Probability Abstracts 71
This document contains abstracts 2040-2093.
They have been mailed on October 24, 2002.
2040. ON PERPETUITIES RELATED TO THE SIZE-BIASED DISTRIBUTIONS
Aleksander M. Iksanov
We study perpetuities of a special type related to the size-biased
distributions. Necessary and sufficient conditions of their existence and
uniqueness are obtained. A crucial point in proving all results is a close
connection between perpetuities treated in the paper and fixed points of
so-called Poisson shot noise transforms.
iksan@unicyb.kiev.ua
2041. ON FIXED POINTS OF POISSON SHOT NOISE TRANSFORMS
Aleksander M. Iksanov and Zbigniew J. Jurek
Distributional fixed points of a Poisson shot noise transform (for
non-negative, non-increasing and bounded by 1, response functions) are
characterized. The tail behavior of fixed points is described. Typically they
have either exponential moments or their tails are proportional to a power
function, with exponent greater than minus one. The uniqueness of fixed points
is also discussed. Finally it is proved that in most cases fixed points are
absolutely continuous, apart from the possible atom at zero.
iksan@unicyb.kiev.ua
2042. ON CHOOSING AND BOUNDING PROBABILITY METRICS
Alison L. Gibbs and Francis Edward Su
When studying convergence of measures, an important issue is the choice of
probability metric. In this review, we provide a summary and some new results
concerning bounds among ten important probability metrics/distances that are
used by statisticians and probabilists. We focus on these metrics because they
are either well-known, commonly used, or admit practical bounding techniques.
We summarize these relationships in a handy reference diagram, and also give
examples to show how rates of convergence can depend on the metric chosen.
su@math.hmc.edu
2043. BEST CHOICE FROM THE PLANAR POISSON PROCESS
Alexander Gnedin
Various best-choice problems related to the planar homogeneous Poisson
process in finite or semi-infinite rectangle are studied. The analysis is
largely based on properties of the one-dimensional box-area process associated
with the sequence of records. We prove a series of distributional identities
involving exponential and uniform random variables, and resolve the
Petruccelli-Porosinski-Samuels paradox on coincidence of asymptotic values in
certain discrete-time optimal stopping problems.
gnedin@math.uu.nl
2044. SAMPLING FROM A COUPLE OF POSITIVELY CORRELATED BETA VARIATES
Mario Catalani
We know that the marginals in a Dirichlet distribution are beta variates
exhibiting a negative correlation. But we can construct two linear combinations
of such marginals in such a way to obtain a positive correlation. We discuss
the restrictions that are to be imposed on the parameters to accomplish such a
result. In the case the sampling from the Dirichlet distribution is performed
through a generalization of Johnk's method we discuss the efficiency of the
algorithm implementing the method.
mario.catalani@unito.it
2045. RANDOM WALKS THAT AVOID THEIR PAST CONVEX HULL
Omer Angel, Itai Benjamini, Balint Virag
We introduce planar random walk conditioned to avoid its past convex hull,
and we show that it escapes at a positive limsup speed. Experimental results
show that fluctuations from a limiting direction are on the order of n^(3/4).
This behavior is also observed for the extremal investor, a natural financial
model related to the planar walk.
balint@math.mit.edu
2046. LARGE DEVIATIONS FOR BROWNIAN MOTION IN A RANDOM SCENERY
A. Asselah, F. Castell
We prove large deviations principles in large time, for the Brownian
occupation time in random scenery. The random scenery is constant on unit
cubes, and consist of i.i.d. bounded variables, independent of the Brownian
motion. This model is a time-continuous version of Kesten and Spitzer's random
walk in random scenery. We prove large deviations principles in ``quenched''
and ``annealed'' settings.
fabienne.castell@cmi.univ-mrs.fr
2047. ON THE LARGEST EIGENVALUE OF A RANDOM SUBGRAPH OF THE HYPERCUBE
Alexander Soshnikov, Benny Sudakov
Let G be a random subgraph of the n-cube where each edge appears randomly and
independently with small probability p. We prove that the largest eigenvalue of
the adjacency matrix of G is almost surely \lambda_1(G)= (1+o(1))
max(\Delta^{1/2}(G),np), where \Delta(G) is the maximum degree of G and o(1)
term tends to zero as max (\Delta^{1/2}(G), np) tends to infinity.
soshniko@math.ucdavis.edu
2048. SPECTRAL THEORY AND LIMIT THEOREMS FOR GEOMETRICALLY ERGODIC MARKOV
PROCESSES
Ioannis Kontoyiannis, Sean Meyn
Consider the partial sums {S_t} of a real-valued functional F(Phi(t)) of a
Markov chain {Phi(t)} with values in a general state space. Assuming only that
the Markov chain is geometrically ergodic and that the functional F is bounded,
the following conclusions are obtained:
1. Spectral theory: Well-behaved solutions can be constructed for the
``multiplicative Poisson equation''.
2. A ``multiplicative'' mean ergodic theorem: For all complex \alpha in a
neighborhood of the origin, the normalized mean of \exp(\alpha S_t) converges
exponentially fast to a solution of the multiplicative Poisson equation.
3. Edgeworth Expansions: Rates are obtained for the convergence of the
distribution function of the normalized partial sums S_t to the standard
Gaussian distribution.
4. Large Deviations: The partial sums are shown to satisfy a large deviations
principle in a neighborhood of the mean. This result, proved under geometric
ergodicity alone, cannot in general be extended to the whole real line.
5. Exact Large Deviations Asymptotics: Rates of convergence are obtained for
the large deviations estimates above.
Extensions of these results to continuous-time Markov processes are also
given.
yiannis@dam.brown.edu
2049. THE ODE METHOD AND SPECTRAL THEORY OF MARKOV OPERATORS
J. Huang, I. Kontoyiannis, S.P. Meyn
We give a development of the ODE method for the analysis of recursive
algorithms described by a stochastic recursion. With variability modelled via
an underlying Markov process, and under general assumptions, the following
results are obtained: 1. Stability of an associated ODE implies that the
stochastic recursion is stable in a strong sense when a gain parameter is
small. 2. The range of gain-values is quantified through a spectral analysis of
an associated linear operator, providing a non-local theory. 3. A second-order
analysis shows precisely how variability leads to sensitivity of the algorithm
with respect to the gain parameter.
All results are obtained within the natural operator-theoretic framework of
geometrically ergodic Markov processes.
yiannis@dam.brown.edu
2050. THE OPTIMAL ORDER FOR THE P-TH MOMENT OF SUMS OF INDEPENDENT RANDOM
VARIABLES WITH RESPECT TO SYMMETRIC NORMS AND RELATED COMBINATORIAL ESTIMATES
Marius Junge
We calculate the p-the moment of the sum of n independent random variables
with respect to symmetric norm in R^n. The order of growth for upper bound p/ln
p obtained in ths estimate is optimal. The result extends to generalized
Lorentz spaces l_{f,w} under mild assumptions on f. Indeed, the key
combinatorial estimate is obtained for the weak l_1 (l_{1,infinity})-norm.
Similar results have been obtained independently by Montgomery-Smith using
different techniques and avoiding the combinatorial estimate.
junge@math.uiuc.edu
2051. SAMPLING FROM A COUPLE OF NEGATIVELY CORRELATED GAMMA VARIATES
Mario Catalani
We propose two algorithms for sampling from two gamma variates possessing a
negative correlation. The case of positive correlation is easily solved, so we
just mention it. The main problem is the lowest value of the correlation
coefficient that can be reached. The starting point of both algorithms is
generation from a bivariate density with uniform negatively correlated
marginals. Actually the first method uses a degenerate bivariate density since
it considers two uniforms related by a linear relationship. Then we resort
essentially to the inverse transform method. For both algorithms we stress
restrictions on the parameters and rigidities.
mario.catalani@unito.it
2052. CONFORMAL RESTRICTION: THE CHORDAL CASE
Gregory Lawler, Oded Schramm, Wendelin Werner
We characterize and describe all random subsets $K$ of a given simply
connected planar domain (the upper half-plane $\H$, say) which satisfy the
``conformal restriction'' property, i.e., $K$ connects two fixed boundary
points (0 and $\infty$, say) and the law of $K$ conditioned to remain in a
simply connected open subset $D$ of $\H$ is identical to that of $\Phi(K)$,
where $\Phi$ is a conformal map from $\H$ onto $D$ with $\Phi(0)=0$ and
$\Phi(\infty)=\infty$. The construction of this family relies on the stochastic
Loewner evolution (SLE) processes with parameter $\kappa \le 8/3$ and on their
distortion under conformal maps. We show in particular that SLE(8/3) is the
only random simple curve satisfying conformal restriction and relate it to the
outer boundaries of planar Brownian motion and SLE(6).
schramm@microsoft.com
2053. STEADY STATE ANALYSIS OF BALANCED-ALLOCATION ROUTING
Aris Anagnostopoulos, Ioannis Kontoyiannis, and Eli Upfal
We compare the long-term, steady-state performance of a variant of the
standard Dynamic Alternative Routing (DAR) technique commonly used in telephone
and ATM networks, to the performance of a path-selection algorithm based on the
"balanced-allocation" principle; we refer to this new algorithm as the Balanced
Dynamic Alternative Routing (BDAR) algorithm. While DAR checks alternative
routes sequentially until available bandwidth is found, the BDAR algorithm
compares and chooses the best among a small number of alternatives.
We show that, at the expense of a minor increase in routing overhead, the
BDAR algorithm gives a substantial improvement in network performance, in terms
both of network congestion and of bandwidth requirement.
yiannis@dam.brown.edu
2054. UPPER BOUND OF A VOLUME EXPONENT FOR DIRECTED POLYMERS IN A RANDOM
ENVIRONMENT
Olivier Mejane
We consider the model of directed polymers in a random environment introduced
by Petermann : the random walk is $\mathbb{R}^d$-valued and has independent
gaussian $N(0,I_d)$-increments, and the random media is a stationary centred
Gaussian process $(g(k,x), k \geq 1, x \in \mathbb{R}^d)$ with covariance
matrix $cov(g(i,x),g(j,y))=\delta_{ij} \Gamma(x-y) ,$ where $\Gamma$ is a
bounded integrable function on $\mathbb{R}^d .$ For this model, we establish an
upper bound of the volume exponent in all dimensions $d$.
olivier.mejane@math.ups-tlse.fr
2055. CONFORMAL FIELDS, RESTRICTION PROPERTIES, DEGENERATE REPRESENTATIONS AND SLE
Roland Friedrich, Wendelin Werner
In this research anouncement, we show how to relate the Schramm-Loewner
Evolution processes (SLE) to highest-weight representations of the Virasoro
Algebra. The conformal restriction properties of SLE that have been recently
studied in the paper arXiv:math.PR/0209343 by G. Lawler, O. Schramm and the
second author play an instrumental role. In this setup, various considerations
from conformal field theory can be interpreted and reformulated via SLE. This
enables to make a concrete link between the two-dimensional discrete critical
systems from statistical physics and conformal field theory.
wendelin.werner@math.u-psud.fr
2056. POISSON TREES, SUCCESSION LINES AND COALESCING RANDOM WALKS
P. A. Ferrari, C. Landim, H. Thorisson
We give a deterministic algorithm to construct a graph with no loops (a tree
or a forest) whose vertices are the points of a d-dimensional stationary
Poisson process S, subset of R^d. The algorithm is independent of the origin of
coordinates. We show that (1) the graph has one topological end --that is, from
any point there is exactly one infinite self-avoiding path; (2) the graph has a
unique connected component if d=2 and d=3 (a tree) and it has infinitely many
components if d\ge 4 (a forest); (3) in d=2 and d=3 we construct a bijection
between the points of the Poisson process and Z using the preorder-traversal
algorithm. To construct the graph we interpret each point in S as a space-time
point (x,r)\in\R^{d-1}\times R. Then a (d-1) dimensional random walk in
continuous time continuous space starts at site x at time r. The first jump of
the walk is to point x', at time r'>r, (x',r')\in S, where r' is the minimal
time after r such that |x-x'|<1. All the walks jumping to x' at time r'
coalesce with the one starting at (x',r'). Calling (x',r') = \alpha(x,r), the
graph has vertex set S and edges {(s,\alpha(s)), s\in S}. This enables us to
shift the origin of S^o = S + \delta_0 (the Palm version of S) to another point
in such a way that the distribution of S^o does not change (to any point if d =
2 and d = 3; point-stationarity).
pablo@ime.usp.br
2057. SEPARATED-OCCURRENCE INEQUALITIES FOR DEPENDENT PERCOLATION AND ISING
MODELS
Kenneth S. Alexander
Separated-occurrence inequalities are variants for dependent lattice models
of the van den Berg-Kesten inequality for independent models. They take the
form $P(A \circ_r B) \leq (1 + ce^{-\epsilon r})P(A)P(B)$, where $A \circ_r B$
is the event that $A$ and $B$ occur at separation $r$ in a configuration
$\omega$, that is, there exist two random sets of bonds or sites separated by
at least distance $r$, one set responsible for the occurrence of the event $A$
in $\omega$, the other for the occurrence of $B$. We establish such
inequalities for certain subcritical FK models, and for certain Ising models
which are at supercritical temperature or have an external field, with $A$ and
$B$ increasing or decreasing events.
alexandr@math.usc.edu
2058. LOWER BOUNDS FOR BOUNDARY ROUGHNESS FOR DROPLETS IN BERNOULLI
PERCOLATION
Hasan B. Uzun and Kenneth S. Alexander
We consider boundary roughness for the ``droplet'' created when supercritical
two-dimensional Bernoulli percolation is conditioned to have an open dual
circuit surrounding the origin and enclosing an area at least $l^2$, for large
$l$. The maximum local roughness is the maximum inward deviation of the droplet
boundary from the boundary of its own convex hull; we show that for large $l$
this maximum is at least of order $l^{1/3}(\log l)^{-2/3}$. This complements
the upper bound of order $l^{1/3}(\log l)^{2/3}$ known for the average local
roughness. The exponent 1/3 on $l$ here is in keeping with predictions from the
physics literature for interfaces in two dimensions.
alexandr@math.usc.edu
2059. ASYMPTOTIC STABILITY OF THE OPTIMAL FILTER FOR NON-ERGODIC SIGNALS
Anastasia Papavasiliou
In this paper, we study the problem of estimating a Markov chain $X$(signal)
from its noisy partial information $Y$, when the transition probability kernel
depends on some unknown parameters. Our goal is to compute the conditional
distribution process ${\mathbb P}\{X_n|Y_n,...,Y_1\}$, referred to hereafter as
the {\it optimal filter}. We rewrite the system, so that the kernel is now
known but the uncertainty is transfered to the initial conditions. We show
that, under certain conditions, the optimal filter will forget any erroneous
initialization. So, starting with a `good' prior distribution on the
parameters, the filter will ultimately choose the correct value. This can also
be seen as an asymptotic stability result, for non-ergodic systems.
pp2102@columbia.edu
2060. A REMARK ON UNIFIED ERROR EXPONENTS: HYPOTHESIS TESTING, DATA
COMPRESSION AND MEASURE CONCENTRATION
Ioannis Kontoyiannis, Ali Devin Sezer
Let A be finite set equipped with a probability distribution P, and let M be
a "mass" function on A. A characterization is given for the most efficient way
in which A^n can be covered using spheres of a fixed radius. A covering is a
subset C_n of A^n with the property that most of the elements of A^n are within
some fixed distance from at least one element of C_n, and "most of the
elements" means a set whose probability is exponentially close to one (with
respect to the product distribution P^n). An efficient covering is one with
small mass M^n(C_n). With different choices for the geometry on A, this
characterization gives various corollaries as special cases, including Marton's
error-exponents theorem in lossy data compression, Hoeffding's optimal
hypothesis testing exponents, and a new sharp converse to some measure
concentration inequalities on discrete spaces.
yiannis@dam.brown.edu
2061. NONLINEAR FILTERING OF DIFFUSION PROCESSES IN CORRELATED NOISE: ANALYSIS
BY SEPARATION OF VARIABLES
Sergey V. Lototsky
An approximation to the solution of a stochastic parabolic equation is
constructed using the Galerkin approximation followed by the Wiener Chaos
decomposition. The result is applied to the nonlinear filtering problem for the
time homogeneous diffusion model with correlated noise. An algorithm is
proposed for computing recursive approximations of the unnormalized filtering
density and filter, and the errors of the approximations are estimated. Unlike
most existing algorithms for nonlinear filtering, the real-time part of the
algorithm does not require solving partial differential equations or evaluating
integrals. The algorithm can be used for both continuous and discrete time
observations.
lototsky@math.usc.edu
2062. ERGODICITY OF THE FINITE DIMENSIONAL APPROXIMATION OF THE 3D
NAVIER-STOKES EQUATIONS FORCED BY A DEGENERATE NOISE
M. Romito
We prove ergodicity of the finite dimensional approximations of the three
dimensional Navier-Stokes equations, driven by a random force. The forcing
noise acts only on a few modes and some algebraic conditions on the forced
modes are found that imply the ergodicity. The convergence rate to the unique
invariant measure is shown to be exponential.
romito@math.unifi.it
2063. FLUID LIMITS OF PURE JUMP MARKOV PROCESSES: A PRACTICAL GUIDE
R. W. R. Darling
A rescaled Markov chain converges uniformly in probability to the solution of
an ordinary differential equation, under carefully specified assumptions.
The presentation is much simpler than those in the outside literature.
The result may be used to build parsimonious models of large random or
pseudo-random systems.
rwrd@afterlife.ncsc.mil
2064. FORMULE D'ITO POUR DES DIFFUSIONS UNIFORMEMENT ELLIPTIQUES ET PROCESSUS
DE DIRICHLET
K.Dupoiron, P.Mathieu, J.San Martin
If X is a d-dimensional uniformly elliptic diffusion, with initial law nu, we
show that F(X) is a Dirichlet process, whenever F satisfies an integrability
condition linking its weak derivative to the coefficients of the diffusion and
the initial law nu.
We then show that F(X) satisfies an Ito formula, giving a construction of the
stochastic integral of grad F(X) with respect to X, provided that the two first
weak derivatives of F satisfy integrability conditions involving the
coefficients of the diffusion and the initial law.
Si X est une diffusion uniformement elliptique d-dimensionnelle, de loi
initiale nu, on montre que F(X) est un processus de Dirichlet, lorsque F
verifie une condition d'integrabilite qui lie ses derivees faibles aux
coefficients de la diffusion et a la loi initiale nu. On montre ensuite qu'on
peut ecrire une formule d'Ito pour F(X), en donnant une construction de
l'integrale stochastique de grad F(X) par rapport a X. Les conditions requises
sur F sont des conditions d'integrabilite liant ses derivees faibles, premiere
et seconde, aux coefficients de la diffusion et a la loi initiale nu.
dupoiron@cmi.univ-mrs.fr
2065. HIGH TEMPERATURE SHERRINGTON-KIRKPATRICK MODEL FOR GENERAL SPINS
Philippe Carmona
Francesco Guerra and Fabio Toninelli have developped a very powerful
technique to study the high temperature behaviour of the
Sherrington-Kirkpatrick mean field spin glass model.
They show that this model is asymptoticaly comparable to a linear model. The
key ingredient is a clever interpolation technique between the two different
Hamiltonians describing the models.
This paper contribution to the subject are the following: (1) The
replica-symmetric solution holds for general spins, not just $\pm 1$ valued.
(2) The proof does not involve cavitation but only first order differential
calculus and Gaussian integration by parts.
philippe.carmona@math.univ-nantes.fr
2066. LIMIT THEOREM FOR SUMS OF RANDOM PRODUCTS
O. Khorunzhiy
We study a generalization of the Random Energy Model to the case when the
number of exponential factors varies at random. Also a relation between REM and
the Erd"os-R'enyi limit theorem for maximums of partial sums is considered.
khorunjy@math.uvsq.fr
2067. THE SERIAL HARNESS INTERACTING WITH A WALL
P.A. Ferrari, L.R.G. Fontes, B. Niederhauser, M. Vachkovskaia
The serial harnesses introduced by Hammersley describe the motion of a
hypersurface of dimension $d$ embedded in a space of dimension $d+1$. The
height assigned to each site $i$ of $\Z^d$ is updated by taking a weighted
average of the heights of some of the neighbors of $i$ plus a ``noise'' (a
centered random variable). The surface interacts by exclusion with a ``wall''
located at level zero: the updated heights are not allowed to go below zero. We
show that for any distribution of the noise variables and in all dimensions,
the surface delocalizes. This phenomenon is related to the so called ``entropic
repulsion''. For some classes of noise distributions, characterized by their
tail, we give explicit bounds on the speed of the repulsion.
pablo@ime.usp.br
2068. A LIMIT THEOREM FOR SHIFTED SCHUR MEASURES
Craig A. Tracy and Harold Widom
To each partition $\lambda$ with distinct parts we assign the probability
$Q_\lambda(x) P_\lambda(y)/Z$ where $Q_\lambda$ and $P_\lambda$ are the Schur
$Q$-functions and $Z$ is a normalization constant. This measure, which we call
the shifted Schur measure, is analogous to the much-studied Schur measure. For
the specialization of the first $m$ coordinates of $x$ and the first $n$
coordinates of $y$ equal to $\alpha$ ($0<\alpha<1$) and the rest equal to zero,
we derive a limit law for $\lambda_1$ as $m,n\ra\infty$ with $\tau=m/n$ fixed.
For the Schur measure the $\alpha$-specialization limit law was derived by
Johansson. Our main result implies that the two limit laws are identical.
tracy@math.ucdavis.edu
2069. A SIMPLE CONSTRUCTION OF THE FRACTIONAL BROWNIAN MOTION
Enriquez Nathanael
In this work we introduce correlated random walks on $\Z$. When picking
suitably at random the coefficient of correlation, and taking the average over
a large number of walks, we obtain a discrete Gaussian process, whose scaling
limit is the fractional Brownian motion. We have to use two radically different
models for both cases ${1\over2}\leq H<1$ and $0<H<{1\over2}$. This result
provides an algorithm for the simulation of the fractional Brownian motion,
which appears to be quite efficient.
enriquez@ccr.jussieu.fr
2070. ALMOST SURE LIMIT THEOREMS FOR EXPANDING MAPS OF THE INTERVAL
J.-R.Chazottes, P. Collet
For a large class of expanding maps of the interval, we prove that partial
sums of Lipschitz observables satisfy an almost sure central limit theorem
(ASCLT). In fact, we provide a speed of convergence in the Kantorovich metric.
Maxima of partial sums are also shown to obey an ASCLT. The key-tool is an
exponential inequality recently obtained. Then we derive almost-sure
convergence rates for the supremum of moving averages of Lipschitz observables
(Erdos-Renyi type law). We end up with an application to entropy estimation
ASCLT's that refi ne Shannon-McMillan-Breiman and Ornstein-Weiss theorems.
jean-rene.chazottes@cpht.polytechnique.fr
2071. THREE SAMPLING FORMULAS
Alexander Gnedin
Sampling formulas describe probability laws of exchangeable combinatorial
structures like partitions and compositions. We give a brief account of two
known parametric families of sampling formulas and add a new family to the
list.
gnedin@math.uu.nl
2072. RANDOM LINEAR COMBINATIONS OF FUNCTIONS FROM $L_1$
Pavel Grigoriev
In the paper I study properties of random polynomials with respect to a
general system of functions. Some lower bounds for the mathematical expectation
of the uniform and recently introduced integral-uniform norms of random
polynomials are established.
{\sc Key words and phrases:} Random polynomial, estimates for maximum of
random process, integral-uniform norm.
grigorev@mccme.ru
2073. ESTIMATES FOR NORMS OF RANDOM POLYNOMIALS
Pavel Grigoriev
This paper contains some estimates for the {\it integral-uniform} norm and
the uniform norm of a wide class of random polynomials. The family of
integral-uniform norms introduced by Kasin and Tzafriri is a natural
generalization of the maximum norm taken over a net. We prove some properties
of the integral-uniform norms. The given application of the established
estimates demonstrates that the integral-uniform norms may be useful whenever
one is interested in the properties of a function distribution.
Key words: integral-uniform norm; random polynomials with respect to a
general function system; trigonometric polynomials with random coefficients.
grigorev@mccme.ru
2074. FINITARY CODING FOR THE 1-D $T,T^{-1}$--PROCESS WITH DRIFT
Michael Keane and Jeffrey E. Steif
We show that there is a finitary isomorphism from a
finite state i.i.d. process to the $T,T^{-1}$--process
associated to 1-d random walk with positive drift.
This contrasts with the situation for simple symmetric
random walk in any dimension, where it cannot be a
finitary factor of any i.i.d. process including in
$d\ge 5$ where it becomes weak Bernoulli.
mkeane@wesleyan.edu steif@math.chalmers.se
- To see a preprint or other
information provided by the author
click here.
2075. OPTIMAL REWARD ON A SPARSE TREE
WITH RANDOM EDGE-WEIGHTSD
D. Khoshnevisan and T. M. Lewis
It is well known that the maximal displacement of a random walk
indexed by an m-ary tree with bounded i.i.d. edge-weights can reliably
yield much larger asymptotics than a classical random walk whose
summands are drawn from the same distribution. Presently we show that if
the edge-weights are mean-zero, then nonclassical asymptotics arise
even when the tree grows much more slowly than subexponentially. Our
conditions are stated in terms of a Minkowski-type logarithmic dimension
of the boundary of the tree.
davar@math.utah.edu tom.lewis@math.furman.edu
- To see a preprint or other
information provided by the author
click here.
2076. SUPER-BROWNIAN MOTION WITH REFLECTING HISTORICAL PATHS. II
CONVERGENCE OF APPROXIMATIONS
Krzysztof Burdzy and Leonid Mytnik
We prove that the sequence of finite reflecting
branching Brownian motion forests defined by Burdzy and Le Gall
converges in probability to the ``super-Brownian
motion with reflecting historical paths.'' This solves an open
problem posed by Burdzy and Le Gall who proved only tightness
for the sequence of approximations.
burdzy@math.washington.edu leonid@ie.technion.ac.il
- To see a preprint or other
information provided by the author
click here.
2077. INVARIANT STATES AND RATES OF CONVERGENCE FOR A CRITICAL
FLUID MODEL OF A PROCESSOR SHARING QUEUE
Amber L. Puha and Ruth J. Williams
This paper contains an asymptotic analysis of a fluid model
for a heavily loaded processor sharing queue. Specifically,
we consider the behavior of solutions of critical fluid models
as time approaches infinity. The main theorems of the paper
provide sufficient conditions for a fluid model solution to
converge to an invariant state and, under slightly more
restrictive assumptions, provide a rate of convergence. These
results are used in a related work by Gromoll for establishing
a heavy traffic diffusion approximation for a processor
sharing queue.
apuha@csusm.edu williams@math.ucsd.edu
- To see a preprint or other
information provided by the author
click here.
2078. DIFFUSION APPROXIMATION FOR A PROCESSOR SHARING QUEUE IN HEAVY TRAFFIC
H. Christian Gromoll
Consider a single server queue with renewal arrivals and
i.i.d. service times in which the server operates under a
processor sharing service discipline. To describe the
evolution of this system, we use a measure valued process that
keeps track of the residual service times of all jobs in the
system at any given time. From this measure valued process,
one can recover the traditional performance processes,
including queue length and workload. We show that under mild
assumptions, including standard heavy traffic assumptions,
the (suitably rescaled) measure valued processes corresponding
to a sequence of processor sharing queues converge in
distribution to a measure valued diffusion process. The
limiting process is characterized as the image under an
appropriate lifting map, of a one dimensional reflected
Brownian motion. As an immediate consequence, one obtains a
diffusion approximation for the queue length process of a
processor sharing queue.
gromoll@eurandom.tue.nl
- To see a preprint or other
information provided by the author
click here.
2079. STATE TAMENESS: A NEW APPROACH FOR CREDIT CONSTRAINS
Jaime Londońo
We propose a new definition for tameness within the model of
security prices as Itô processes that is risk-aware. We give
a new definition for arbitrage and characterize it. We then
prove a theorem that can be seen as an extension of the
second fundamental theorem of asset pricing, and a theorem
for valuation of contingent claims of the American type.
The valuation of European contingent claims and American
contingent claims that we obtain does not require the full
range of the volatility matrix. The formulas obtain to price
American contingent claims are closer in spirit to a
computational approach.
jalondon@sigma.eafit.edu.co
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
2080. HEAT KERNEL ESTIMATES FOR STABLE-LIKE PROCESSES ON D-SETS
Zhen-Qing Chen and Takashi Kumagai
The notion of $d$-set arises in the theory of function spaces and in fractal
geometry. Geometrically self-similar sets are typical examples of $d$-sets.
In this paper stable-like processes on $d$-sets are investigated, which
include reflected stable processes in Euclidean domains as a special case.
More precisely, we establish parabolic Harnack principle and derive
sharp two-sided heat kernel estimate for such stable-like processes.
Results on the exact Hausdorff dimensions for the range of stable-like
processes are also obtained.
zchen@math.washington.edu kumagai@kurims.kyoto-u.ac.jp
- To see a preprint or other
information provided by the author
click here.
- Or
here.
2081. DESIGNING A CONTACT PROCESS : THE PIECEWISE-HOMOGENEOUS PROCESS ON A FINITE SET
Aaron B. Wagner and Venkat Anantharam
We consider how to choose the reproduction rates in a one-dimensional contact process on a
finite set to maximize the growth rate of the extinction time with the population size.
The constraints are an upper bound on the average reproduction rate, and that the rate profile
must be piecewise constant. We show that the optimum
growth rate is achieved by a rate profile with at most
two rates, and we characterize the solution in terms of a “spatial correlation length”
of the supercritical process. We examine the
analogous problem for the simpler biased voter model,
for which we completely characterize the optimum
profile. The contact process proofs make use of a
planar-graph duality in the graphical representation, due to Durrett and Schonmann.
ananth@eecs.berkeley.edu
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information provided by the author
click here.
2082. A NEW LOOK AT THE GENERALIZED DISTRIBUTIVE LAW
Payam Pakzad and Venkat Anantharam
In this paper we develop a measure-theoretic version
of the Junction Tree algorithm to compute the
marginalizations of a product function. We
reformulate the problem in a measure-theoretic
framework, where the desired marginalizations are
viewed as conditional expectations of a product
function given certain sigma fields. We generalize
the notions of independence and junction trees at the
level of these sigma fields and produce algorithms
to find or construct a junction tree on a given set
of sigma fields.
ananth@eecs.berkeley.edu
- To see a preprint or other
information provided by the author
click here.
2083. STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY
STABLE PROCESSES FOR WHICH PATHWISE UNIQUENESS FAILS
Richard F. Bass, Krzysztof Burdzy and Zhen-Qing Chen
Let $Z_t$ be a one-dimensional symmetric stable
process of order $\al$ with $\al\in(0,2)$ and consider the
stochastic differential equation $$dX_t=\phi (X_{t-}) dZ_t.$$
For $\beta< \frac{1}{\al}\land 1$, we show there exists a function
$\phi$ that is bounded above and below by positive constants and
which is Holder continuous of order $\beta$ but for which
pathwise uniqueness of the stochastic differential equation does
not hold. This result is sharp.
bass@math.uconn.edu burdzy@math.washington.edu zchen@math.washington.edu
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
2084. HEAT KERNEL ESTIMATES FOR SYMMETRIC RANDOM WALKS ON A CLASS OF
FRACTAL GRAPHS AND STABILITY UNDER ROUGH ISOMETRIES
Ben M. Hambly and Takashi Kumagai
We examine a class of fractal graphs which arise from a subclass of
finitely ramified fractals. The two-sided heat kernel estimates for
these graphs are obtained in terms of an effective resistance metric
and they are best possible up to constants. If the graph has symmetry,
these estimates can be expressed as the usual Gaussian or sub-Gaussian
estimates. However, without symmetry, the off-diagonal terms show
different decay in different directions. We also discuss the
stability of the sub-Gaussian heat kernel estimates under rough
isometries.
hambly@maths.ox.ac.uk kumagai@kurims.kyoto-u.ac.jp
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information provided by the author
click here.
- Or
here.
2085. STOCHASTIC FINANCE: AN INTRODUCTION IN DISCRETE TIME
Hans Föllmer and Alexander Schied
This book is an introduction to financial mathematics.
It is intended for graduate students in mathematics and
for researchers working in academia and industry.
The focus on stochastic models in discrete time has two
immediate benefits. First, the probabilistic machinery
is simpler, and one can discuss right away some of the
key problems in the theory of pricing and hedging of
financial derivatives. Second, the paradigm of a complete
financial market, where all derivatives admit a perfect
hedge, becomes the exception rather than the rule. Thus,
the need to confront the problems arising in incomplete
financial market models appears at a very early stage.
The first part of the book studies a simple one-period
model which serves as a building stone for later
developments. Topics include the characterization of
arbitrage-free markets, the representation of preferences
on asset profiles by expected utility theory and its
robust extensions, monetary measures of risk, and an
introduction to equilibrium analysis.
In the second part, the idea of dynamic hedging of
contingent claims is developed in a multi-period
framework. Such models are typically incomplete: They
involve intrinsic risks which cannot be hedged away
completely. Topics include martingale measures, pricing
formulas for derivatives, American options, superhedging,
and hedging strategies with minimal shortfall risk.
Markets are modeled on general probability spaces. Thus,
the text captures the interplay between probability
theory and functional analysis which has been crucial for
recent advances in mathematical finance.
Further information , along with the table of contents
and ordering information, can be found at the
- To see a preprint or other
information provided by the author
click here.
- Or
here.
2086. A FOURIER FORMULATION OF THE FROSTMAN CRITERION FOR
RANDOM GRAPHS AND ITS APPLICATIONS TO WAVELET SERIES
Antoine Ayache and Francois Roueff
Lower bounds of the Hausdorff dimension of the graphs of random wavelet
series have been obtained, essentially under the hypothesis that the
wavelet coefficients have a bounded probability density function with
respect to the Lebesgue measure. In this article we extend these lower
bounds to classes of Random Wavelet Series that do not satisfy this
hypothesis. Our results are obtained thanks to a Fourier formulation
of the Frostman criterion for random graphs that we first present in our
paper.
At last, we believe that this formulation of the Frostman criterion would
be applied to the study of the Hausdorff dimension of the graphs of
stochastic processes, that are not necessarily random wavelet series.
ayache@cict.fr roueff@tsi.enst.fr
2087. AVERAGING VERSUS CHAOS IN TURBULENCE?
Houman Owhadi
The turbulence phenomenon has been addressed through two
main theoretical axes. Along the first one, the flow is
considered as a random vector field and averaging (space,
time, ensemble) is employed to obtain statistical order.
With the second one the flow is assumed to be stationary
and deterministic, then its evolution obtained by
renormalizing Navier Stokes equations from its transport
properties along the framework of dynamical systems and
chaos. In this paper we start from the second point of view
by examining deterministic stationary incompressible flows
which can be decomposed over an infinite number of spatial
scales without separation between them. It appears that a
low order dynamical system related to local Reynolds
numbers can be extracted from these flows and it controls
their transport properties. Its analysis shows that these
flows are strongly self-averaging and super-diffusive: the
delay $\tau(r)$ for any finite number of passive tracers
initially close to separate till a distance $r$ is almost
surely anomalously fast ($\tau(r)\sim r^{2-\nu}$, with
$\nu>0$). This strong self-averaging property is such that
the dissipative power of the flow compensates its
convective power at every scale. However as the circulation
increase in the eddies the transport behavior of the flow
may (discontinuously) bifurcate and become ruled by
deterministic chaos: the self-averaging property collapses
and advection dominates dissipation. When the flow is
anisotropic a new law describing turbulent viscosity is
identified.
owhadi@cmi.univ-mrs.fr
2088. THE DISTRIBUTION OF THE MAXIMUM OF A LEVY PROCESS
WITH POSITIVE JUMPS OF PHASE-TYPE
Ernesto Mordecki
Consider a Lévy process with finite intensity positive jumps of the
phase-type and arbitrary negative jumps. Assume that the process either is killed
at a constant rate or drifts to $-\infty$. We show that the distribution of the
overall maximum of this process is also of phase-type, and find the distribution of
this random variable. Previous results (hyperexpontial positive jumps) are
obtained as a particular case.
mordecki@cmat.edu.uy
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information provided by the author
click here.
2089. LARGE VOID ZONES AND OCCUPATION TIMES
FOR COALESCING RANDOM WALKS
Endre Csaki, Pal Revesz and Zhan Shi
The basic coalescing random walk is a system of interacting
particles. These particles start from every site of $Z^d$,
and each moves independently as a continuous-time random
walk. When two particles visit the same site, they coalesce
into a single particle. We describe the almost sure
asymptotic behaviours of: (a) the radius of the largest ball
centered at the origin which does not contain any particle at
time $T$; and (b) the amount of time when the origin is
occupied during $[0,T]$.
csaki@renyi.hu revesz@ci.tuwien.ac.at zhan@proba.jussieu.fr
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information provided by the author
click here.
2090. LOWER FUNCTIONS OF EMPIRICAL PROCESSES AND BROWNIAN SHEET
M.A.Lifshits and Z.Shi
We give a complete characterization of the lower functions for
the two-parameter Brownian sheet and for the uniform empirical
process via integral criteria. Our result for the Brownian sheet
can be viewed as a solution, in a very special case (the general
case remains an open question), of the following problem: find the
escape rate of infinite-dimensional Brownian motion. Our result for
the empirical process disproves (and provides the correct form of)
a conjecture in the book of Shorack and Wellner.
e-mail: lifts@mail.rcom.ru zhan@proba.jussieu.fr
2091. THE FIRST EXIT TIME OF BROWNIAN MOTION FROM
A PARABOLIC DOMAIN
M.A.Lifshits and Z.Shi
Consider a planar Brownian motion starting from an interior
point of the parabolic domain $D=\{ (x,y): \; y>x^2\}$,
and let $\tau_D$ denote the first time the Brownian motion
exits from $D$. The tail behaviour of $\tau_D$ is
somewhat exotic since it arises from an interference of large
deviation and small deviation events. Our main result implies
that the limit of $T^{-1/3} \log\p\{ \tau_D >T\}$
[as $T\to \infty$] exists and equals $-3\pi^2/8$, thus improving
previous estimates by Ba\~nuelos et al.~(2001) and W.Li~(2001+).
Our result actually applies to more general parabolic domains
in the space of arbitrary finite dimension.
e-mail: lifts@mail.rcom.ru zhan@proba.jussieu.fr
2092. SMALL DEVIATIONS OF WEIGHTED FRACTIONAL PROCESSES AND
AVERAGE NON-LINEAR APPROXIMATION
M.A.Lifshits and W.Linde
We investigate the small deviation problem for weighted fractional
Brownian motions in $L_q$--norm, $1\le q\le\infty$.
Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1$.
If $1/r:=H+1/q$, then our main result asserts
$$\lim_{\e\to 0} \e^{1/H}\log\pr{\norm{\rho\,B^H}_{L_q(0,\infty)}<\e} =
-c(H,q)\cdot\norm{\rho}_{L_r(0,\infty)}^{1/H}$$
provided the weight function $\rho$ satisfies a condition
slightly stronger than $r$-integrability. Thus we extend earlier results
for Brownian motion. Our basic tools are entropy estimates for fractional
integration operators, a non-linear approximation technique for
Gaussian processes as well as sharp entropy estimates for $l_q$--sums
of linear operators defined on a Hilbert space.
e-mail: lifts@mail.rcom.ru lindew@minet.uni-jena.de
2093. ON THE TAIL BEHAVIOR OF UNISOTROPIC NORMS
FOR GAUSSIAN RANDOM FIELDS
M.A.Lifshits, A.I.Nazarov, and Ya.Yu.Nikitin
We investigate the logarithmic large deviation asymptotics
for anisotropic norms of Gaussian random functions of
two variables. The problem is solved by evaluation
of the anisotropic norm of the corresponding integral
covariance operator. We find the exact values of such norms
for some important classes of Gaussian fields on the square
generalizing and extending the results of Nazarov and Nikitin.
e-mail: lifts@mail.rcom.ru
