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Probability Abstracts 85
This document contains abstracts 3072-3204.
They have been mailed on March 1, 2005.
Author(s): Ph. Barbe and W.P. McCormick (CNRS and University of Georgia)
Abstract: We establish some asymptotic expansions for infinite weighted convolution of
distributions having regular varying tails. Various applications to statistics
and probability are developed.
http://arXiv.org/abs/math/0412537
http://front.math.ucdavis.edu/math.PR/0412537
(alternate)
Author(s): Rami Atar and Amarjit Budhiraja
Abstract: We consider a class of jump-diffusion processes, constrained to a polyhedral
cone $G\subset\R^n$, where the constraint vector field is constant on each face
of the boundary. The constraining mechanism corrects for ``attempts'' of the
process to jump outside the domain. Under Lipschitz continuity of the Skorohod
map \Gamma, it is known that there is a cone \mathcalC such that the image
\Gamma\phi of a deterministic linear trajectory \phi remains bounded if and
only if \dot\phi\in\mathcalC. Denoting the generator of a corresponding
unconstrained jump-diffusion by \cll, we show that a key condition for the
process to admit an invariant probability measure is that for x\in G, \cll
\id(x) belongs to a compact subset of \mathcalC^o.
http://arXiv.org/abs/math/0501014
http://front.math.ucdavis.edu/math.PR/0501014
(alternate)
Author(s): Krzysztof Burdzy and Zhen-Qing Chen and Peter Jones
Abstract: For every bounded planar domain $D$ with a smooth boundary, we define a
`Lyapunov exponent' $\Lambda(D)$ using a fairly explicit formula. We consider
two reflected Brownian motions in $D$, driven by the same Brownian motion
(i.e., a `synchronous coupling'). If $\Lambda(D)>0$ then the distance between
the two Brownian particles goes to 0 exponentially fast with rate $\Lambda
(D)/(2|D|)$ as time goes to infinity. The exponent $\Lambda(D)$ is strictly
positive if the domain has at most one hole. It is an open problem whether
there exists a domain with $\Lambda(D)<0$.
http://arXiv.org/abs/math/0501486
http://front.math.ucdavis.edu/math.PR/0501486
(alternate)
Author(s): Marek Biskup and Lincoln Chayes and Nicholas Crawford
Abstract: We consider a class of spin systems on $\Z^d$ with vector valued spins
$(S_x)$ that interact via the pair-potentials $J_{x,y}S_x\cdot S_y$. The
interactions are generally spread-out in the sense that the $J_{x,y}$'s exhibit
either exponential or power-law fall-off. Under the technical condition of
reflection positivity and for sufficiently spread out interactions, we prove
that the model exhibits a first-order phase transition whenever the associated
mean-field theory signals such a transition. As a consequence, e.g., in
dimensions $d\ge3$, we can finally provide examples of the 3-state Potts model
with spread-out, exponentially decaying interactions, which undergoes a
first-order phase transition as the temperature varies. Similar transitions are
established in dimensions $d=1,2$ for power-law decaying interactions and in
high dimensions for next-nearest neighbor couplings. In addition, we also
investigate the limit of infinitely spread-out interactions. Specifically, we
show that once the mean-field theory is in a unique "state," then in any
sequence of translation-invariant Gibbs states various observables converge to
their mean-field values and the states themselves converge to product measure.
http://arXiv.org/abs/math-ph/0501067
http://front.math.ucdavis.edu/math-ph/0501067
(alternate)
Author(s): Michael Kleber
Abstract: This "Mathematical Entertainments" column from the Intelligencer is an
exposition of current investigations, rooted in recent work of Jim Propp, into
"quasirandom" analogues of random walk and random aggregation processes.
Featured are the "Goldbugs" and the "Rotor-router". These are deterministic
processes which simulate the random ones, for example having the same limiting
states, but with faster convergence.
The paper includes three large illustrations, which appear twice in the
submission, as both raster image (.png) and postscript (.eps) files. The latter
are much larger but needed for latex inclusion; the former are smaller, used by
pdflatex, and better for pixel-level viewing.
http://arXiv.org/abs/math/0501497
http://front.math.ucdavis.edu/math.CO/0501497
(alternate)
Author(s): Giovanni Peccati (LSTA) and Marc Yor (PMA)
Abstract: We present three new identities in law for quadratic functionals of
conditioned bivariate Gaussian processes. In particular, our results provide a
two-parameter generalization of a celebrated identity in law, involving the
path variance of a Brownian bridge, due to Watson (1961). The proof is based on
ideas from a recent note by J. R. Pycke (2005) and on the stochastic Fubini
theorem for general Gaussian measures proved in Deheuvels et al. (2004).
http://arXiv.org/abs/math/0501506
http://front.math.ucdavis.edu/math.PR/0501506
(alternate)
Author(s): Lukianov Vladimir
Abstract: We define a weighted multiplicity function for closed geodesics of given
length on a finite area Riemann surface. These weighted multiplicities appear
naturally in the Selberg trace formula, and in particular their mean square
plays an important role in the study of statistics of the eigenvalues of the
Laplacian on the surface.
In the case of the modular domain, E. Bogomolny, F. Leyvraz and C. Schmit
gave a formula for the mean square, which was rigorously proved by M. Peter. In
this paper we calculate the mean square of weighted multiplicities for some
surfaces associated to congruence subgroups of the unit group of a rational
quaternion algebra, in particular for congruence subgroups of the modular
group. Remarkably, the result turns out to be a rational multiple of the mean
square for the modular domain.
http://arXiv.org/abs/math/0501519
http://front.math.ucdavis.edu/math.NT/0501519
(alternate)
Author(s): Yuval Peres and Gabor Pete and Ariel Scolnicov
Abstract: An important conjecture in percolation theory is that almost surely no
infinite cluster exists in critical percolation on any transitive graph for
which the critical probability is less than 1. Earlier work has established
this for the amenable cases Z^2 and Z^d for large d, as well as for all
non-amenable graphs with unimodular automorphism groups. We show that the
conjecture holds for several classes of non-amenable graphs with non-unimodular
automorphism groups: for decorated trees, for the non-unimodular Diestel-Leader
graphs, and for direct products of these graphs with an arbitrary transitive
graph. We also show that, in any of these graphs, the connection probability
between two vertices decay exponentially in their distance. Finally, we prove
that critical percolation on the positive part of the lamplighter group has no
infinite clusters.
http://arXiv.org/abs/math/0501532
http://front.math.ucdavis.edu/math.PR/0501532
(alternate)
Author(s): Maury Bramson and Ofer Zeitouni and Martin P. W. Zerner
Abstract: We construct forests spanning $\Z^d, d\geq 2,$ that are stationary and
directed, and whose trees are infinite but are as short as possible. For $d\geq
3$, two independent copies of such forests, pointing into opposite directions,
can be pruned so as to become disjoint. From this, we construct in $d\geq 3$ a
stationary, polynomially mixing and uniformly elliptic environment of
nearest-neighbor transition probabilities on $\Z^d$, for which the
corresponding random walk (RWRE) disobeys a certain zero-one law for
directional transience.
http://arXiv.org/abs/math/0501533
http://front.math.ucdavis.edu/math.PR/0501533
(alternate)
Author(s): Rami Atar and Amarjit Budhiraja
Abstract: We study a class of stochastic control problems where a cost of the form
\E\int_{[0,\infty)}e^{-\beta s}[\ell(X_s)ds+h(Y^\circ_s)d|Y|_s] is to be
minimized over control processes Y whose increments take values in a cone \YY
of \R^p, keeping the state process X=x+B+GY in a cone \bS of \R^k, k\le p.
Here, x\in\bS, B is a Brownian motion with drift b and covariance \Sigma, G is
a fixed matrix, and Y^\circ is the Radon-Nikodym derivative dY/d|Y|. Let
\calL=-(1/2)\trace(\Sig D^2)-b\cd D where D denotes the gradient. Solutions to
the corresponding dynamic programming PDE [(\calL+\beta) f-\ell]
\vee\sup_{y\in\YY:|Gy|=1}[-Gy\cd Df - h(y)]=0, on \bS^o are considered with a
polynomial growth condition and are required to be supersolution up to the
boundary (corresponding to a ``state constraint'' boundary condition on
\pl\XX). Under suitable conditions on the problem data, including continuity
and nonnegativity of \ell and h, and polynomial growth of \ell, our main result
is the unique viscosity-sense solvability of the PDE by the control problem's
value function in appropriate classes of functions. In some cases where
uniqueness generally fails to hold in the class of functions that grow at most
polynomially (e.g., when h=0), our methods provide uniqueness within the class
of functions that, in addition, have compact level sets. The results are new
even in the following special cases: (1) The one-dimensional case k=p=1,
\bS=\YY=\R_+; (2) The first order case \Sigma=0; (3) The case where \ell and h
are linear. The proofs combine probabilistic arguments and viscosity solution
methods. Our framework covers a wide range of diffusion control problems that
arise from queueing networks in heavy traffic.
http://arXiv.org/abs/math/0501016
http://front.math.ucdavis.edu/math.PR/0501016
(alternate)
Author(s): Lo\"{i}c Chaumont (LPMA) and Ron A. Doney
Abstract: We construct the law of L\'{e}vy processes conditioned to stay positive under
general hypotheses. We obtain a Williams type path decomposition at the minimum
of these processes. This result is then applied to prove the weak convergence
of the law of L\'{e}vy processes conditioned to stay positive as their initial
state tends to 0. We describe an absolute continuity relationship between the
limit law and the measure of the excursions away from 0 of the underlying
L\'{e}vy process reflected at its minimum. Then, when the L\'{e}vy process
creeps upwards, we study the lower tail at 0 of the law of the height this
excursion.
http://arXiv.org/abs/math/0502012
http://front.math.ucdavis.edu/math.PR/0502012
(alternate)
Author(s): Keye Martin
Abstract: We give an algorithm for calculating the maximum entropy state as the least
fixed point of a Scott continuous mapping on the domain of classical states in
their Bayesian order.
http://arXiv.org/abs/math/0502024
http://front.math.ucdavis.edu/math.PR/0502024
(alternate)
Author(s): Christian Borgs and Jennifer Chayes and Stephan Mertens and Chandra Nair
Abstract: The number partitioning problem is a classic problem of combinatorial
optimization in which a set of $n$ numbers is partitioned into two subsets such
that the sum of the numbers in one subset is as close as possible to the sum of
the numbers in the other set. When the $n$ numbers are i.i.d. variables drawn
from some distribution, the partitioning problem turns out to be equivalent to
a mean-field antiferromagnetic Ising spin glass. In the spin glass
representation, it is natural to define energies -- corresponding to the costs
of the partitions, and overlaps -- corresponding to the correlations between
partitions. Although the energy levels of this model are {\em a priori} highly
correlated, a surprising recent conjecture asserts that the energy spectrum of
number partitioning is locally that of a random energy model (REM): the
spacings between nearby energy levels are uncorrelated. In other words, the
properly scaled energies converge to a Poisson process. The conjecture also
asserts that the corresponding spin configurations are uncorrelated, indicating
vanishing overlaps in the spin glass representation. In this paper, we prove
these two claims, collectively known as the local REM conjecture.
http://arXiv.org/abs/cond-mat/0501760
http://front.math.ucdavis.edu/cond-mat/0501760
(alternate)
Author(s): Majid Hosseini
Abstract: Let $U$ be a domain, convex in $x$ and symmetric about the y-axis, which is
contained in a centered and oriented rectangle $R$. \linebreak If $\tau_A$ is
the first exit time of Brownian motion from $A$ and $A^+=A\cap \{(x,y):x>0\}$,
it is proved that $P^z(\tau_{U^+}>s\mid \tau_{R^+}>t)\leq P^z(\tau_{U}>s\mid
\tau_{R}>t)$ for every $s,t>0$ and every $z\in U^+$.
http://arXiv.org/abs/math/0502057
http://front.math.ucdavis.edu/math.PR/0502057
(alternate)
Author(s): Rami Atar and Amarjit Budhiraja and P. Dupuis
Abstract: Let G \subset \R^k be a convex polyhedral cone with vertex at the origin
given as the intersection of half spaces {G_i, i= 1, ..., N}, where n_i and d_i
denote the inward normal and direction of constraint associated with G_i,
respectively. Stability properties of a class of diffusion processes,
constrained to take values in G, are studied under the assumption that the
Skorokhod problem defined by the data {(n_i, d_i), i = 1, ..., N} is well posed
and the Skorokhod map is Lipschitz continuous. Explicit conditions on the drift
coefficient, b(\cdot), of the diffusion process are given under which the
constrained process is positive recurrent and has a unique invariant measure.
Define \C \Df{- \sum_{i=1}^N \alpha_i d_i; \alpha_i \ge 0, i \in \{1, ..., N}}.
Then the key condition for stability is that there exists \delta \in (0,
\infty) and a bounded subset A of G such that for all x \in G\backslash A, b(x)
\in \C and \dist(b(x), \partial \C) \ge \delta, where \partial \C denotes the
boundary of \C.
http://arXiv.org/abs/math/0501018
http://front.math.ucdavis.edu/math.PR/0501018
(alternate)
Author(s): Alexander Yu. Veretennikov
Abstract: We establish the large deviation principle for stochastic differential
equations with averaging in the case when all coefficients of the fast
component depend on the slow one, including diffusion.
http://arXiv.org/abs/math/0502098
http://front.math.ucdavis.edu/math.PR/0502098
(alternate)
Author(s): Mohamed El Machkouri (LMRS)
Abstract: We investigate the nonparametric estimation for regression in a fixed-design
setting when the errors are given by a field of dependent random variables.
Sufficient conditions for kernel estimators to converge uniformly are obtained.
These estimators can attain the optimal rates of uniform convergence and the
results apply to a large class of random fields which contains
martingale-difference random fields and mixing random fields.
http://arXiv.org/abs/math/0502091
http://front.math.ucdavis.edu/math.ST/0502091
(alternate)
Author(s): Philippe Robert (RAP UR-R)
Abstract: A simple approach is presented to study the asymptotic behavior of some
algorithms with an underlying tree structure. It is shown that some asymptotic
oscillating behaviors can be precisely analyzed without resorting to complex
analysis techniques as it is usually done in this context. A new explicit
representation of periodic functions involved is obtained at the same time.
http://arXiv.org/abs/cs/0502014
http://front.math.ucdavis.edu/cs.DS/0502014
(alternate)
Author(s): Alexander Yu. Veretennikov
Abstract: We establish the large deviation principle for stochastic differential
equations with averaging in the case when all coefficients of the fast
component depend on the slow one, including diffusion.
http://arXiv.org/abs/math/0502098
http://front.math.ucdavis.edu/math.PR/0502098
(alternate)
Author(s): Mohamed El Machkouri (LMRS)
Abstract: We investigate the nonparametric estimation for regression in a fixed-design
setting when the errors are given by a field of dependent random variables.
Sufficient conditions for kernel estimators to converge uniformly are obtained.
These estimators can attain the optimal rates of uniform convergence and the
results apply to a large class of random fields which contains
martingale-difference random fields and mixing random fields.
http://arXiv.org/abs/math/0502091
http://front.math.ucdavis.edu/math.ST/0502091
(alternate)
Author(s): Philippe Robert (RAP UR-R)
Abstract: A simple approach is presented to study the asymptotic behavior of some
algorithms with an underlying tree structure. It is shown that some asymptotic
oscillating behaviors can be precisely analyzed without resorting to complex
analysis techniques as it is usually done in this context. A new explicit
representation of periodic functions involved is obtained at the same time.
http://arXiv.org/abs/cs/0502014
http://front.math.ucdavis.edu/cs.DS/0502014
(alternate)
Author(s): Ted Theodosopoulos and Ming Yuen
Abstract: In this short paper we define the wealth process in a spin model for market
microstructure, for individual agents and in aggregate. The agents in our model
try to balance their desire to belong to the local majority (herding behavior),
defined over random network neighborhoods, and the occasional advantage of
belonging to the global minority (contrarian trading). We arrive at a
classification of the martingale properties of this wealth process and use it
to determine the strategic stability of the agents' interactions. Our goal is
to add a behavioral interpretation to this stochastic agent-based model for
market fluctuations.
http://arXiv.org/abs/math/0502105
http://front.math.ucdavis.edu/math.PR/0502105
(alternate)
Author(s): Jean Bertoin (PMA)
Abstract: This text surveys different probabilistic aspects of a model which is used to
describe the evolution of an object that falls apart randomly as time passes.
Each point of view yields useful techniques to establish properties of such
random fragmentation processes.
http://arXiv.org/abs/math/0502132
http://front.math.ucdavis.edu/math.PR/0502132
(alternate)
Author(s): Mohamed El Machkouri (LMRS) and Lahcen Ouchti (LMRS)
Abstract: We investigate the invariance principle for set-indexed partial sums of a
stationary field $(X\_{k})\_{k\in\mathbb{Z}^{d}}$ of martingale-difference or
independent random variables under standard-normalization or self-normalization
respectively.
http://arXiv.org/abs/math/0502135
http://front.math.ucdavis.edu/math.PR/0502135
(alternate)
Author(s): Kenneth S. Alexander and Vladas Sidoravicius
Abstract: We consider a polymer, with monomer locations modeled by the trajectory of a
Markov chain, in the presence of a potential that interacts with the polymer
when it visits a particular site 0. Disorder is introduced by, for example,
having the interaction vary from one monomer to another, as a constant $u$ plus
i.i.d. mean-0 randomness. There is a critical value of $u$ above which the
polymer is pinned, placing a positive fraction of its monomers at 0 with high
probability. This critical point may differ for the quenched, annealed and
deterministic cases. We show that self-averaging occurs, we evaluate the
critical point for a deterministic interaction and establish our main result
that the critical point in the quenched case is strictly smaller. We show that
for every fixed $u \in \mathbb{R}$, pinning occurs at sufficiently low
temperatures. If the excursion length distribution has polynomial tails and the
interaction does not have a finite exponential moment, then pinning occurs for
all $u \in \mathbb{R}$ at arbitrary temperature. Our results apply to other
mathematically similar situations as well, such as a directed polymer that
interacts with a random potential located in a one-dimensional defect, or an
interface in two dimensions interacting with a random potential along a wall.
http://arXiv.org/abs/math/0501028
http://front.math.ucdavis.edu/math.PR/0501028
(alternate)
Author(s): Radford M. Neal
Abstract: I show how Markov chain sampling with the Metropolis-Hastings algorithm can
be modified so as to take bigger steps when the distribution being sampled from
has the characteristic that its density can be quickly recomputed for a new
point if this point differs from a previous point only with respect to a subset
of 'fast' variables. I show empirically that when using this method, the
efficiency of sampling for the remaining 'slow' variables can approach what
would be possible using Metropolis updates based on the marginal distribution
for the slow variables.
http://arXiv.org/abs/math/0502099
http://front.math.ucdavis.edu/math.ST/0502099
(alternate)
Author(s): Gilles Pag\`{e}s (PMA)
Abstract: In a recent paper, M. Bena\"{i}m and G. Ben Arous solve a multi-armed bandit
problem arising in the theory of learning in games. We propose an short
elementary proof of this result based on a variant of the Kronecker Lemma.
http://arXiv.org/abs/math/0502182
http://front.math.ucdavis.edu/math.PR/0502182
(alternate)
Author(s): Bruno Bouchard (PMA) and Emmanuel Temam (PMA)
Abstract: In this note, we consider a general discrete time financial market with
proportional transaction costs as in Kabanov and Stricker (2001), Kabanov et
al. (2002), Kabanov et al. (2003) and Schachermayer (2004). We provide a dual
formulation for the set of initial endowments which allow to super-hedge some
American claim. We show that this extends the result of Chalasani and Jha
(2001) which was obtained in a model with constant transaction costs and risky
assets which evolve on a finite dimensional tree. We also provide fairly
general conditions under which the expected formulation in terms of stopping
times does not work.
http://arXiv.org/abs/math/0502189
http://front.math.ucdavis.edu/math.PR/0502189
(alternate)
Author(s): M. R. Pistorius
Abstract: Consider the problem to explicitly calculate the law of the first passage
time T(a) of a general Levy process Z above a positive level a. In this paper
it is shown that the law of T(a) can be approximated arbitrarily closely by the
laws of T^n(a), the corresponding first passages time for X^n, where (X^n)_n is
a sequence of Levy processes whose positive jumps follow a phase-type
distribution. Subsequently, explicit expressions are derived for the laws of
T^n(a) and the upward ladder process of X^n.
The derivation is based on an embedding of X^n into a class of Markov
additive processes and on the solution of the fundamental (matrix) Wiener-Hopf
factorisation for this class.
This Wiener-Hopf factorisation can be computed explicitly by solving
iteratively a certain fixed point equation. It is shown that, typically, this
iteration converges geometrically fast.
http://arXiv.org/abs/math/0502192
http://front.math.ucdavis.edu/math.PR/0502192
(alternate)
Author(s): Thomas Garel and Cecile Monthus
Abstract: We numerically study the wetting (adsorption) transition of a polymer chain
on a disordered substrate in 1+1 dimension.Following the Poland-Scheraga model
of DNA denaturation, we use a Fixman-Freire scheme for the entropy of loops.
This allows us to consider chain lengths of order $N \sim 10^5 $ to $10^6$,
with $10^4$ disorder realizations. Our study is based on the statistics of
loops between two contacts with the substrate, from which we define Binder-like
parameters: their crossings for various sizes $N$ allow a precise determination
of the critical temperature, and their finite size properties yields a
crossover exponent $\phi=1/(2-\alpha) \simeq 0.5$.We then analyse at
criticality the distribution of loop length $l$ in both regimes $l \sim O(N)$
and $1 \ll l \ll N$, as well as the finite-size properties of the contact
density and energy. Our conclusion is that the critical exponents for the
thermodynamics are the same as those of the pure case, except for strong
logarithmic corrections to scaling. The presence of these logarithmic
corrections in the thermodynamics is related to a disorder-dependent
logarithmic singularity that appears in the critical loop distribution in the
rescaled variable $\lambda=l/N$ as $\lambda \to 1$.
http://arXiv.org/abs/cond-mat/0502195
http://front.math.ucdavis.edu/cond-mat/0502195
(alternate)
Author(s): N. Berger and C. Borgs and J. T. Chayes and R. M. D'Souza and R. D. Kleinberg
Abstract: We introduce a family of one-dimensional geometric growth models, constructed
iteratively by locally optimizing the tradeoffs between two competing metrics,
and show that this family is equivalent to a family of preferential attachment
random graph models with upper cutoffs. This is the first explanation of how
preferential attachment can arise from a more basic underlying mechanism of
local competition. We rigorously determine the degree distribution for the
family of random graph models, showing that it obeys a power law up to a finite
threshold and decays exponentially above this threshold.
We also rigorously analyze a generalized version of our graph process, with
two natural parameters, one corresponding to the cutoff and the other a
``fertility'' parameter. We prove that the general model has a power-law degree
distribution up to a cutoff, and establish monotonicity of the power as a
function of the two parameters. Limiting cases of the general model include the
standard preferential attachment model without cutoff and the uniform
attachment model.
http://arXiv.org/abs/cond-mat/0502205
http://front.math.ucdavis.edu/cond-mat/0502205
(alternate)
Author(s): K. Hamza and F.C. Klebaner
Abstract: We prove that if the Black-Scholes formula holds with the spot volatility for
call options with all strikes, then the volatility parameter is constant. The
proof relies some result on semimartingales (Theorem 2) of independent
interest.
http://arXiv.org/abs/math/0502201
http://front.math.ucdavis.edu/math.PR/0502201
(alternate)
Author(s): Giovanni Peccati (LSTA) and Mich\`{e}le Thieullen (PMA) and Ciprian A. Tudor (SAMOS)
Abstract: Let the process Y(t) be a Skorohod integral process with respect to Brownian
motion. We use a recent result by Tudor (2004), to prove that Y(t) can be
represented as the limit of linear combinations of processes that are products
of forward and backward Brownian martingales. Such a result is a further step
towards the connection between the theory of continuous-time (semi)martingales,
and that of anticipating stochastic integration. We establish an explicit link
between our results and the classic characterization, due to Duc and Nualart
(1990), of the chaotic decomposition of Skorohod integral processes. We also
explore the case of Skorohod integral processes that are time-reversed Brownian
martingales, and provide an "anticipating" counterpart to the classic Optional
Sampling Theorem for It\^{o} stochastic integrals.
http://arXiv.org/abs/math/0502208
http://front.math.ucdavis.edu/math.PR/0502208
(alternate)
Author(s): Jean Bertoin (PMA) and Gr\'{e}gory Marc Miermont (LM-Orsay)
Abstract: We consider a generalized version of Knuth's parking problem, in which
caravans consisting of a number of cars arrive at random on the unit circle.
Then each car turns clockwise until it finds a free space to park. Extending a
recent work by Chassaing and Louchard, we relate the asymptotics for the sizes
of blocks formed by occupied spots with the dynamics of the additive
coalescent. According to the behavior of the caravan's size tail distribution,
several qualitatively different versions of eternal additive coalescent are
involved.
http://arXiv.org/abs/math/0502220
http://front.math.ucdavis.edu/math.PR/0502220
(alternate)
Author(s): Rami Atar and Paul Dupuis and Adam Shwartz
Abstract: We consider the problem of risk-sensitive control of a stochastic network. In
controlling such a network, an escape time criterion can be useful if one
wishes to regulate the occurrence of large buffers and buffer overflow. In this
paper a risk-sensitive escape time criterion is formulated, which in comparison
to the ordinary escape time criteria penalizes exits which occur on short time
intervals more heavily. The properties of the risk-sensitive problem are
studied in the large buffer limit, and related to the value of a deterministic
differential game with constrained dynamics. We prove that the game has value,
and that the value is the (viscosity) solution of a PDE. For a simple network,
the value is computed, demonstrating the applicability of the approach.
http://arXiv.org/abs/math/0501031
http://front.math.ucdavis.edu/math.PR/0501031
(alternate)
Author(s): Yuval Peres and K\'aroly Simon and Boris Solomyak
Abstract: We consider linear iterated function systems with a random multiplicative
error on the real line. Our system is $\{x\mapsto d_i + \lambda_i Y
x\}_{i=1}^m$, where $d_i\in \R$ and $\lambda_i>0$ are fixed and $Y> 0$ is a
random variable with an absolutely continuous distribution. The iterated maps
are applied randomly according to a stationary ergodic process, with the
sequence of i.i.d. errors $y_1,y_2,...$, distributed as $Y$, independent of
everything else. Let $h$ be the entropy of the process, and let $\chi =
E[\log(\lambda Y)]$ be the Lyapunov exponent. Assuming that $\chi < 0$, we
obtain a family of conditional measures $\nu_y$ on the line, parametrized by $y
= (y_1,y_2,...)$, the sequence of errors. Our main result is that if $h >
|\chi|$, then $\nu_y$ is absolutely continuous with respect to the Lebesgue
measure for a.e. $y$. We also prove that if $h < |\chi|$, then the measure
$\nu_y$ is singular and has dimension $h/|\chi|$ for a.e. $y$. These results
are applied to a randomly perturbed IFS suggested by Y. Sinai, and to a class
of random sets considered by R. Arratia, motivated by probabilistic number
theory.
http://arXiv.org/abs/math/0502200
http://front.math.ucdavis.edu/math.DS/0502200
(alternate)
Author(s): Steven N. Evans and Anita Winter
Abstract: We use Dirichlet form methods to construct and analyze a reversible Markov
process, the stationary distribution of which is the Brownian continuum random
tree. This process is inspired by the subtree prune and re-graft (SPR) Markov
chains that appear in phylogenetic analysis. A key technical ingredient in this
work is the use of a novel Gromov--Hausdorff type distance to metrize the space
whose elements are compact real trees equipped with a probability measure.
Also, the investigation of the Dirichlet form hinges on a new path
decomposition of the Brownian excursion.
http://arXiv.org/abs/math/0502226
http://front.math.ucdavis.edu/math.PR/0502226
(alternate)
Author(s): Svante Janson
Abstract: We study the asymptotic distribution of the displacements in hashing with
coalesced chains, for both late-insertion and early-insertion. Asymptotic
formulas for means and variances follow. The method uses Poissonization and
some stochastic calculus.
http://arXiv.org/abs/math/0502232
http://front.math.ucdavis.edu/math.PR/0502232
(alternate)
Author(s): Christina Goldschmidt and James B. Martin
Abstract: We describe a representation of the Bolthausen-Sznitman coalescent in terms
of the cutting of random recursive trees. Using this representation, we prove
results concerning the final collision of the coalescent restricted to [n]: we
show that the distribution of the number of blocks involved in the final
collision converges as n tends to infinity, and obtain a scaling law for the
sizes of these blocks. We also consider the discrete-time Markov chain giving
the number of blocks after each collision of the coalescent restricted to [n];
we show that the transition probabilities of the time-reversal of this Markov
chain have limits as n tends to infinity. These results can be interpreted as
describing a ``post-gelation'' phase of the Bolthausen-Sznitman coalescent, in
which a giant cluster containing almost all of the mass has already formed and
the remaining small blocks are being absorbed.
http://arXiv.org/abs/math/0502263
http://front.math.ucdavis.edu/math.PR/0502263
(alternate)
Author(s): Nicholas Pippenger
Abstract: We consider crossbar switching networks with base $b$ (that is, constructed
from $b\times b$ crossbar switches), scale $k$ (that is, with $b^k$ inputs,
$b^k$ outputs and $b^k$ links between each consecutive pair of stages) and
depth $l$ (that is, with $l$ stages). We assume that the crossbars are
interconnected according to the spider-web pattern, whereby two diverging paths
reconverge only after at least $k$ stages. We assume that each vertex is
independently idle with probability $q$, the vacancy probability. We assume
that $b\ge 2$ and the vacancy probability $q$ are fixed, and that $k$ and $l =
ck$ tend to infinity with ratio a fixed constant $c>1$. We consider the linking
probability $Q$ (the probability that there exists at least one idle path
between a given idle input and a given idle output). In a previous paper it was
shown that if $c\le 2$, then the linking probability $Q$ tends to 0 if
$01$. This is done by using generating functions
and complex-variable techniques to estimate the second moments of various
random variables involved in the analysis of the networks.
http://arXiv.org/abs/math/0502294
http://front.math.ucdavis.edu/math.PR/0502294
(alternate)
Author(s): Rami Atar and Paul Dupuis and Adam Shwartz
Abstract: We consider optimal control of a stochastic network,where service is
controlled to prevent buffer overflow. We use a risk-sensitive escape time
criterion, which in comparison to the ordinary escape time criteria heavily
penalizes exits which occur on short time intervals. A limit as the buffer
sizes tend to infinity is considered. In [2] we showed that, for a large class
of networks, the limit of the normalized cost agrees with the value function of
a differential game. The game's value is characterized in [2] as the unique
solution to a Hamilton-Jacobi-Bellman Partial Differential Equation (PDE). In
the current paper we apply this general theory to the important case of a
network of queues in tandem. Our main results are: (i) the construction of an
explicit solution to the corresponding PDE, and (ii) drawing out the
implications for optimal risk-sensitive and robust regulation of the network.
In particular, the following general principle can be extracted. To avoid
buffer overflow there is a natural competition between two tendencies. One may
choose to serve a particular queue, since that will help prevent its own buffer
from overflowing, or one may prefer to stop service, with the goal of
preventing overflow of buffers further down the line. The solution to the PDE
indicates the optimal choice between these two, specifying the parts of the
state space where each queue must be served (so as not to lose optimality), and
where it can idle.
http://arXiv.org/abs/math/0501035
http://front.math.ucdavis.edu/math.PR/0501035
(alternate)
Author(s): Giambattista Giacomin and Fabio Lucio Toninelli
Abstract: We consider a directed random walk model of a random heterogeneous polymer in
the proximity of an interface separating two selective solvents. This model
exhibits a localization/delocalization transition. A positive value of the free
energy corresponds to the localized regime and strong results on the polymer
path behavior are known in this case. We focus on the interior of the
delocalized phase, which is characterized by the free energy equal to zero, and
we show in particular that in this regime there are O(log N) monomers in the
unfavorable solvent (N is the length of the polymer). The previously known
result was o(N). Our approach is based on concentration bounds on suitably
restricted partition functions. The same idea allows also to interpolate
between different types of disorder in the weak coupling limit. In this way we
show the universal nature of this limit, previously considered only for binary
disorder.
http://arXiv.org/abs/math/0502304
http://front.math.ucdavis.edu/math.PR/0502304
(alternate)
Author(s): Alexis Devulder (PMA)
Abstract: In this paper, we are interested in some questions of Greven and den
Hollander about the rate function $I\_{\eta}^q$ of quenched large deviations
for random walk in random environment. By studying the hitting times of RWRE,
we prove that in the recurrent case, $\lim\_{\theta\to
0^+}(I\_{\eta}^q)''(\theta)=+\infty$, which gives an affirmative answer to a
conjecture of Greven and den Hollander. We also establish a comparison result
between the rate function of quenched large deviations for a diffusion in a
drifted Brownian potential, and the rate function for a drifted Brownian motion
with the same speed.
http://arXiv.org/abs/math/0502316
http://front.math.ucdavis.edu/math.PR/0502316
(alternate)
Author(s): Vincent Lemaire
Abstract: We propose a new scheme for the long time approximation of a diffusion when
the drift vector field is not globally Lipschitz. Under this assumption,
regular explicit Euler scheme --with constant or decreasing step-- may explode
and implicit Euler scheme are CPU-time expensive. The algorithm we introduce is
explicit and we prove that any weak limit of the weighted empirical measures of
this scheme is a stationary distribution of the stochastic differential
equation. Several examples are presented including gradient dissipative systems
and Hamiltonian dissipative systems.
http://arXiv.org/abs/math/0502317
http://front.math.ucdavis.edu/math.PR/0502317
(alternate)
Author(s): J. Heffernan & S. Resnick
Abstract: Models based on assumptions of multivariate regular variation and hidden
regular variation provide ways to describe a broad range of extremal dependence
structures when marginal distributions are heavy tailed. Multivariate regular
variation provides a rich description of extremal dependence in the case of
asymptotic dependence, but fails to distinguish between exact independence and
asymptotic independence. Hidden regular variation addresses this problem by
requiring components of the random vector to be simultaneously large but on a
smaller scale than the scale for the marginal distributions. In doing so,
hidden regular variation typically restricts attention to that part of the
probability space where all variables are simultaneously large. However, since
under asymptotic independence the largest values do not occur in the same
observation, the region where variables are simultaneously large may not be of
primary interest. A different philosophy was offered in the paper of Heffernan
and Tawn (2004) which allows examination of distributional tails other than the
joint tail. This approach used an asymptotic argument which conditions on one
component of the random vector and finds the limiting conditional distribution
of the remaining components as the conditioning variable becomes large. In this
paper, we provide a thorough mathematical examination of the limiting arguments
building on the orientation of Heffernan and Tawn (2004). We examine the
conditions required for the assumptions made by the conditioning approach to
hold, and highlight similarities and differences between the new and
established methods.
http://arXiv.org/abs/math/0502324
http://front.math.ucdavis.edu/math.PR/0502324
(alternate)
Author(s): Jan Poland and Marcus Hutter
Abstract: We study the properties of the MDL (or maximum penalized complexity)
estimator for Regression and Classification, where the underlying model class
is countable. We show in particular a finite bound on the Hellinger losses
under the only assumption that there is a "true" model contained in the class.
This implies almost sure convergence of the predictive distribution to the true
one at a fast rate. It corresponds to Solomonoff's central theorem of universal
induction, however with a bound that is exponentially larger.
http://arXiv.org/abs/math/0502315
http://front.math.ucdavis.edu/math.ST/0502315
(alternate)
Author(s): Magdalena Musat
Abstract: We study the operator space UMD property, introduced by Pisier in the context
of noncommutative vector-valued Lp-spaces. It is unknown whether the property
is independent of p in this setting. We prove that for 1
http://arXiv.org/abs/math/0501033
http://front.math.ucdavis.edu/math.OA/0501033
(alternate)
Author(s): Marcus Hutter
Abstract: Various optimality properties of universal sequence predictors based on
Bayes-mixtures in general, and Solomonoff's prediction scheme in particular,
will be studied. The probability of observing $x_t$ at time $t$, given past
observations $x_1...x_{t-1}$ can be computed with the chain rule if the true
generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If
$\mu$ is unknown, but known to belong to a countable or continuous class $\M$
one can base ones prediction on the Bayes-mixture $\xi$ defined as a
$w_\nu$-weighted sum or integral of distributions $\nu\in\M$. The cumulative
expected loss of the Bayes-optimal universal prediction scheme based on $\xi$
is shown to be close to the loss of the Bayes-optimal, but infeasible
prediction scheme based on $\mu$. We show that the bounds are tight and that no
other predictor can lead to significantly smaller bounds. Furthermore, for
various performance measures, we show Pareto-optimality of $\xi$ and give an
Occam's razor argument that the choice $w_\nu\sim 2^{-K(\nu)}$ for the weights
is optimal, where $K(\nu)$ is the length of the shortest program describing
$\nu$. The results are applied to games of chance, defined as a sequence of
bets, observations, and rewards. The prediction schemes (and bounds) are
compared to the popular predictors based on expert advice. Extensions to
infinite alphabets, partial, delayed and probabilistic prediction,
classification, and more active systems are briefly discussed.
http://arXiv.org/abs/cs/0311014
http://front.math.ucdavis.edu/cs.LG/0311014
(alternate)
Author(s): Jan Poland and Marcus Hutter
Abstract: We consider the Minimum Description Length principle for online sequence
prediction. If the underlying model class is discrete, then the total expected
square loss is a particularly interesting performance measure: (a) this
quantity is bounded, implying convergence with probability one, and (b) it
additionally specifies a `rate of convergence'. Generally, for MDL only
exponential loss bounds hold, as opposed to the linear bounds for a Bayes
mixture. We show that this is even the case if the model class contains only
Bernoulli distributions. We derive a new upper bound on the prediction error
for countable Bernoulli classes. This implies a small bound (comparable to the
one for Bayes mixtures) for certain important model classes. The results apply
to many Machine Learning tasks including classification and hypothesis testing.
We provide arguments that our theorems generalize to countable classes of
i.i.d. models.
http://arXiv.org/abs/cs/0407039
http://front.math.ucdavis.edu/cs.LG/0407039
(alternate)
Author(s): Marcus Hutter and Andrej Muchnik
Abstract: Solomonoff's central result on induction is that the posterior of a universal
semimeasure M converges rapidly and with probability 1 to the true sequence
generating posterior mu, if the latter is computable. Hence, M is eligible as a
universal sequence predictor in case of unknown mu. Despite some nearby results
and proofs in the literature, the stronger result of convergence for all
(Martin-Loef) random sequences remained open. Such a convergence result would
be particularly interesting and natural, since randomness can be defined in
terms of M itself. We show that there are universal semimeasures M which do not
converge for all random sequences, i.e. we give a partial negative answer to
the open problem. We also provide a positive answer for some non-universal
semimeasures. We define the incomputable measure D as a mixture over all
computable measures and the enumerable semimeasure W as a mixture over all
enumerable nearly-measures. We show that W converges to D and D to mu on all
random sequences. The Hellinger distance measuring closeness of two
distributions plays a central role.
http://arXiv.org/abs/cs/0407057
http://front.math.ucdavis.edu/cs.LG/0407057
(alternate)
Author(s): Harald Luschgy and Gilles Pag\`{e}s (PMA)
Abstract: We derive a high-resolution formula for the $L^2$-quantization errors of
Riemann-Liouville processes and the sharp Kolmogorov entropy asymptotics for
related Sobolev balls. We describe a quantization procedure which leads to
asymptotically optimal functional quantizers. Regular variation of the
eigenvalues of the covariance operator plays a crucial role.
http://arXiv.org/abs/math/0502375
http://front.math.ucdavis.edu/math.PR/0502375
(alternate)
Author(s): Marcus Hutter
Abstract: The problem of making sequential decisions in unknown probabilistic
environments is studied. In cycle $t$ action $y_t$ results in perception $x_t$
and reward $r_t$, where all quantities in general may depend on the complete
history. The perception $x_t$ and reward $r_t$ are sampled from the (reactive)
environmental probability distribution $\mu$. This very general setting
includes, but is not limited to, (partial observable, k-th order) Markov
decision processes. Sequential decision theory tells us how to act in order to
maximize the total expected reward, called value, if $\mu$ is known.
Reinforcement learning is usually used if $\mu$ is unknown. In the Bayesian
approach one defines a mixture distribution $\xi$ as a weighted sum of
distributions $\nu\in\M$, where $\M$ is any class of distributions including
the true environment $\mu$. We show that the Bayes-optimal policy $p^\xi$ based
on the mixture $\xi$ is self-optimizing in the sense that the average value
converges asymptotically for all $\mu\in\M$ to the optimal value achieved by
the (infeasible) Bayes-optimal policy $p^\mu$ which knows $\mu$ in advance. We
show that the necessary condition that $\M$ admits self-optimizing policies at
all, is also sufficient. No other structural assumptions are made on $\M$. As
an example application, we discuss ergodic Markov decision processes, which
allow for self-optimizing policies. Furthermore, we show that $p^\xi$ is
Pareto-optimal in the sense that there is no other policy yielding higher or
equal value in {\em all} environments $\nu\in\M$ and a strictly higher value in
at least one.
http://arXiv.org/abs/cs/0204040
http://front.math.ucdavis.edu/cs.AI/0204040
(alternate)
Author(s): Bruno Bouchard (PMA and Crest and Lfa)
Abstract: We discuss the no-arbitrage conditions in a general framework for
discrete-time models of financial markets with proportional transaction costs
and general information structure. We extend the results of Kabanov and al.
(2002), Kabanov and al. (2003) and Schachermayer (2004) to the case where
bid-ask spreads are not known with certainty. In the "no-friction" case, we
retrieve the result of Kabanov and Stricker (2003).
http://arXiv.org/abs/math/0501045
http://front.math.ucdavis.edu/math.PR/0501045
(alternate)
Author(s): Marcus Hutter
Abstract: Solomonoff's uncomputable universal prediction scheme $\xi$ allows to predict
the next symbol $x_k$ of a sequence $x_1...x_{k-1}$ for any Turing computable,
but otherwise unknown, probabilistic environment $\mu$. This scheme will be
generalized to arbitrary environmental classes, which, among others, allows the
construction of computable universal prediction schemes $\xi$. Convergence of
$\xi$ to $\mu$ in a conditional mean squared sense and with $\mu$ probability 1
is proven. It is shown that the average number of prediction errors made by the
universal $\xi$ scheme rapidly converges to those made by the best possible
informed $\mu$ scheme. The schemes, theorems and proofs are given for general
finite alphabet, which results in additional complications as compared to the
binary case. Several extensions of the presented theory and results are
outlined. They include general loss functions and bounds, games of chance,
infinite alphabet, partial and delayed prediction, classification, and more
active systems.
http://arXiv.org/abs/cs/0106036
http://front.math.ucdavis.edu/cs.LG/0106036
(alternate)
Author(s): Marcus Hutter
Abstract: The probability of observing $x_t$ at time $t$, given past observations
$x_1...x_{t-1}$ can be computed with Bayes' rule if the true generating
distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If $\mu$ is
unknown, but known to belong to a class $M$ one can base ones prediction on the
Bayes mix $\xi$ defined as a weighted sum of distributions $\nu\in M$. Various
convergence results of the mixture posterior $\xi_t$ to the true posterior
$\mu_t$ are presented. In particular a new (elementary) derivation of the
convergence $\xi_t/\mu_t\to 1$ is provided, which additionally gives the rate
of convergence. A general sequence predictor is allowed to choose an action
$y_t$ based on $x_1...x_{t-1}$ and receives loss $\ell_{x_t y_t}$ if $x_t$ is
the next symbol of the sequence. No assumptions are made on the structure of
$\ell$ (apart from being bounded) and $M$. The Bayes-optimal prediction scheme
$\Lambda_\xi$ based on mixture $\xi$ and the Bayes-optimal informed prediction
scheme $\Lambda_\mu$ are defined and the total loss $L_\xi$ of $\Lambda_\xi$ is
bounded in terms of the total loss $L_\mu$ of $\Lambda_\mu$. It is shown that
$L_\xi$ is bounded for bounded $L_\mu$ and $L_\xi/L_\mu\to 1$ for $L_\mu\to
\infty$. Convergence of the instantaneous losses are also proven.
http://arXiv.org/abs/cs/0301014
http://front.math.ucdavis.edu/cs.LG/0301014
(alternate)
Author(s): Marcus Hutter and Marco Zaffalon
Abstract: Given the joint chances of a pair of random variables one can compute
quantities of interest, like the mutual information. The Bayesian treatment of
unknown chances involves computing, from a second order prior distribution and
the data likelihood, a posterior distribution of the chances. A common
treatment of incomplete data is to assume ignorability and determine the
chances by the expectation maximization (EM) algorithm. The two different
methods above are well established but typically separated. This paper joins
the two approaches in the case of Dirichlet priors, and derives efficient
approximations for the mean, mode and the (co)variance of the chances and the
mutual information. Furthermore, we prove the unimodality of the posterior
distribution, whence the important property of convergence of EM to the global
maximum in the chosen framework. These results are applied to the problem of
selecting features for incremental learning and naive Bayes classification. A
fast filter based on the distribution of mutual information is shown to
outperform the traditional filter based on empirical mutual information on a
number of incomplete real data sets.
http://arXiv.org/abs/cs/0306126
http://front.math.ucdavis.edu/cs.LG/0306126
(alternate)
Author(s): Ivan Werner
Abstract: In this paper we calculate Kolmogorov-Sinai entropy $h_M(S)$ of the
generalized Markov shift associated with a contractive Markov system (CMS)
\cite{Wer1} using the coding map constructed in \cite{Wer3}. We show that
\[h_M(S)=-\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_e\log p_ed\mu\] where
$\mu$ is a unique invariant Borel probability measure of the CMS. I. Werner,
Contractive Markov systems, J. London Math. Soc. (2005) 236-258. I. Werner,
Coding map for a contractive Markov system, Math. Proc. Camb. Phil. Soc. to
appear 140 (2), March 2006.
http://arXiv.org/abs/math/0502389
http://front.math.ucdavis.edu/math.DS/0502389
(alternate)
Author(s): B. Rider and Jack W. Silverstein
Abstract: Consider an ensemble of $N \times N$ non-Hermitian matrices in which all
entries are independent identically distributed complex random variables of
mean zero and absolute mean-square one. If the entry distributions also possess
bounded densities and finite $(4+\ep)$ moments, then Z.D. Bai has shown the
ensemble to satisfy the circular law: after scaling by a factor of $1/\sqrt{N}$
and letting $N \ra \infty$, the empirical measure of the eigenvalues converges
weakly to the uniform measure on the unit disk in the complex plane. In this
note we investigate fluctuations from the circular law in a more restrictive
class of non-Hermitian matrices for which higher moments of the entries obey a
growth condition. The main result is a central limit theorem for linear
statistics of type $X_N(f) = \sum_{k=1}^N f(\ld_k)$ where $\lambda_1,
\lambda_2, ..., \lambda_N$ denote the ensemble eigenvalues and the test
function $f$ is analytic on an appropriate domain.
http://arXiv.org/abs/math/0502400
http://front.math.ucdavis.edu/math.PR/0502400
(alternate)
Author(s): James Allen Fill and Nevin Kapur
Abstract: Using recent results on singularity analysis for Hadamard products of
generating functions, we obtain the limiting distributions for additive
functionals on $m$-ary search trees on $n$ keys with toll sequence (i)
$n^\alpha$ with $\alpha \geq 0$ ($\alpha=0$ and $\alpha=1$ correspond roughly
to the space requirement and total path length, respectively); (ii) $\ln
\binom{n}{m-1}$, which corresponds to the so-called shape functional; and (iii)
$\mathbf{1}_{n=m-1}$, which corresponds to the number of leaves.
http://arXiv.org/abs/math/0502422
http://front.math.ucdavis.edu/math.PR/0502422
(alternate)
Author(s): Philippe Chassaing (IEC) and Guy Louchard (ULB)
Abstract: In this paper, we consider hashing with linear probing for a hashing table
with m places, n items (n < m), and l = m
http://arXiv.org/abs/math/0501060
http://front.math.ucdavis.edu/math.PR/0501060
(alternate)
Author(s): Domenico Marinucci
Abstract: In this paper, we study the asymptotic behaviour of the angular bispectrum of
spherical random fields. Here, the asymptotic theory is developed in the
framework of fixed-radius fields, which are observed with increasing resolution
as the sample size grows. The results we present are then exploited in a set of
procedures aimed at testing non-Gaussianity; for these statistics, we are able
to show convergence to functionals of standard Brownian motion under the null
hypothesis. Analytic results are also presented on the behaviour of the tests
in the presence of a broad class of non-Gaussian alternatives. The issue of
testing for non-Gaussianity on spherical random fields has recently gained an
enormous empirical importance, especially in connection with the statistical
analysis of Cosmic Microwave Background radiation.
http://arXiv.org/abs/math/0502434
http://front.math.ucdavis.edu/math.PR/0502434
(alternate)
Author(s): David Aldous
Abstract: In a network where the cost of flow across an edge is nonlinear in the volume
of flow, and where sources and destinations are uniform, one can consider the
relationship between total volume $v$ of flow through the network and the
minimum cost $c = Psi(v)$ of any flow with volume $v$. Under a simple
probability model (locally tree-like directed network, independent cost-volume
functions or different edges) we show how to compute $\Psi(v)$ in the
infinite-size limit. The argument uses a probabilistic reformulation of the
cavity method from statistical physics, and is not rigorous as presented here.
The methodology seems potentially useful for many problems concerning flows on
this class of random networks.
http://arXiv.org/abs/cond-mat/0502346
http://front.math.ucdavis.edu/cond-mat/0502346
(alternate)
Author(s): Leonhard Euler
Abstract: This seems to be the first English translation of this paper from the French
original, ``Sur les rentes viageres''. In the paper, Euler gives a general
formula for calculating the price of a life annuity that yields a certain
amount per year, assuming the annuity manager can get a 5 percent return, for
people of different ages. He also gives formulas to calculate the price of
annuities that only start to pay out a certain number of years after they are
purchased. He gives many numerical examples, giving tables for the prices of
annuities for annuitants up to 90 years old.
http://arXiv.org/abs/math/0502421
http://front.math.ucdavis.edu/math.HO/0502421
(alternate)
Author(s): Nate Harvey and Alexander E. Holroyd and Yuval Peres and Dan Romik
Abstract: In 1977, Keane and Smorodinsky showed that there exists a finitary
homomorphism from any finite-alphabet Bernoulli process to any other
finite-alphabet Bernoulli process of strictly lower entropy. In 1996, Serafin
proved the existence of a finitary homomorphism with finite expected coding
length. In this paper, we construct such a homomorphism in which the coding
length has exponential tails. Our construction is source-universal, in the
sense that it does not use any information on the source distribution other
than the alphabet size and a bound on the entropy gap between the source and
target distributions. We also indicate how our methods can be extended to prove
a source-specific version of the result for Markov chains.
http://arXiv.org/abs/math/0502484
http://front.math.ucdavis.edu/math.PR/0502484
(alternate)
Author(s): Jean-Francois Le Gall (ENS Paris) and Mathilde Weill (ENS Paris)
Abstract: We consider a Brownian tree consisting of a collection of one-dimensional
Brownian paths started from the origin, whose genealogical structure is given
by the Continuum Random Tree (CRT). This Brownian tree may be generated from
the Brownian snake driven by a normalized Brownian excursion, and thus yields a
convenient representation of the so-called Integrated Super-Brownian Excursion
(ISE), which can be viewed as the uniform probability measure on the tree of
paths. We discuss different approaches that lead to the definition of the
Brownian tree conditioned to stay on the positive half-line. We also establish
a Verwaat-like theorem showing that this conditioned Brownian tree can be
obtained by re-rooting the unconditioned one at the vertex corresponding to the
minimal spatial position. In terms of ISE, this theorem yields the following
fact: Conditioning ISE to put no mass on $]-\infty,-\epsilon[$ and letting
$\epsilon$ go to 0 is equivalent to shifting the unconditioned ISE to the right
so that the left-most point of its support becomes the origin. We derive a
number of explicit estimates and formulas for our conditioned Brownian trees.
In particular, the probability that ISE puts no mass on $]-\infty,-\epsilon[$
is shown to behave like $2\epsilon^4/21$ when $\epsilon$ goes to 0. Finally,
for the conditioned Brownian tree with a fixed height $h$, we obtain a
decomposition involving a spine whose distribution is absolutely continuous
with respect to that of a nine-dimensional Bessel process on the time interval
$[0,h]$, and Poisson processes of subtrees originating from this spine.
http://arXiv.org/abs/math/0501066
http://front.math.ucdavis.edu/math.PR/0501066
(alternate)
Author(s): Ferenc Igloi and Cecile Monthus
Abstract: There is a large variety of quantum and classical systems in which the
quenched disorder plays a dominant r\^ole over quantum, thermal, or stochastic
fluctuations : these systems display strong spatial heterogeneities, and many
averaged observables are actually governed by rare regions. A unifying approach
to treat the dynamical and/or static singularities of these systems has emerged
recently, following the pioneering RG idea by Ma and Dasgupta and the detailed
analysis by Fisher who showed that the Ma-Dasgupta RG rules yield asymptotic
exact results if the broadness of the disorder grows indefinitely at large
scales. Here we report these new developments by starting with an introduction
of the main ingredients of the strong disorder RG method. We describe the basic
properties of infinite disorder fixed points, which are realized at critical
points, and of strong disorder fixed points, which control the singular
behaviors in the Griffiths-phases. We then review in detail applications of the
RG method to various disordered models, either (i) quantum models, such as
random spin chains, ladders and higher dimensional spin systems, or (ii)
classical models, such as diffusion in a random potential, equilibrium at low
temperature and coarsening dynamics of classical random spin chains, trap
models, delocalization transition of a random polymer from an interface, driven
lattice gases and reaction diffusion models in the presence of quenched
disorder. For several one-dimensional systems, the Ma-Dasgupta RG rules yields
very detailed analytical results, whereas for other, mainly higher dimensional
problems, the RG rules have to be implemented numerically. If available, the
strong disorder RG results are compared with another, exact or numerical
calculations.
http://arXiv.org/abs/cond-mat/0502448
http://front.math.ucdavis.edu/cond-mat/0502448
(alternate)
Author(s): W. Hachem and P. Loubaton and J. Najim
Abstract: Consider a $N\times n$ random matrix $Z_n=(Z^n_{j_1 j_2})$ where the
individual entries are a realization of a properly rescaled stationary gaussian
random field.
The purpose of this article is to study the limiting empirical distribution
of the eigenvalues of Gram random matrices such as $Z_n Z_n ^*$ and $(Z_n
+A_n)(Z_n +A_n)^*$ where $A_n$ is a deterministic matrix with appropriate
assumptions in the case where $n\to \infty$ and $\frac Nn \to c \in
(0,\infty)$.
The proof relies on related results for matrices with independent but not
identically distributed entries and substantially differs from related works in
the literature (Boutet de Monvel et al., Girko, etc.).
http://arXiv.org/abs/math/0502535
http://front.math.ucdavis.edu/math.PR/0502535
(alternate)
Author(s): Oliver Johnson and Christina Goldschmidt
Abstract: We extend Hoggar's result that the sum of two independent discrete-valued
log-concave random variables is itself log-concave. Firstly, we weaken the
assumption of independence, and introduce conditions under which the result
still holds for dependent variables. Secondly, we introduce a wider class of
random variables such that in the independent case the sum is still
log-concave, and prove simple results concerning this class.
http://arXiv.org/abs/math/0502548
http://front.math.ucdavis.edu/math.PR/0502548
(alternate)
Author(s): S Satheesh and E Sandhya
Abstract: In this note we identify Semi-Selfdecomposable Laws as the class of
distributions that can generate a linear, additive, first order auto-regressive
scheme, that is marginally stationary. We give a method to construct these
distributions. Its implications in selfsimilar and semi-selfsimilar processes
with additive increments and their subordination are given. The discrete
analogues of these processes are also discussed.
http://arXiv.org/abs/math/0412546
http://front.math.ucdavis.edu/math.PR/0412546
(alternate)
Author(s): Amine Asselah and Fabienne Castell
Abstract: We consider a d-dimensional random walk in random scenery X(n), where the
scenery consists of i.i.d. with exponential moments but a tail decay of the
form exp(-c t^a) with any}. We show that this probability is of order
exp(-(ny)^b) with b=a/(a+1).
http://arXiv.org/abs/math/0501068
http://front.math.ucdavis.edu/math.PR/0501068
(alternate)
Author(s): Thomas Duquesne (Paris 11) and Jean-Francois Le Gall (ENS Paris)
Abstract: We investigate the random continuous trees called L\'evy trees, which are
obtained as scaling limits of discrete Galton-Watson trees. We give a
mathematically precise definition of these random trees as random variables
taking values in the set of equivalence classes of compact rooted R-trees,
which is equipped with the Gromov-Hausdorff distance. To construct L\'evy
trees, we make use of the coding by the height process which was studied in
detail in previous work. We then investigate various probabilistic properties
of L\'evy trees. In particular we establish a branching property analogous to
the well-known property for Galton-Watson trees: Conditionally given the tree
below level a, the subtrees originating from that level are distributed as the
atoms of a Poisson point measure whose intensity involves a local time measure
supported on the vertices at distance a from the root. We study regularity
properties of local times in the space variable, and prove that the support of
local time is the full level set, except for certain exceptional values of a
corresponding to local extinctions. We also compute several fractal dimensions
of L\'evy trees, including Hausdorff and packing dimensions, in terms of lower
and upper indices for the branching mechanism function $\psi$ which
characterizes the distribution of the tree. We finally discuss some
applications to super-Brownian motion with a general branching mechanism.
http://arXiv.org/abs/math/0501079
http://front.math.ucdavis.edu/math.PR/0501079
(alternate)
Author(s): Magnus Bordewich and Martin Dyer and Marek Karpinski
Abstract: We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and
the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the
Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4
and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the
hardness of exact and approximate counting for both problems.
http://arXiv.org/abs/math/0501081
http://front.math.ucdavis.edu/math.PR/0501081
(alternate)
Author(s): Anne-Laure Basdevant (LPMA)
Abstract: In this paper, we study Ruelle's probability cascades in the framework of
time-inhomogeneous fragmentation processes. We describe Ruelle's cascades
mechanism exhibiting a family of measures $(\nu_t,t\in [0,1[)$ that
characterizes its infinitesimal evolution. To this end, we will first extend
the time-homogeneous fragmentation theory to the inhomogeneous case. In the
last section, we will study the behavior for small and large times of Ruelle's
fragmentation process.
http://arXiv.org/abs/math/0501088
http://front.math.ucdavis.edu/math.PR/0501088
(alternate)
Author(s): Thierry Levy (DMA)
Abstract: We construct one Yang-Mills measure on a compact surface for each isomorphism
class of principal bundles over this surface. For this, we define a new
discrete gauge theory which is essentially a covering of the usual one. We
prove that the measures correponding to different isomorphism classes of
bundles or to different total areas of the surface are mutually singular. We
give also a combinatorial computation of the partition functions based on the
formalism of fat graphs.
http://arXiv.org/abs/math-ph/0501014
http://front.math.ucdavis.edu/math-ph/0501014
(alternate)
Author(s): Yu Zhang
Abstract: Consider the first passage percolation model on ${\bf Z}^d$ for $d\geq 2$. In
this model we assign independently to each edge the value zero with probability
$p$ and the value one with probability $1-p$. We denote by $T({\bf 0}, v)$ the
passage time from the origin to $v$ for $v\in {\bf R}^d$ and $$B(t)=\{v\in {\bf
R}^d: T({\bf 0}, v)\leq t\}{and} G(t)=\{v\in {\bf R}^d: ET({\bf 0}, v)\leq
t\}.$$ It is well known that if $p < p_c$, there exists a compact shape
$B_d\subset {\bf R}^d$ such that for all $\epsilon >0$ $$t B_d(1-\epsilon)
\subset {B(t)} \subset tB_d(1+\epsilon){and} G(t)(1-{\epsilon}) \subset {B(t)}
\subset G(t)(1+{\epsilon}) {eventually w.p.1.}$$ We denote the fluctuations of
$B(t)$ from $tB_d$ and $G(t)$ by &&F(B(t), tB_d)=\inf \{l:tB_d(1-{l\over
t})\subset B(t)\subset tB_d(1+{l\over t})\} && F(B(t),
G(t))=\inf\{l:G(t)(1-{l\over t})\subset B(t)\subset G(t)(1+{l\over t})\}.
The means of the fluctuations $E[F(B(t), tB_d]$ and $E[F(B(t), G(t))]$ have
been conjectured ranging from divergence to non-divergence for large $d\geq 2$
by physicists. In this paper, we show that for all $d\geq 2$ with a high
probability, the fluctuations $F(B(t), G(t))$ and $F(B(t), tB_d)$ diverge with
a rate of at least $C \log t$ for some constant $C$.
The proof of this argument depends on the linearity between the number of
pivotal edges of all minimizing paths and the paths themselves. This linearity
is also independently interesting.
http://arXiv.org/abs/math/0501095
http://front.math.ucdavis.edu/math.PR/0501095
(alternate)
Author(s): Uffe Haagerup and Hanne Schultz and Steen Thorbjornsen
Abstract: In 1982 Pimsner and Voiculescu computed the K_0- and K_1-groups of the
reduced group C*-algebra C*_red(F_k) of the free group F_k on k generators and
settled thereby a long standing conjecture: C*_red(F_k) has no projections
except for the trivial projections 0 and 1. Later simpler proofs of this
conjecture were found by methods from K-theory or from non-commutative
differential geometry. In this paper we provide a new proof of the fact that
C*_red(F_k) is projectionless. The new proof is based on random matrices and is
obtained by a refinement of the methods recently used by the first and the
third named author to show that the semigroup Ext(C*_red(F_k)) is not a group
for k >= 2. By the same type of methods we also obtain that two phenomena
proved by Bai and Silverstein for certain classes of random matrices: ``no
eigenvalues outside (a small neighbourhood of) the support of the limiting
distribution'' and ``exact separation of eigenvalues by gaps in the limiting
distribution'' also hold for arbitrary non-commutative selfadjoint polynomials
of independent GUE, GOE or GSE random matrices with matrix coefficients.
http://arXiv.org/abs/math/0412545
http://front.math.ucdavis.edu/math.OA/0412545
(alternate)
Author(s): G. Ben Arous and S. Molchanov and A.F. Ramirez
Abstract: In this work we study a natural transition mechanism describing the passage
from a quenched (almost sure) regime to an annealed (in average) one, for a
symmetric simple random walk on random obstacles on sites having an identical
and independent law. The transition mechanism we study was first proposed in
the context of sums of identical independent random exponents by Ben Arous,
Bogachev and Molchanov in \cite{bbm}. Let $p(x,t)$ be the survival probability
at time $t$ of the random walk, starting from site $x$, and $L(t)$ be some
increasing function of time. We show that the empirical average of $p(x,t)$
over a box of side $L(t)$ has different asymptotic behaviors depending on
$L(t)$. There are constants $0<\gamma_1<\gamma_2$ such that if $ L(t)\ge
e^{\gamma t^{d/(d+2)}}$, with $\gamma>\gamma_1$, a law of large numbers is
satisfied and the empirical survival probability decreases like the annealed
one; if $ L(t)\ge e^{\gamma t^{d/(d+2)}}$, with $\gamma>\gamma_2$, also a
central limit theorem is satisfied. If $L(t)\ll t$, the averaged survival
probability decreases like the quenched survival probability. If $t\ll L(t)$
and $\log L(t)\ll t^{d/(d+2)}$ we obtain an intermediate regime. Furthermore,
when the dimension $d=1$ it is possible to describe the fluctuations of the
averaged survival probability when $L(t)=e^{\gamma t^{d/(d+2)}}$ with
$\gamma<\gamma_2$: it is shown that they are infinitely divisible laws with a
L\'evy spectral function which explodes when $x\to 0$ as stable laws of
characteristic exponent $\alpha<2$. These results show that the quenched and
annealed survival probabilities correspond to a low and high temperature
behavior of a mean field type phase transition mechanism.
http://arXiv.org/abs/math/0501107
http://front.math.ucdavis.edu/math.PR/0501107
(alternate)
Author(s): Elise Janvresse (LMRS) and Thierry De La Rue (LMRS) and Yvan Velenik (LMRS)
Abstract: We consider a directed polymer interacting with a diluted pinning potential
restricted to a line. We characterize explicitely the set of disorder
configurations that give rise to localization of the polymer. We study both
relevant cases of dimension 1+1 and 1+2. We also discuss the case of massless
effective interface models in dimension 2+1.
http://arXiv.org/abs/math/0501135
http://front.math.ucdavis.edu/math.PR/0501135
(alternate)
Author(s): Franz Merkl and Silke Rolles
Abstract: We prove that the edge-reinforced random walk on the ladder Z x {1,2} with
initial weights a > 3/4 is recurrent. The proof uses a known representation of
the edge-reinforced random walk on a finite piece of the ladder as a random
walk in a random environment. This environment is given by a marginal of a
multi-component Gibbsian process. A transfer operator technique and entropy
estimates from statistical mechanics are used to analyse this Gibbsian process.
Furthermore, we prove spatially exponentially fast decreasing bounds for
normalized local times of the edge-reinforced random walk on a finite piece of
the ladder, uniformly in the size of the finite piece.
http://arXiv.org/abs/math/0501137
http://front.math.ucdavis.edu/math.PR/0501137
(alternate)
Author(s): Rongfeng Sun
Abstract: The Brownian Web (BW) is a family of coalescing Brownian motions starting
from every point in space and time $\R\times\R$. It was first introduced by
Arratia, and later analyzed in detail by T\'{o}th and Werner. More recently,
Fontes, Isopi, Newman and Ravishankar gave a characterization of the BW, and
general convergence criteria allowing either crossing or noncrossing paths,
which they verified for coalescing simple random walks. Later Ferrari, Fontes,
and Wu verified these criteria for a two dimensional Poisson Tree. In both
cases, the paths are noncrossing. In this thesis, we formulate new convergence
criteria for crossing paths, and verify them for non-simple coalescing random
walks (both discrete and continuous time) satisfying a finite fifth moment
condition. This is the first time convergence to the BW has been proved for
models with crossing paths. Several corollaries are presented, including an
analysis of the scaling limit of voter model interfaces that extends a result
of Cox and Durrett.
http://arXiv.org/abs/math/0501141
http://front.math.ucdavis.edu/math.PR/0501141
(alternate)
Author(s): Rainer Gottlob
Abstract: Some drawbacks of the formalism of Bayes Theorem can be avoided by the
rMPE-Method, a modification of the cMPE-Method that permits (i): Adding
probabilities in spite of non-linearity. (ii): Taking into account extensional
evidence and weight-bearing evidence that are mutually dependent, but opposed
in their effects. (iii): Arriving at higher probabilities than by Bayes Theorem
and (iv): Confirming also hypotheses that imply certain evidence.
http://arXiv.org/abs/math/0501134
http://front.math.ucdavis.edu/math.ST/0501134
(alternate)
Author(s): Francis Comets (PMA) and Francesco Guerra (Fisica and Roma 1) and Fabio Lucio Toninelli (Phys-ENS)
Abstract: We study a spin system on a large box with both Ising interaction and
Sherrington-Kirpatrick couplings, in the presence of an external field. Our
results are: (i) existence of the pressure in the limit of an infinite box.
When both Ising and Sherrington-Kirpatrick temperatures are high enough, we
prove that: (ii) the value of the pressure is given by a suitable replica
symmetric solution, and (iii) the fluctuations of the pressure are of order of
the inverse of the square of the volume with a normal distribution in the
limit. In this regime, the pressure can be expressed in terms of random field
Ising models.
http://arXiv.org/abs/math/0501164
http://front.math.ucdavis.edu/math.PR/0501164
(alternate)
Author(s): Paavo Salminen and Marc Yor (PMA)
Abstract: Starting from the potential theoretic definition of the local times of a
Markov process - when these exist - we obtain a Tanaka formula for the local
times of symmetric L\'{e}vy processes. The most interesting case is that of the
symmetric $\al$-stable L\'{e}vy process (for $\al\in[1,2]$) which is studied in
detail. In particular, we determine which powers of such a process are
semimartingales. These results complete, in a sense, the works by K. Yamada
\cite{yamada02} and Fitzsimmons and Getoor \cite{fitzsimmonsgetoor92a}.
http://arXiv.org/abs/math/0501182
http://front.math.ucdavis.edu/math.PR/0501182
(alternate)
Author(s): S. Albeverio and H. Gottschalk and J.-L. Wu
Abstract: Given a probability distribution $\mu$ a set $\Lambda (\mu)$ of positive real
numbers is introduced, so that $\Lambda (\mu)$ measures the "divisibility" of
$\mu$. The basic properties of $\Lambda (\mu)$ are described and examples of
probability distributions are given, which exhibit the existence of a continuum
of situations interpolating the extreme cases of infinitely and minimally
divisible probability distributions.
http://arXiv.org/abs/math/0501183
http://front.math.ucdavis.edu/math.PR/0501183
(alternate)
Author(s): S. Albeverio and H. Gottschalk and J.-L. Wu
Abstract: A notion of admissible probability measures $\mu$ on a locally compact
Abelian group (LCA-group) $G$ with connected dual group $\hat G=\R^d\times
\T^n$ is defined. To such a measure $\mu$, a closed semigroup
$\Lambda(\mu)\subseteq (0,\infty)$ can be associated, such that, for $t\in
\Lambda(\mu)$, the Fourier transform to the power $t$, $(\hat \mu)^t$, is a
characteristic function. We prove that the existence of roots for non
admissible probability measures underlies some restrictions, which do not hold
in the admissible case. As we show for the example $\Z_2$, in the case of
LCA-groups with non connected dual group, there is no canonical definition of
the set $\Lambda(\mu)$.
http://arXiv.org/abs/math/0501185
http://front.math.ucdavis.edu/math.PR/0501185
(alternate)
Author(s): Michael J. Kozdron (University of Regina) and Gregory F. Lawler (Cornell University)
Abstract: We prove an estimate for the probability that a simple random walk in a
simply connected subset A of Z^2 starting on the boundary exits A at another
specified boundary point. The estimates are uniform over all domains of a given
inradius. We apply these estimates to prove a conjecture of S. Fomin in 2001
concerning a relationship between crossing probabilities of loop-erased random
walk and Brownian motion.
http://arXiv.org/abs/math/0501189
http://front.math.ucdavis.edu/math.PR/0501189
(alternate)
Author(s): E. Ben-Naim and P.L. Krapivsky
Abstract: We study the evolution of percolation with freezing. Specifically, we
consider cluster formation via two competing processes: irreversible
aggregation and freezing. We find that when the freezing rate exceeds a certain
threshold, the percolation transition is suppressed. Below this threshold, the
system undergoes a series of percolation transitions with multiple giant
clusters ("gels") formed. Giant clusters are not self-averaging as their total
number and their sizes fluctuate from realization to realization. The size
distribution F_k, of frozen clusters of size k, has a universal tail, F_k ~
k^{-3}. We propose freezing as a practical mechanism for controlling the gel
size.
http://arXiv.org/abs/cond-mat/0501218
http://front.math.ucdavis.edu/cond-mat/0501218
(alternate)
Author(s): Laure Coutin and Peter Friz and Nicolas Victoir
Abstract: We consider anticipative Stratonovich stochastic differential equations
driven by some stochastic process (not necessarily a semi-martingale). No
adaptedness of initial point or vector fields is assumed. Under a simple
condition on the stochastic process, we show that the unique solution of the
above SDE understood in the rough path sense is actually a Stratonovich
solution. This condition is satisfied by the Brownian motion and the fractional
Brownian motion with Hurst parameter greater than 1/4. As application, we
obtain rather flexible results such as support theorems, large deviation
principles and Wong-Zakai approximations for SDEs driven by fractional Brownian
Motion along anticipating vectorfields. In particular, this unifies many
results on anticipative SDEs.
http://arXiv.org/abs/math/0501197
http://front.math.ucdavis.edu/math.PR/0501197
(alternate)
Author(s): Endre Cs\'aki and Yueyun Hu
Abstract: Let $W$ be a one-dimensional Brownian motion starting from 0. Define $Y(t)=
\int_0^t{\d s \over W(s)} := \lim_{\epsilon\to0} \int_0^t 1_{(|W(s)|>
\epsilon)} {\d s \over W(s)} $ as Cauchy's principal value related to local
time. We prove limsup and liminf results for the increments of $Y$.
http://arXiv.org/abs/math/0501199
http://front.math.ucdavis.edu/math.PR/0501199
(alternate)
Author(s): Makoto Katori and Hideki Tanemura
Abstract: The system of one-dimensional symmetric simple random walks, in which none of
walkers have met others in a given time period, is called the vicious walker
model. It was introduced by Michael Fisher and applications of the model to
various wetting and melting phenomena were described in his Boltzmann medal
lecture. In the present report, we explain interesting connections among
representation theory, probability theory, and random matrix theory using this
simple diffusion particle system. Each vicious walk of $N$ walkers is
represented by an $N$-tuple of nonintersecting lattice paths on the
spatio-temporal plane. There is established a simple bijection between
nonintersecting lattice paths and semistandard Young tableaux. Based on this
bijection and some knowledge of symmetric polynomials called the Schur
functions, we can give a determinantal expression to the partition function of
vicious walks, which is regarded as a special case of the Karlin-McGregor
formula in the probability theory (or the Lindstr\"om-Gessel-Viennot formula in
the enumerative combinatorics). Due to a basic property of Schur function, we
can take the diffusion scaling limit of the vicious walks and define a
noncolliding system of Brownian particles. This diffusion process solves the
stochastic differential equations with the drift terms acting as the repulsive
two-body forces proportional to the inverse of distances between particles, and
thus it is identified with Dyson's Brownian motion model. In other words, the
obtained noncolliding system of Brownian particles is equivalent in
distribution with the eigenvalue process of a Hermitian matrix-valued process.
http://arXiv.org/abs/math/0501218
http://front.math.ucdavis.edu/math.PR/0501218
(alternate)
Author(s): Tomasz Schreiber
Abstract: We construct random dynamics on collections of non-intersecting planar
contours, leaving invariant the distributions of length- and area-interacting
polygonal Markov fields with V-shaped nodes. The first of these dynamics is
based on the dynamic construction of consistent polygonal fields, as presented
in the original articles by Arak (1982) and Arak and Surgailis (1989, 1991),
and it provides an easy-to-implement Metropolis-type simulation algorithm. The
second dynamics leads to a graphical construction in the spirit of Fernandez,
Ferrari and Garcia (1998,2002) and it yields a perfect simulation scheme in a
finite window from the infinite-volume limit. This algorithm seems difficult to
implement, yet its value lies in that it allows for theoretical analysis of
thermodynamic limit behaviour of length-interacting polygonal fields. The
results thus obtained include the uniqueness and exponential $\alpha$-mixing of
the thermodynamic limit of such fields in the low temperature region, in the
class of infinite-volume Gibbs measures without infinite contours. Outside this
class we conjecture the existence of an infinite number of extreme phases
breaking both the translational and rotational symmetries
http://arXiv.org/abs/math/0501228
http://front.math.ucdavis.edu/math.PR/0501228
(alternate)
Author(s): Robert B. Ellis and Jeremy L. Martin and and Catherine Yan
Abstract: The unit ball random geometric graph $G=G^d_p(\lambda,n)$ has as its vertices
$n$ points distributed independently and uniformly in the $d$-dimensional unit
ball, with two vertices adjacent if and only if their $l_p$-distance is at most
$\lambda$. Like its cousin the Erdos-Renyi random graph, $G$ has a connectivity
threshold: an asymptotic value for $\lambda$ in terms of $n$, above which $G$
is connected and below which $G$ is disconnected (and in fact has isolated
vertices in most cases). In the disconnected zone, we discuss the number of
isolated vertices. In the connected zone, we determine upper and lower bounds
for the graph diameter of $G$. We employ a combination of methods from
probabilistic combinatorics and stochastic geometry.
http://arXiv.org/abs/math/0501214
http://front.math.ucdavis.edu/math.CO/0501214
(alternate)
Author(s): Muffasir Badshah and Robert Boyer and Ted Theodosopoulos
Abstract: Increased day-trading activity and the subsequent jump in intraday volatility
and trading volume fluctuations has raised considerable interest in models for
financial market microstructure. We investigate the random transitions between
two phases of an agent-based spin market model on a random network. The
objective of the agents is to balance their desire to belong to the global
minority and simultaneously to the local majority. We show that transitions
between the "ordered" and "disordered" phases follow a Poisson process with a
rate that is a monotonically decreasing function of the network connectivity.
http://arXiv.org/abs/math/0501244
http://front.math.ucdavis.edu/math.PR/0501244
(alternate)
Author(s): Muffasir Badshah and Robert Boyer and Ted Theodosopoulos
Abstract: In this short note we investigate the natur of the phase transitions in a
spin market model as a function of the interaction strength between local and
global effects. We find that the stochastic dynamics of this stylized market
model exhibit a periodicity whose dependence on the coupling constant in the
Ising-like Hamiltonian is robust to changes in the temperature and the size of
the market.
http://arXiv.org/abs/math/0501248
http://front.math.ucdavis.edu/math.PR/0501248
(alternate)
Author(s): Fumio Hiai and Yoshimichi Ueda
Abstract: We prove the free analogue of the transportation cost inequality for tracial
distributions of non-commutative self-adjoint (also unitary) multi-variables
based on random matrix approximation procedure.
http://arXiv.org/abs/math/0501238
http://front.math.ucdavis.edu/math.OA/0501238
(alternate)
Author(s): Frederik S Herzberg
Abstract: Non-perpetual American option prices shall be approximated by non-perpetual
Bermudan option prices, which in turn can be computed in a recombining tree of
European options. It will be proven that perpetual and non-perpetual Bermudan
option prices have comparable analytic behaviour when perceived as functions of
the exercise mesh size. Using a Wiener-Hopf factorisation, a theoretical
formula for perpetual Bermudan option prices is derived. Based on this formula,
some rather elementary semigroup analysis gives rise to a power series for the
perpetual Bermudan price as a function of the exercise mesh size, paving the
way to understand the limiting behaviour as the exercise mesh size tends to
naught. Results by Feller that are based on Fourier analytic deliberations will
enable us -- for a number of models, including the Black-Scholes and Merton's
jump-diffusion models, -- to prove order estimates on the behaviour of Bermudan
option prices on stocks with a start price at the exercise boundary. As a
consequence, one obtains a natural scaling for the computation of American
option prices by means of a non-polynomial extrapolation of Bermudan prices.
http://arXiv.org/abs/math/0501261
http://front.math.ucdavis.edu/math.PR/0501261
(alternate)
Author(s): Fabrice Blache
Abstract: The problem of finding a martingale on a manifold with a fixed random
terminal value can be solved by considering BSDEs with a generator with
quadratic growth. We study here a generalization of these equations and we give
uniqueness and existence results in two different frameworks, using
differential geometry tools. Applications to PDEs are given, including a
certain class of Dirichlet problems on manifolds.
http://arXiv.org/abs/math/0501265
http://front.math.ucdavis.edu/math.PR/0501265
(alternate)
Author(s): Rafa{\l} Lata{\l}a and Krzysztof Oleszkiewicz
Abstract: A certain inequality conjectured by Vershynin is studied. It is proved that
for any $n$-dimensional symmetric convex body $K$ with inradius $w$ and
$\gamma_{n}(K) \leq 1/2$ there is $\gamma_{n}(sK) \leq
(2s)^{w^{2}/4}\gamma_{n}(K)$ for any $s \in [0,1]$. Some natural corollaries
are deduced. Another conjecture of Vershynin is proved to be false.
http://arXiv.org/abs/math/0501268
http://front.math.ucdavis.edu/math.PR/0501268
(alternate)
Author(s): Fumio Hiai
Abstract: The free analog of the pressure is introduced for multivariate noncommutative
random variables and its Legendre transform is compared with Voiculescu's
microstate free entropy.
http://arXiv.org/abs/math/0403210
http://front.math.ucdavis.edu/math.OA/0403210
(alternate)
Author(s): Gerard Ben Arous and Dan Virgil Voiculescu
Abstract: Free probability analogues of the basics of extreme value theory are
obtained, based on Ando's spectral order. This includes classification of
freely max-stable laws and their domains of attraction, using ``free extremal
convolutions'' on the distributions. These laws coincide with the limit laws in
the classical peaks-over-threshold approach. A free extremal projection-valued
process over a measure-space is constructed, which is related to the free
Poisson point process.
http://arXiv.org/abs/math/0501274
http://front.math.ucdavis.edu/math.OA/0501274
(alternate)
Author(s): A. Gamburd
Abstract: Brooks and Makover introduced an approach to studying the global geometric
quantities (in particular, the first eigenvalue of the Laplacian, injectivity
radius and diameter) of a "typical" compact Riemann surface of large genus
based on compactifying finite-area Riemann surfaces associated with random
cubic graphs; by a theorem of Belyi these are "dense" in the space of compact
Riemann surfaces. The question as to how these surfaces are distributed in the
Teichm\"{u}ller spaces depends on the study of oriented cycles in random cubic
graphs with random orientation; Brooks and Makover conjectured that
asymptotically normalized cycles lengths follow Poisson-Dirichlet distribution.
We present a proof of this conjecture using representation theory of the
symmetric group. Consequently we also make progress towards a conjecture of
Pippenger and Schleich which arose in the study of topological characteristics
of random surfaces generated by cubic interactions.
http://arXiv.org/abs/math/0501283
http://front.math.ucdavis.edu/math.PR/0501283
(alternate)
Author(s): Pablo A. Ferrari and James B. Martin
Abstract: We consider totally asymmetric simple exclusion processes with n types of
particle and holes (n-TASEPs) on Z and on the cycle Z_N. Angel recently gave an
elegant construction of the stationary measures for the 2-TASEP, based on a
pair of independent product measures. We show that Angel's construction can be
interpreted in terms of the operation of a discrete-time M/M/1 queueing server;
the two product measures correspond to the arrival and service processes of the
queue. We extend this construction to represent the stationary measures of an
n-TASEP in terms of a system of queues in tandem. The proof of stationarity
involves a system of n 1-TASEPs, whose evolutions are coupled but whose
distributions at any fixed time are independent. Using the queueing
representation, we give quantitative results for stationary probabilities of
states of the n-TASEP on Z_N, and simple proofs of various independence and
regeneration properties for systems on Z.
http://arXiv.org/abs/math/0501291
http://front.math.ucdavis.edu/math.PR/0501291
(alternate)
Author(s): Inder Jeet Taneja
Abstract: Using Blackwell's definition of comparing two experiments, a comparison is
made with \textit{generalized AG - divergence} measure having one and two
scalar parameters. Connection of \textit{generalized AG - divergence} measure
with \textit{Fisher measure of information} is also presented. A unified
\textit{generalization of AG - divergence }and\textit{ Jensen-Shannon
divergence measures} is also presented.
http://arXiv.org/abs/math/0501297
http://front.math.ucdavis.edu/math.PR/0501297
(alternate)
Author(s): Inder Jeet Taneja
Abstract: \textit{Arithmetic, geometric and harmonic means} are the three classical
means famous in the literature. Another mean such as \textit{square-root mean}
is also known. In this paper, we have constructed divergence measures based on
nonnegative differences among these means, and established an interesting
inequality by use of properties of Csisz\'{a}r $f-$\textit{divergence}.
Connections of new \textit{mean divergences} measures with classical divergence
measures such as Jeffreys-Kullback-Leiber \cite{jef}, \cite{kul}
\textit{J-divergence}, Sibson-Burbea-Rao \cite{sib}, \cite{bra} \textit{Jensen
difference divergence measure} and Taneja \cite{tan2} \textit{AG -- divergence}
are also established.
http://arXiv.org/abs/math/0501298
http://front.math.ucdavis.edu/math.PR/0501298
(alternate)
Author(s): Pranesh Kumar and Inder Jeet Taneja
Abstract: In this paper we shall consider one parametric generalization of some
non-symmetric divergence measures. The \textit{non-symmetric divergence
measures} are such as: Kullback-Leibler \textit{relative information}, $\chi
^2-$\textit{divergence}, \textit{relative J -- divergence}, \textit{relative
Jensen -- Shannon divergence} and \textit{relative Arithmetic -- Geometric
divergence}. All the generalizations considered can be written as particular
cases of Csisz\'{a}r's \textit{f | |
