estimation methods for means of
bounded
random variables based on a sequential procedure such that the
sampling is
continued until the sample sum is no less than a prescribed threshold.
http://arxiv.org/abs/0802.3539
---------------------------------------------------------------
6739. ON THE LOCALITY OF THE PR\"UFER CODE
Craig Lennon
The Pr\"ufer code is a bijection between trees on the vertex set $[n]$
and
strings on the set $[n]$ of length $n-2$ (Pr\"ufer strings of order $n
$). In
this paper we examine the `locality' properties of the Pr\"ufer code,
i.e. the
effect of changing an element of the Pr\"ufer string on the structure
of the
corresponding tree. Our measure for the distance between two trees
$T,T^*$ is
$\Delta(T,T^*)=n-1-| E(T)\cap E(T^*)|$. We randomly mutate the $\mu$th
element
of the Pr\"ufer string of the tree $T$, changing it to the tree $T^*$,
and we
asymptotically estimate the probability that this results in a change
of $\ell$
edges, i.e. $P(\Delta=\ell | \mu).$ We find that P(\Delta=\ell | \mu)$
is on
the order of $ n^{-1/3+o(1)}$ for any integer $\ell>1,$ and that
$P(\Delta=1 |
\mu)=(1-\mu/n)^2+o(1).$ This result implies that the probability of a
`perfect'
mutation in the Pr\"ufer code (one for which $\Delta(T,T^*)=1$) is
$1/3.$
http://arxiv.org/abs/0802.3514
---------------------------------------------------------------
6740. STRONG SOLUTIONS FOR STOCHASTIC POROUS MEDIA EQUATIONS WITH JUMPS
Viorel Barbu and Carlo Marinelli
We prove global well-posedness in the strong sense for stochastic
generalized
porous media equations driven by locally square integrable martingales
with
stationary independent increments.
http://arxiv.org/abs/0802.3594
---------------------------------------------------------------
6741. ASYMPTOTICALLY OPTIMAL QUANTIZATION SCHEMES FOR GAUSSIAN PROCESSES
Harald Luschgy and Gilles Pag\`es (PMA) and Benedikt Wilbertz
We describe quantization designs which lead to asymptotically and order
optimal functional quantizers. Regular variation of the eigenvalues of
the
covariance operator plays a crucial role to achieve these rates. For the
development of a constructive quantization scheme we rely on the
knowledge of
the eigenvectors of the covariance operator in order to transform the
problem
into a finite dimensional quantization problem of normal distributions.
Furthermore we derive a high-resolution formula for the $L^2$-
quantization
errors of Riemann-Liouville processes.
http://arxiv.org/abs/0802.3761
---------------------------------------------------------------
6742. STOCHASTIC TAMED 3D NAVIER-STOKES EQUATIONS: EXISTENCE,
UNIQUENESS AND ERGODICITY
Michael R\"ockner and Xicheng Zhang
In this paper, we prove the existence of a unique strong solution to a
stochastic tamed 3D Navier-Stokes equation in the whole space as well
as in the
periodic boundary case. Then, we also study the Feller property of
solutions,
and prove the existence of invariant measures for the corresponding
Feller
semigroup in the case of periodic conditions. Moreover, in the case of
periodic
boundary and degenerated additive noise, using the notion of
asymptotic strong
Feller property proposed by Hairer and Mattingly \cite{Ha-Ma}, we
prove the
uniqueness of invariant measures for the corresponding transition
semigroup.
http://arxiv.org/abs/0802.3934
---------------------------------------------------------------
6743. A LINK BETWEEN BINOMIAL PARAMETERS AND MEANS OF BOUNDED RANDOM
VARIABLES
Xinjia Chen
In this paper, we establish a fundamental connection between binomial
parameters and means of bounded random variables. Such connection finds
applications in statistical inference of means of bounded variables.
http://arxiv.org/abs/0802.3946
---------------------------------------------------------------
6744. THE SMALLEST SINGULAR VALUE OF A RANDOM RECTANGULAR MATRIX
Mark Rudelson and Roman Vershynin
We prove an optimal estimate on the smallest singular value of a random
subgaussian matrix, valid for all fixed dimensions. For an N by n
matrix A with
independent and identically distributed subgaussian entries, the
smallest
singular value of A is at least of the order \sqrt{N} - \sqrt{n-1}
with high
probability. A sharp estimate on the probability is also obtained.
http://arxiv.org/abs/0802.3956
---------------------------------------------------------------
6745. EXIT PROBLEM OF A TWO-DIMENSIONAL RISK PROCESS FROM THE
QUADRANT: EXACT AND ASYMPTOTIC RESULTS
Florin Avram and Zbigniew Palmowski and Martijn Pistorius
Consider two insurance companies (or two branches of the same company)
that
divide between them both claims and premia in some specified
proportions. We
model the occurrence of claims according to a renewal process. One
ruin problem
considered is that of the corresponding two-dimensional risk process
first
leaving the positive quadrant; another is that of entering the negative
quadrant. When the claims arrive according to a Poisson process we
obtain a
closed form expression for the ultimate ruin probability. In the
general case
we analyze the asymptotics of the ruin probability when the initial
reserves of
both companies tend to infinity under a Cram\'er light-tail assumption
on the
claim size distribution.
http://arxiv.org/abs/0802.4060
---------------------------------------------------------------
6746. GOOD DEAL BOUNDS INDUCED BY SHORTFALL RISK
Takuji Arai
We shall provide in this paper good deal pricing bounds for contingent
claims
induced by the shortfall risk with some loss function. Assumptions we
impose on
loss functions and contingent claims are very mild. We prove that the
upper and
lower bounds of good deal pricing bounds are expressed by convex risk
measures
on Orlicz hearts. In addition, we obtain its representation with the
minimal
penalty function. Moreover, we give a representation, for two simple
cases, of
good deal bounds and calculate the optimal strategies when a claim is
traded at
the upper or lower bounds of its good deal pricing bound.
http://arxiv.org/abs/0802.4141
---------------------------------------------------------------
6747. ORBIT MEASURES AND INTERLACED DETERMINANTAL POINT PROCESSES
Manon Defosseux (PMA)
We study some random interlaced configurations considering the
eigenvalues of
the main minors of Hermitian random matrices of the classical complex
Lie
algebras. We claim that these random configurations are determinantal
and give
their correlation kernels.
http://arxiv.org/abs/0802.4183
---------------------------------------------------------------
6748. THE HEIGHT OF WATERMELONS WITH WALL
Thomas Feierl
We derive asymptotics for the moments as well as the weak limit of the
height
distribution of watermelons with p branches with wall. This
generalises a
famous result of de Bruijn, Knuth and Rice on the average height of
planted
plane trees, and results by Fulmek and Katori et al. on the expected
value,
respectively the higher moments, of the height distribution of
watermelons with
two branches.
The asymptotics for the moments depend on the analytic behaviour of
certain
multidimensional Dirichlet series. In order to obtain this information
we prove
a reciprocity relation satisfied by the derivatives of one of Jacobi's
theta
functions, which generalises the well known reciprocity law for
Jacobi's theta
functions.
http://arxiv.org/abs/0802.2691
---------------------------------------------------------------
6749. LIMITS LAWS FOR GEOMETRIC MEANS OF FREE POSITIVE RANDOM VARIABLES
Gabriel H. Tucci
Let $\{a_{k}\}_{k=1}^{\infty}$ be free identically distributed positive
non--commuting random variables with probability measure distribution $
\mu$. In
this paper we proved a multiplicative version of the Free Central Limit
Theorem. More precisely, let $b_{n}=a_{1}^{1/2}a_{2}^{1/2}... a_{n}...
a_{2}^{1/2}a_{1}^{1/2}$ then $b_{n}$ is a positive operator with the
same
moments as $x_{n}=a_{1}a_{2}... a_{n}$ and $b_{n}^{1/2n}$ converges in
distribution to positive operator $\Lambda$. We completely determined
the
probability measure distribution $\nu$ of $\Lambda$ from the
distribution
$\mu$. This gives us a natural map $\mathcal{G}:\mathcal{M_{+}}\to
\mathcal{M_{+}}$ with $\mu\mapsto \mathcal{G}(\mu)=\nu.$ We study how
this map
behaves with respect to additive and multiplicative free convolution.
As an
interesting consequence of our results, we illustrate the relation
between the
probability distribution $\nu$ and the distribution of the Lyapunov
exponents
for the sequence $\{a_{k}\}_{k=1}^{\infty}$ introduced in \cite{LyaV}.
http://arxiv.org/abs/0802.4226
---------------------------------------------------------------
6750. MULTISTEP BAYESIAN STRATEGY IN COIN-TOSSING GAMES AND ITS
APPLICATION TO ASSET TRADING GAMES IN CONTINUOUS TIME
Kei Takeuchi and Masayuki Kumon and Akimichi Takemura
We study multistep Bayesian betting strategies in coin-tossing games
in the
framework of game-theoretic probability of Shafer and Vovk (2001). We
show that
by a countable mixture of these strategies, a gambler or an investor can
exploit arbitrary patterns of deviations of nature's moves from
independent
Bernoulli trials. We then apply our scheme to asset trading games in
continuous
time and derive the exponential growth rate of the investor's capital
when the
variation exponent of the asset price path deviates from two.
http://arxiv.org/abs/0802.4311
---------------------------------------------------------------
6751. POSITIVE STOCHASTIC VOLATILITY SIMULATION
William Halley and Simon J.A. Malham and Anke Wiese
We present a positivity preserving numerical scheme for the pathwise
solution
of nonlinear stochastic differential equations driven by a multi-
dimensional
Wiener process and governed by non-commutative linear and non-
Lipschitz vector
fields. This strong order one scheme uses: (i) Strang exponential
splitting, an
approximation that decomposes the stochastic flow separately into the
drift
flow, and the pure diffusion flow governed by the diffusion vector
fields; (ii)
an implicit Euler method to approximate the drift flow; and (iii) an
implicit
Milstein method to approximate the pure diffusion flow. The separate
approximations for the drift and pure diffusion flows preserve
positivity.
Therefore the Strang exponential splitting approximation does also. We
demonstrate the efficacy of our method by applying it to the Heston
model and a
variance curve model, and compare it against well-established positivity
preserving schemes.
http://arxiv.org/abs/0802.4411
---------------------------------------------------------------
6752. ASYMPTOTIC ANALYSIS OF A FLUID MODEL MODULATED BY AN $M/M/1$ QUEUE
Charles Knessl and Diego Dominici
We analyze asymptotically a differential-difference equation, that
arises in
a Markov-modulated fluid model. We use singular perturbation methods
to analyze
the problem with appropriate scalings of the two state variables. In
particular, the ray method and asymptotic matching are used.
http://arxiv.org/abs/0802.4434
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