[Pas] Probability Abstract 85
pas at www.economia.unimi.it
pas at www.economia.unimi.it
Tue Mar 1 18:29:22 CET 2005
March 1, 2005
Letter 85
Dear Colleagues,
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3074. ASYMPTOTIC EXPANSIONS FOR INFINITE WEIGHTED CONVOLUTIONS OF HEAVY
TAIL DISTRIBUTIONS AND APPLICATIONS
Ph. Barbe and W.P. McCormick (CNRS and University of Georgia)
We establish some asymptotic expansions for infinite weighted
convolution of
distributions having regular varying tails. Various applications to
statistics
and probability are developed.
http://front.math.ucdavis.edu/math.PR/0412537
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3075. STABILITY PROPERTIES OF CONSTRAINED JUMP-DIFFUSION PROCESSES
Rami Atar and Amarjit Budhiraja
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://front.math.ucdavis.edu/math.PR/0501014
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3076. SYNCHRONOUS COUPLINGS OF REFLECTED BROWNIAN MOTIONS IN SMOOTH
DOMAINS
Krzysztof Burdzy and Zhen-Qing Chen and Peter Jones
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://front.math.ucdavis.edu/math.PR/0501486
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3077. MEAN-FIELD DRIVEN FIRST-ORDER PHASE TRANSITIONS IN SYSTEMS WITH
LONG-RANGE INTERACTIONS
Marek Biskup and Lincoln Chayes and Nicholas Crawford
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://front.math.ucdavis.edu/math-ph/0501067
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3078. GOLDBUG VARIATIONS
Michael Kleber
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://front.math.ucdavis.edu/math.CO/0501497
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3079. IDENTITIES IN LAW BETWEEN QUADRATIC FUNCTIONALS OF BIVARIATE
GAUSSIAN PROCESSES, THROUGH FUBINI THEOREMS AND SYMMETRIC PROJECTIONS
Giovanni Peccati (LSTA) and Marc Yor (PMA)
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://front.math.ucdavis.edu/math.PR/0501506
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3080. THE MEAN SQUARE OF WEIGHTED MULTIPLICITIES FUNCTION
Lukianov Vladimir
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://front.math.ucdavis.edu/math.NT/0501519
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3081. CRITICAL PERCOLATION ON CERTAIN NON-UNIMODULAR GRAPHS
Yuval Peres and Gabor Pete and Ariel Scolnicov
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://front.math.ucdavis.edu/math.PR/0501532
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3082. SHORTEST SPANNING TREES AND A COUNTEREXAMPLE FOR RANDOM WALKS IN
RANDOM ENVIRONMENTS
Maury Bramson and Ofer Zeitouni and Martin P. W. Zerner
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://front.math.ucdavis.edu/math.PR/0501533
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3083. SINGULAR CONTROL WITH STATE CONSTRAINTS ON UNBOUNDED DOMAIN
Rami Atar and Amarjit Budhiraja
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://front.math.ucdavis.edu/math.PR/0501016
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3084. ON L\'{E}VY PROCESSES CONDITIONED TO STAY POSITIVE
Lo\"{i}c Chaumont (LPMA) and Ron A. Doney
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://front.math.ucdavis.edu/math.PR/0502012
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3085. THE MAXIMUM ENTROPY STATE
Keye Martin
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://front.math.ucdavis.edu/math.PR/0502024
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3086. PROOF OF THE LOCAL REM CONJECTURE FOR NUMBER PARTITIONING
Christian Borgs and Jennifer Chayes and Stephan Mertens and Chandra
Nair
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://front.math.ucdavis.edu/cond-mat/0501760
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3087. A SHARP INEQUALITY FOR CONDITIONAL DISTRIBUTION OF THE FIRST EXIT
TIME OF BROWNIAN MOTION
Majid Hosseini
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://front.math.ucdavis.edu/math.PR/0502057
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3088. ON POSITIVE RECURRENCE OF CONSTRAINED DIFFUSION PROCESSES
Rami Atar and Amarjit Budhiraja and P. Dupuis
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://front.math.ucdavis.edu/math.PR/0501018
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3089. ON LARGE DEVIATIONS IN THE AVERAGING PRINCIPLE FOR SDE'S WITH A
``FULL DEPENDENCE'', CORRECTION
Alexander Yu. Veretennikov
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://front.math.ucdavis.edu/math.PR/0502098
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3090. NONPARAMETRIC REGRESSION ESTIMATION FOR RANDOM FIELDS IN A
FIXED-DESIGN
Mohamed El Machkouri (LMRS)
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://front.math.ucdavis.edu/math.ST/0502091
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3091. ON THE ASYMPTOTIC BEHAVIOR OF SOME ALGORITHMS
Philippe Robert (RAP UR-R)
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://front.math.ucdavis.edu/cs.DS/0502014
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3092. ON LARGE DEVIATIONS IN THE AVERAGING PRINCIPLE FOR SDE'S WITH A
``FULL DEPENDENCE'', CORRECTION
Alexander Yu. Veretennikov
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://front.math.ucdavis.edu/math.PR/0502098
---------------------------------------------------------------
3093. NONPARAMETRIC REGRESSION ESTIMATION FOR RANDOM FIELDS IN A
FIXED-DESIGN
Mohamed El Machkouri (LMRS)
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://front.math.ucdavis.edu/math.ST/0502091
---------------------------------------------------------------
3094. ON THE ASYMPTOTIC BEHAVIOR OF SOME ALGORITHMS
Philippe Robert (RAP UR-R)
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://front.math.ucdavis.edu/cs.DS/0502014
---------------------------------------------------------------
3095. PROPERTIES OF THE WEALTH PROCESS IN A MARKET MICROSTRUCTURE MODEL
Ted Theodosopoulos and Ming Yuen
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://front.math.ucdavis.edu/math.PR/0502105
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3096. DIFFERENT ASPECTS OF A MODEL FOR RANDOM FRAGMENTATION PROCESSES
Jean Bertoin (PMA)
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://front.math.ucdavis.edu/math.PR/0502132
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3097. INVARIANCE PRINCIPLES FOR STANDARD-NORMALIZED AND SELF-NORMALIZED
RANDOM FIELDS
Mohamed El Machkouri (LMRS) and Lahcen Ouchti (LMRS)
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://front.math.ucdavis.edu/math.PR/0502135
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3098. PINNING OF POLYMERS AND INTERFACES BY RANDOM POTENTIALS
Kenneth S. Alexander and Vladas Sidoravicius
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://front.math.ucdavis.edu/math.PR/0501028
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3099. TAKING BIGGER METROPOLIS STEPS BY DRAGGING FAST VARIABLES
Radford M. Neal
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://front.math.ucdavis.edu/math.ST/0502099
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3100. A TWO ARMED BANDIT TYPE PROBLEM REVISITED
Gilles Pag\`{e}s (PMA)
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://front.math.ucdavis.edu/math.PR/0502182
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3101. ON THE HEDGING OF AMERICAN OPTIONS IN DISCRETE TIME MARKETS WITH
PROPORTIONAL TRANSACTION COSTS
Bruno Bouchard (PMA) and Emmanuel Temam (PMA)
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://front.math.ucdavis.edu/math.PR/0502189
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3102. ON MAXIMA AND LADDER PROCESSES FOR A DENSE CLASS OF LEVY PROCESSES
M. R. Pistorius
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://front.math.ucdavis.edu/math.PR/0502192
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3103. TWO-DIMENSIONAL WETTING WITH BINARY DISORDER: A NUMERICAL STUDY
OF THE LOOP STATISTICS
Thomas Garel and Cecile Monthus
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://front.math.ucdavis.edu/cond-mat/0502195
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3104. DEGREE DISTRIBUTION OF COMPETITION-INDUCED PREFERENTIAL
ATTACHMENT GRAPHS
N. Berger and C. Borgs and J. T. Chayes and R. M. D'Souza and R. D.
Kleinberg
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://front.math.ucdavis.edu/cond-mat/0502205
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3105. ON NONEXISTENCE OF NON-CONSTANT VOLATILITY IN THE BLACK-SCHOLES
FORMULA
K. Hamza and F.C. Klebaner
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://front.math.ucdavis.edu/math.PR/0502201
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3106. MARTINGALE STRUCTURE OF SKOROHOD INTEGRAL PROCESSES
Giovanni Peccati (LSTA) and Mich\`{e}le Thieullen (PMA) and Ciprian
A. Tudor (SAMOS)
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://front.math.ucdavis.edu/math.PR/0502208
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3107. ASYMPTOTICS IN KNUTH'S PARKING PROBLEM FOR CARAVANS
Jean Bertoin (PMA) and Gr\'{e}gory Marc Miermont (LM-Orsay)
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://front.math.ucdavis.edu/math.PR/0502220
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3108. AN ESCAPE TIME CRITERION FOR QUEUEING NETWORKS: ASYMPTOTIC
RISK-SENSITIVE CONTROL VIA DIFFERENTIAL GAMES
Rami Atar and Paul Dupuis and Adam Shwartz
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://front.math.ucdavis.edu/math.PR/0501031
---------------------------------------------------------------
3109. ABSOLUTE CONTINUITY FOR RANDOM ITERATED FUNCTION SYSTEMS WITH
OVERLAPS
Yuval Peres and K\'aroly Simon and Boris Solomyak
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://front.math.ucdavis.edu/math.DS/0502200
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3110. SUBTREE PRUNE AND RE-GRAFT: A REVERSIBLE REAL TREE VALUED MARKOV
PROCESS
Steven N. Evans and Anita Winter
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://front.math.ucdavis.edu/math.PR/0502226
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3111. INDIVIDUAL DISPLACEMENTS IN HASHING WITH COALESCED CHAINS
Svante Janson
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://front.math.ucdavis.edu/math.PR/0502232
---------------------------------------------------------------
3112. RANDOM RECURSIVE TREES AND THE BOLTHAUSEN-SZNITMAN COALESCENT
Christina Goldschmidt and James B. Martin
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://front.math.ucdavis.edu/math.PR/0502263
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3113. THE LINKING PROBABILITY OF DEEP SPIDER-WEB NETWORKS
Nicholas Pippenger
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
$0<q<q_c$ (where $q_c = 1/b^{(c-1)/c}$ is the critical vacancy
probability),
and tends to $(1-\xi)^2$ (where $\xi$ is the unique solution of the
equation
$(1-q (1-x))^b=x$ in the range $0<x<1$) if $q_c<q<1$. In this paper we
extend
this result to all rational $c>1$. 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://front.math.ucdavis.edu/math.PR/0502294
---------------------------------------------------------------
3114. EXPLICIT SOLUTION FOR A NETWORK CONTROL PROBLEM IN THE LARGE
DEVIATION REGIME
Rami Atar and Paul Dupuis and Adam Shwartz
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://front.math.ucdavis.edu/math.PR/0501035
---------------------------------------------------------------
3115. ESTIMATES ON PATH DELOCALIZATION FOR COPOLYMERS AT SELECTIVE
INTERFACES
Giambattista Giacomin and Fabio Lucio Toninelli
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://front.math.ucdavis.edu/math.PR/0502304
---------------------------------------------------------------
3116. SOME PROPERTIES OF THE RATE FUNCTION OF QUENCHED LARGE DEVIATIONS
FOR RANDOM WALK IN RANDOM ENVIRONMENT
Alexis Devulder (PMA)
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://front.math.ucdavis.edu/math.PR/0502316
---------------------------------------------------------------
3117. AN ADAPTIVE SCHEME FOR THE APPROXIMATION OF DISSIPATIVE SYSTEMS
Vincent Lemaire
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://front.math.ucdavis.edu/math.PR/0502317
---------------------------------------------------------------
3118. LIMIT LAWS FOR RANDOM VECTORS WITH AN EXTREME COMPONENT
J. Heffernan & S. Resnick
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://front.math.ucdavis.edu/math.PR/0502324
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3119. STRONG ASYMPTOTIC ASSERTIONS FOR DISCRETE MDL IN REGRESSION AND
CLASSIFICATION
Jan Poland and Marcus Hutter
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://front.math.ucdavis.edu/math.ST/0502315
---------------------------------------------------------------
3120. ON THE OPERATOR SPACE UMD PROPERTY FOR NONCOMMUTATIVE LP-SPACES
Magdalena Musat
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<p,q<\infty, the
Schatten q-classes Sq are OUMDp. The proof relies on properties of the
Haagerup
tensor product and complex interpolation. Using ultraproduct
techniques, we
extend this result to a large class of noncommutative Lq-spaces.
Namely, we
show that if M is a QWEP von Neumann algebra (i.e., a quotient of a
C^*-algebra
with Lance's weak expectation property) equipped with a normal, faithful
tracial state \tau, then Lq(M,\tau) is OUMDp for 1<p,q<\infty.
http://front.math.ucdavis.edu/math.OA/0501033
---------------------------------------------------------------
3121. OPTIMALITY OF UNIVERSAL BAYESIAN SEQUENCE PREDICTION FOR GENERAL
LOSS AND ALPHABET
Marcus Hutter
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://front.math.ucdavis.edu/cs.LG/0311014
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3122. ON THE CONVERGENCE SPEED OF MDL PREDICTIONS FOR BERNOULLI
SEQUENCES
Jan Poland and Marcus Hutter
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://front.math.ucdavis.edu/cs.LG/0407039
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3123. UNIVERSAL CONVERGENCE OF SEMIMEASURES ON INDIVIDUAL RANDOM
SEQUENCES
Marcus Hutter and Andrej Muchnik
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://front.math.ucdavis.edu/cs.LG/0407057
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3124. FUNCTIONAL QUANTIZATION AND METRIC ENTROPY FOR RIEMANN-LIOUVILLE
PROCESSES
Harald Luschgy and Gilles Pag\`{e}s (PMA)
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://front.math.ucdavis.edu/math.PR/0502375
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3125. SELF-OPTIMIZING AND PARETO-OPTIMAL POLICIES IN GENERAL
ENVIRONMENTS BASED ON BAYES-MIXTURES
Marcus Hutter
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://front.math.ucdavis.edu/cs.AI/0204040
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3126. NO-ARBITRAGE IN DISCRETE-TIME MARKETS WITH PROPORTIONAL
TRANSACTION COSTS AND GENERAL INFORMATION STRUCTURE
Bruno Bouchard (PMA and Crest and Lfa)
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://front.math.ucdavis.edu/math.PR/0501045
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3127. CONVERGENCE AND ERROR BOUNDS FOR UNIVERSAL PREDICTION OF
NONBINARY SEQUENCES
Marcus Hutter
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://front.math.ucdavis.edu/cs.LG/0106036
---------------------------------------------------------------
3128. CONVERGENCE AND LOSS BOUNDS FOR BAYESIAN SEQUENCE PREDICTION
Marcus Hutter
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://front.math.ucdavis.edu/cs.LG/0301014
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3129. BAYESIAN TREATMENT OF INCOMPLETE DISCRETE DATA APPLIED TO MUTUAL
INFORMATION AND FEATURE SELECTION
Marcus Hutter and Marco Zaffalon
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://front.math.ucdavis.edu/cs.LG/0306126
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3130. KOLMOGOROV-SINAI ENTROPY OF A GENERALIZED MARKOV SHIFT
Ivan Werner
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://front.math.ucdavis.edu/math.DS/0502389
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3131. GAUSSIAN FLUCTUATIONS FOR NON-HERMITIAN RANDOM MATRIX ENSEMBLES
B. Rider and Jack W. Silverstein
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://front.math.ucdavis.edu/math.PR/0502400
---------------------------------------------------------------
3132. A REPERTOIRE FOR ADDITIVE FUNCTIONALS OF UNIFORMLY DISTRIBUTED
M-ARY SEARCH TREES
James Allen Fill and Nevin Kapur
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://front.math.ucdavis.edu/math.PR/0502422
---------------------------------------------------------------
3133. PHASE TRANSITION FOR PARKING BLOCKS, BROWNIAN EXCURSION AND
COALESCENCE
Philippe Chassaing (IEC) and Guy Louchard (ULB)
In this paper, we consider hashing with linear probing for a hashing
table
with m places, n items (n < m), and l = m<n empty places. For a non
computer
science-minded reader, we shall use the metaphore of n cars parking on m
places: each car chooses a place at random, and if this place k is
occupied,
the car tries successively k+1, k+2, ... until it finds an empty place
(with
the convention that place m+1 is actually place 1). Pittel [42] proves
that
when l/m goes to some positive limit a < 1, the size of the largest
block of
consecutive cars is O(log m). In this paper we examine at which level
for n a
phase transition occurs for the largest block of consecutive cars
between o(m)
and O(m). The intermediate case reveals an interesting behaviour of
sizes of
blocks, related to the standard additive coalescent in the same way as
the
sizes of connected components of the random graph are related to the
multiplicative coalescent.
http://front.math.ucdavis.edu/math.PR/0501060
---------------------------------------------------------------
3134. HIGH RESOLUTION ASYMPTOTICS FOR THE ANGULAR BISPECTRUM OF
SPHERICAL RANDOM FIELDS
Domenico Marinucci
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://front.math.ucdavis.edu/math.PR/0502434
---------------------------------------------------------------
3135. COST-VOLUME RELATIONSHIPS FOR FLOWS THROUGH A DISORDERED NETWORK
David Aldous
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://front.math.ucdavis.edu/cond-mat/0502346
---------------------------------------------------------------
3136. CONCERNING LIFE ANNUITIES
Leonhard Euler
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://front.math.ucdavis.edu/math.HO/0502421
---------------------------------------------------------------
3137. UNIVERSAL FINITARY CODES WITH EXPONENTIAL TAILS
Nate Harvey and Alexander E. Holroyd and Yuval Peres and Dan Romik
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://front.math.ucdavis.edu/math.PR/0502484
---------------------------------------------------------------
3138. CONDITIONED BROWNIAN TREES
Jean-Francois Le Gall (ENS Paris) and Mathilde Weill (ENS Paris)
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://front.math.ucdavis.edu/math.PR/0501066
---------------------------------------------------------------
3139. STRONG DISORDER RG APPROACH OF RANDOM SYSTEMS
Ferenc Igloi and Cecile Monthus
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://front.math.ucdavis.edu/cond-mat/0502448
---------------------------------------------------------------
3140. THE EMPIRICAL EIGENVALUE DISTRIBUTION OF A GRAM MATRIX: FROM
INDEPENDENCE TO STATIONARITY
W. Hachem and P. Loubaton and J. Najim
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://front.math.ucdavis.edu/math.PR/0502535
---------------------------------------------------------------
3141. PRESERVATION OF LOG-CONCAVITY ON SUMMATION
Oliver Johnson and Christina Goldschmidt
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://front.math.ucdavis.edu/math.PR/0502548
---------------------------------------------------------------
3142. SEMI-SELFDECOMPOSABLE LAWS AND RELATED PROCESSES
S Satheesh and E Sandhya
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://front.math.ucdavis.edu/math.PR/0412546
---------------------------------------------------------------
3143. A NOTE ON RANDOM WALK IN RANDOM SCENERY
Amine Asselah and Fabienne Castell
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 a<d/2. We study the probability, when averaged
over both
randomness, that {X(n)>ny}. We show that this probability is of order
exp(-(ny)^b) with b=a/(a+1).
http://front.math.ucdavis.edu/math.PR/0501068
---------------------------------------------------------------
3144. PROBABILISTIC AND FRACTAL ASPECTS OF LEVY TREES
Thomas Duquesne (Paris 11) and Jean-Francois Le Gall (ENS Paris)
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://front.math.ucdavis.edu/math.PR/0501079
---------------------------------------------------------------
3145. PATH COUPLING USING STOPPING TIMES AND COUNTING INDEPENDENT SETS
AND COLOURINGS IN HYPERGRAPHS
Magnus Bordewich and Martin Dyer and Marek Karpinski
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://front.math.ucdavis.edu/math.PR/0501081
---------------------------------------------------------------
3146. RUELLE'S PROBABILITY CASCADES SEEN AS A FRAGMENTATION PROCESS
Anne-Laure Basdevant (LPMA)
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://front.math.ucdavis.edu/math.PR/0501088
---------------------------------------------------------------
3147. DISCRETE AND CONTINUOUS YANG-MILLS MEASURE FOR NON-TRIVIAL
BUNDLES OVER COMPACT SURFACES
Thierry Levy (DMA)
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://front.math.ucdavis.edu/math-ph/0501014
---------------------------------------------------------------
3148. THE DIVERGENCE OF FLUCTUATIONS FOR THE SHAPE ON FIRST PASSAGE
PERCOLATION
Yu Zhang
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://front.math.ucdavis.edu/math.PR/0501095
---------------------------------------------------------------
3149. A RANDOM MATRIX APPROACH TO THE LACK OF PROJECTIONS IN C*_RED(F_2)
Uffe Haagerup and Hanne Schultz and Steen Thorbjornsen
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://front.math.ucdavis.edu/math.OA/0412545
---------------------------------------------------------------
3150. TRANSITION FROM THE ANNEALED TO THE QUENCHED ASYMPTOTICS FOR A
RANDOM WALK ON RANDOM OBSTACLES
G. Ben Arous and S. Molchanov and A.F. Ramirez
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://front.math.ucdavis.edu/math.PR/0501107
---------------------------------------------------------------
3151. PINNING BY A SPARSE POTENTIAL
Elise Janvresse (LMRS) and Thierry De La Rue (LMRS) and Yvan Velenik
(LMRS)
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://front.math.ucdavis.edu/math.PR/0501135
---------------------------------------------------------------
3152. EDGE-REINFORCED RANDOM WALK ON A LADDER
Franz Merkl and Silke Rolles
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://front.math.ucdavis.edu/math.PR/0501137
---------------------------------------------------------------
3153. CONVERGENCE OF COALESCING NONSIMPLE RANDOM WALKS TO THE BROWNIAN
WEB
Rongfeng Sun
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://front.math.ucdavis.edu/math.PR/0501141
---------------------------------------------------------------
3154. ALTERNATIVES TO THE NEOBAYESIAN THEOREM, AVOIDING SEVERAL OF ITS
INCONSISTENCIES: THE RMPE-METHOD
Rainer Gottlob
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://front.math.ucdavis.edu/math.ST/0501134
---------------------------------------------------------------
3155. THE ISING-SHERRINGTON-KIRPATRICK MODEL IN A MAGNETIC FIELD AT
HIGH TEMPERATURE
Francis Comets (PMA) and Francesco Guerra (Fisica and Roma 1) and
Fabio Lucio Toninelli (Phys-ENS)
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://front.math.ucdavis.edu/math.PR/0501164
---------------------------------------------------------------
3156. TANAKA FORMULA FOR SYMMETRIC L\'{E}VY PROCESSES
Paavo Salminen and Marc Yor (PMA)
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://front.math.ucdavis.edu/math.PR/0501182
---------------------------------------------------------------
3157. PARTLY DIVISIBLE PROBABILITY DISTRIBUTIONS
S. Albeverio and H. Gottschalk and J.-L. Wu
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://front.math.ucdavis.edu/math.PR/0501183
---------------------------------------------------------------
3158. PARTLY DIVISIBLE PROBABILITY MEASURES ON LOCALLY COMPACT ABELIAN
GROUPS
S. Albeverio and H. Gottschalk and J.-L. Wu
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://front.math.ucdavis.edu/math.PR/0501185
---------------------------------------------------------------
3159. ESTIMATES OF RANDOM WALK EXIT PROBABILITIES AND APPLICATION TO
LOOP-ERASED RANDOM WALK
Michael J. Kozdron (University of Regina) and Gregory F. Lawler
(Cornell University)
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://front.math.ucdavis.edu/math.PR/0501189
---------------------------------------------------------------
3160. PERCOLATION WITH MULTIPLE GIANT CLUSTERS
E. Ben-Naim and P.L. Krapivsky
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://front.math.ucdavis.edu/cond-mat/0501218
---------------------------------------------------------------
3161. GOOD ROUGH PATH SEQUENCES AND APPLICATIONS TO ANTICIPATING &
FRACTIONAL STOCHASTIC CALCULUS
Laure Coutin and Peter Friz and Nicolas Victoir
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://front.math.ucdavis.edu/math.PR/0501197
---------------------------------------------------------------
3162. ON THE INCREMENTS OF THE PRINCIPAL VALUE OF BROWNIAN LOCAL TIME
Endre Cs\'aki and Yueyun Hu
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://front.math.ucdavis.edu/math.PR/0501199
---------------------------------------------------------------
3163. NONINTERSECTING PATHS, NONCOLLIDING DIFFUSION PROCESSES AND
REPRESENTATION THEORY
Makoto Katori and Hideki Tanemura
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://front.math.ucdavis.edu/math.PR/0501218
---------------------------------------------------------------
3164. RANDOM DYNAMICS AND THERMODYNAMIC LIMITS FOR POLYGONAL MARKOV
FIELDS IN THE PLANE
Tomasz Schreiber
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://front.math.ucdavis.edu/math.PR/0501228
---------------------------------------------------------------
3165. RANDOM GEOMETRIC GRAPH DIAMETER IN THE UNIT BALL
Robert B. Ellis and Jeremy L. Martin and and Catherine Yan
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://front.math.ucdavis.edu/math.CO/0501214
---------------------------------------------------------------
3166. STATISTICAL PROPERTIES OF THE PHASE TRANSITIONS IN A SPIN MODEL
FOR MARKET MICROSTRUCTURE
Muffasir Badshah and Robert Boyer and Ted Theodosopoulos
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://front.math.ucdavis.edu/math.PR/0501244
---------------------------------------------------------------
3167. PROPERTIES OF A RENEWAL PROCESS APPROXIMATION FOR A SPIN MARKET
MODEL
Muffasir Badshah and Robert Boyer and Ted Theodosopoulos
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://front.math.ucdavis.edu/math.PR/0501248
---------------------------------------------------------------
3168. FREE TRANSPORTATION COST INEQUALITIES FOR NON-COMMUTATIVE
MULTI-VARIABLES
Fumio Hiai and Yoshimichi Ueda
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://front.math.ucdavis.edu/math.OA/0501238
---------------------------------------------------------------
3169. MULTIDIMENSIONAL BERMUDAN OPTION PRICING VIA CUBATURE AND HOW TO
EXTRAPOLATE TO PRICE AMERICAN OPTIONS
Frederik S Herzberg
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://front.math.ucdavis.edu/math.PR/0501261
---------------------------------------------------------------
3170. BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS ON MANIFOLDS
Fabrice Blache
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://front.math.ucdavis.edu/math.PR/0501265
---------------------------------------------------------------
3171. SMALL BALL PROBABILITY ESTIMATES IN TERMS OF WIDTH
Rafa{\l} Lata{\l}a and Krzysztof Oleszkiewicz
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://front.math.ucdavis.edu/math.PR/0501268
---------------------------------------------------------------
3172. FREE ANALOG OF PRESSURE AND ITS LEGENDRE TRANSFORM
Fumio Hiai
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://front.math.ucdavis.edu/math.OA/0403210
---------------------------------------------------------------
3173. FREE EXTREME VALUES
Gerard Ben Arous and Dan Virgil Voiculescu
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://front.math.ucdavis.edu/math.OA/0501274
---------------------------------------------------------------
3174. POISSON-DIRICHLET DISTRIBUTION FOR RANDOM BELYI SURFACES
A. Gamburd
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://front.math.ucdavis.edu/math.PR/0501283
---------------------------------------------------------------
3175. STATIONARY DISTRIBUTIONS OF MULTI-TYPE TOTALLY ASYMMETRIC
EXCLUSION PROCESSES
Pablo A. Ferrari and James B. Martin
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://front.math.ucdavis.edu/math.PR/0501291
---------------------------------------------------------------
3176. GENERALIZED ARITHMETIC AND GEOMETRIC MEAN DIVERGENCE MEASURE AND
THEIR STATISTICAL ASPECTS
Inder Jeet Taneja
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://front.math.ucdavis.edu/math.PR/0501297
---------------------------------------------------------------
3177. ON MEAN DIVERGENCE MEASURES
Inder Jeet Taneja
\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://front.math.ucdavis.edu/math.PR/0501298
---------------------------------------------------------------
3178. ON UNIFIED GENERALIZATIONS OF RELATIVE JENSEN--SHANNON AND
ARITHMETIC--GEOMETRIC DIVERGENCE MEASURES, AND THEIR PROPERTIES
Pranesh Kumar and Inder Jeet Taneja
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-divergence}. By putting some
conditions on the
probability distribution, the aim here is to develop bounds on these
measures
and their parametric generalizations.
http://front.math.ucdavis.edu/math.PR/0501299
---------------------------------------------------------------
3179. GENERALIZED NON-SYMMETRIC DIVERGENCE MEASURES AND INEQUALITIES
Inder Jeet Taneja and Pranesh Kumar
In this paper we consider one parameter generalizations of some non -
symmetric divergence measures. Measures are \textit{relative
information},
$\chi ^2 - $\textit{divergence}, \textit{relative J-divergence},
\textit{relative Jensen-Shannon divergence }and \textit{relative
arithmetic and
geometric divergence}. All the generalizations considered can be
written as
particular cases of Csisz\'{a}r \textit{f-divergence}. By conditioning
the
probability distributions, relationships among the \textit{relative
divergence
measures }are obtained.
http://front.math.ucdavis.edu/math.PR/0501300
---------------------------------------------------------------
3180. ORNSTEIN-UHLENBECK-CAUCHY PROCESS
Piotr Garbaczewski and Robert Olkiewicz
We combine earlier investigations of linear systems with L\'{e}vy
fluctuations [Physica {\bf 113A}, 203, (1982)] with recent discussions
of
L\'{e}vy flights in external force fields [Phys.Rev. {\bf E 59},2736,
(1999)].
We give a complete construction of the Ornstein-Uhlenbeck-Cauchy
process as a
fully computable model of an anomalous transport and a paradigm example
of
Doob's stable noise-supported Ornstein-Uhlenbeck process. Despite the
nonexistence of all moments, we determine local characteristics
(forward drift)
of the process, generators of forward and backward dynamics, relevant
(pseudodifferential) evolution equations. Finally we prove that this
random
dynamics is not only mixing (hence ergodic) but also exact. The induced
nonstationary spatial process is proved to be Markovian and quite apart
from
its inherent discontinuity defines an associated velocity process in a
probabilistic sense.
http://front.math.ucdavis.edu/chao-dyn/9910028
---------------------------------------------------------------
3181. BURGERS VELOCITY FIELDS AND DYNAMICAL TRANSPORT PROCESSES
P. Garbaczewski and G. Kondrat
We explore a connection of the forced Burgers equation with the
Schr\"{o}dinger (diffusive) interpolating dynamics in the presence of
deterministic external forces. This entails an exploration of the
consistency
conditions that allow to interpret dispersion of passive contaminants
in the
Burgers flow as a Markovian diffusion process. In general, the usage of
a
continuity equation $\partial_t\rho =-\nabla (\vec{v}\rho)$, where
$\vec{v}=\vec{v}(\vec{x},t)$ stands for the Burgers field and $\rho $
is the
density of transported matter, is at variance with the explicit
diffusion
scenario. Under these circumstances, we give a complete
characterisation of the
diffusive matter transport that is governed by Burgers velocity fields.
The
result extends both to the approximate description of the transport
driven by
an incompressible fluid and to motions in an infinitely compressible
medium.
http://front.math.ucdavis.edu/cond-mat/9802060
---------------------------------------------------------------
3182. CAUCHY NOISE AND AFFILIATED STOCHASTIC PROCESSES
P. Garbaczewski and R. Olkiewicz
By departing from the previous attempt (Phys. Rev. {\bf E 51}, 4114,
(1995))
we give a detailed construction of conditional and perturbed Markov
processes,
under the assumption that the Cauchy law of probability replaces the
Gaussian
law (appropriate for the Wiener process) as the model of primordial
noise. All
considered processes are regarded as probabilistic solutions of the
so-called
Schr\"{o}dinger interpolation problem, whose validity is thus extended
to the
jump-type processes and their step process approximants.
http://front.math.ucdavis.edu/math-ph/9804014
---------------------------------------------------------------
3183. THE SCHROEDINGER PROBLEM, LEVY PROCESSES NOISE IN RELATIVISTIC
QUANTUM MECHANICS
P. Garbaczewski and J. R. Klauder and R. Olkiewicz
The main purpose of the paper is an essentially probabilistic analysis
of
relativistic quantum mechanics. It is based on the assumption that
whenever
probability distributions arise, there exists a stochastic process that
is
either responsible for temporal evolution of a given measure or
preserves the
measure in the stationary case. Our departure point is the so-called
Schr\"{o}dinger problem of probabilistic evolution, which provides for
a unique
Markov stochastic interpolation between any given pair of boundary
probability
densities for a process covering a fixed, finite duration of time,
provided we
have decided a priori what kind of primordial dynamical semigroup
transition
mechanism is involved. In the nonrelativistic theory, including quantum
mechanics, Feyman-Kac-like kernels are the building blocks for suitable
transition probability densities of the process. In the standard "free"
case
(Feynman-Kac potential equal to zero) the familiar Wiener noise is
recovered.
In the framework of the Schr\"{o}dinger problem, the "free noise" can
also be
extended to any infinitely divisible probability law, as covered by the
L\'{e}vy-Khintchine formula. Since the relativistic Hamiltonians
$|\nabla |$
and $\sqrt {-\triangle +m^2}-m$ are known to generate such laws, we
focus on
them for the analysis of probabilistic phenomena, which are shown to be
associated with the relativistic wave (D'Alembert) and matter-wave
(Klein-Gordon) equations, respectively. We show that such stochastic
processes
exist and are spatial jump processes. In general, in the presence of
external
potentials, they do not share the Markov property, except for stationary
situations. A concrete example of the pseudodifferential
Cauchy-Schr\"{o}dinger
evolution is analyzed in detail. The relativistic covariance of related
wave
http://front.math.ucdavis.edu/quant-ph/9505003
---------------------------------------------------------------
3184. FEYNMAN-KAC KERNELS IN MARKOVIAN REPRESENTATIONS OF THE
SCHROEDINGER INTERPOLATING DYNAMICS
Piotr Garbaczewski and Robert Olkiewicz
Probabilistic solutions of the so called Schr\"{o}dinger boundary data
problem provide for a unique Markovian interpolation between any two
strictly
positive probability densities designed to form the input-output
statistics
data for the process taking place in a finite-time interval. The key
issue is
to select the jointly continuous in all variables positive Feynman-Kac
kernel,
appropriate for the phenomenological (physical) situation. We extend the
existing formulations of the problem to cases when the kernel is \it
not \rm a
fundamental solution of a parabolic equation, and prove the existence
of a
continuous Markov interpolation in this case. Next, we analyze the
compatibility of this stochastic evolution with the original parabolic
dynamics, while assumed to be governed by the temporally adjoint pair of
(parabolic) partial differential equations, and prove that the
pertinent random
motion is a diffusion process. In particular, in conjunction with Born's
statistical interpretation postulate in quantum theory, we consider
stochastic
processes which are compatible with the Schr\"{o}dinger picture quantum
evolution.
http://front.math.ucdavis.edu/quant-ph/9505012
---------------------------------------------------------------
3185. INFORMATION THEORY, RELATIVE VERSIONS OF THE HYPERGRAPH
REGULARITY AND REMOVAL LEMMAS, THE SZEMER\'EDI-FURSTENBERG-KATZNELSON
THEOREM, AND PRIME
CONSTELLATIONS IN NUMBER FIELDS
Terence Tao
We present a proof of the Szemer\'edi-Furstenberg-Katznelson theorem
concerning multidimensional arithmetic progressions, using the Shannon
entropy
inequalities to establish an information-theoretic analogue of the
Szemer\'edi-Gowers-R\"odl-Skokan hypergraph regularity lemma. In
particular we
give (yet another) a self-contained proof of Szemer\'edi's famous
theorem on
arithmetic progressions, as well as the extension to pseudorandom
measures
obtained recently by Green and the author. As an application, we
combine these
methods with the Goldston-Y{\i}ld{\i}r{\i}m type analysis in that paper
to
reprove that the primes contain arbitrarily long arithmetic
progressions, and
in fact (partially) extend this result to higher dimensions and to
other number
fields, establishing in particular that there are infinitely many
constellations of primes of a prescribed shape in the Gaussian integers
$\Z[i]$.
http://front.math.ucdavis.edu/math.CO/0501314
---------------------------------------------------------------
3186. PERCOLATION-LIKE SCALING EXPONENTS FOR MINIMAL PATHS AND TREES IN
THE STOCHASTIC MEAN FIELD MODEL
David J. Aldous
In the mean field (or random link) model there are $n$ points and
inter-point
distances are independent random variables. For $0 < \ell < \infty$ and
in the
$n \to \infty$ limit, let $\delta(\ell) = 1/n \times$ (maximum number
of steps
in a path whose average step-length is $\leq \ell$). The function
$\delta(\ell)$ is analogous to the percolation function in percolation
theory:
there is a critical value $\ell_* = e^{-1}$ at which $\delta(\cdot)$
becomes
non-zero, and (presumably) a scaling exponent $\beta$ in the sense
$\delta(\ell) \asymp (\ell - \ell_*)^\beta$. Recently developed
probabilistic
methodology (in some sense a rephrasing of the cavity method of
Mezard-Parisi)
provides a simple albeit non-rigorous way of writing down such
functions in
terms of solutions of fixed-point equations for probability
distributions.
Solving numerically gives convincing evidence that $\beta = 3$. A
parallel
study with trees instead of paths gives scaling exponent $\beta = 2$.
The new
exponents coincide with those found in a different context (comparing
optimal
and near-optimal solutions of mean-field TSP and MST) and reinforce the
suggestion that these scaling exponents determine universality classes
for
optimization problems on random points.
http://front.math.ucdavis.edu/cond-mat/0501473
---------------------------------------------------------------
3187. GENERALISED SIFTING IN BLACK-BOX GROUPS
Sophie Ambrose and Max Neunhoeffer and Cheryl E. Praeger and Csaba
Schneider
We present a generalisation of the sifting procedure introduced
originally by
Sims for computation with finite permutation groups, and now used for
many
computational procedures for groups, such as membership testing and
finding
group orders. Our procedure is a Monte Carlo algorithm, and is
presented and
analysed in the context of black-box groups. It is based on a chain of
subsets
instead of a subgroup chain. Two general versions of the procedure are
worked
out in detail, and applications are given for membership tests for
several of
the sporadic simple groups.
Our major objective was that the procedures could be proved to be
Monte Carlo
algorithms, and their costs computed. In addition we explicitly
determined
suitable subset chains for six of the sporadic groups, and we
implemented the
algorithms involving these chains in the {\sf GAP} computational algebra
system. It turns out that sample implementations perform well in
practice. The
implementations will be made available publicly in the form of a {\sf
GAP}
package.
http://front.math.ucdavis.edu/math.GR/0501346
---------------------------------------------------------------
3188. THE STATIONARY MEASURE OF A 2-TYPE TOTALLY ASYMMETRIC EXCLUSION
PROCESS
Omer Angel
We give a combinatorial description of the stationary measure for a
totally
asymmetric exclusion process (TASEP) with second class particles, on
either Z
or on the cycle Z_N. The measure is the image by a simple operation of
the
uniform measure on some larger finite state space. This reveals a
combinatorial
structure at work behind several results on the TASEP with second class
particles.
http://front.math.ucdavis.edu/math.PR/0501005
---------------------------------------------------------------
3189. SIGNAL SIGNIFICANCE IN THE PRESENCE OF SYSTEMATIC AND STATISTICAL
UNCERTAINTIES
S.I. Bityukov (IHEP and Protvino)
The incorporation of uncertainties to calculations of signal
significance in
planned experiments is an actual task. Several approaches to this
problem are
discussed. We present a procedure for taking into account the systematic
uncertainty related to nonexact knowledge of signal and background cross
sections. A method of a treatment of statistical errors of the expected
signal
and background rates is proposed. The interrelation between Gamma- and
Poisson
distributions is demonstrated.
http://front.math.ucdavis.edu/hep-ph/0207130
---------------------------------------------------------------
3190. THE PROBABILITY OF MAKING A CORRECT DECISION IN HYPOTHESES
TESTING AS ESTIMATOR OF QUALITY OF PLANNED EXPERIMENTS
S.I. Bityukov (IHEP and Protvino) and N.V. Krasnikov (INR RAS and
Moscow)
In the report the approach to estimation of quality of planned
experiments is
considered. This approach is based on the analysis of uncertainty,
which will
take place under the future hypotheses testing about the existence of a
new
phenomenon in Nature. The probability of making a correct decision in
hypotheses testing is proposed as estimator of quality of planned
experiments.
This estimator allows to take into account systematics and statistical
uncertainties in determination of signal and background rates.
http://front.math.ucdavis.edu/physics/0309031
---------------------------------------------------------------
3191. A REVERSION OF THE CHERNOFF BOUND
Ted Theodosopoulos
This paper describes the construction of a lower bound for the tails of
general random variables, using solely knowledge of their moment
generating
function. The tilting procedure used allows for the construction of
lower
bounds that are tighter and more broadly applicable than existing tail
approximations.
http://front.math.ucdavis.edu/math.PR/0501360
---------------------------------------------------------------
3192. THE OVERHAND SHUFFLE MIXES IN $\THETA(N^2 \LOG N)$ STEPS
Johan Jonasson
The overhand shuffle is one of the ``real'' card shuffling methods in
the
sense that some people actually use it to mix a deck of cards. A
mathematical
model was constructed and analyzed by Pemantle [\ref{Pemantle}] who
showed that
the mixing time with respect to variation distance is at least of order
$n^2$
and at most of order $n^2\log n$. In this paper we use an extension of
a lemma
of Wilson [\ref{Wilson}] to establish a lower bound of order $n^2 \log
n$,
thereby showing that $n^2 \log n$ is indeed the correct order of the
mixing
time. It is our hope that the extension of Wilson's Lemma will prove
useful
also in other situations; it is demonstrated how it may be used to give
a
simplified proof of the $\Theta(n^3\log n)$ lower bound of Wilson
[\ref{Wilson2}] for the Rudvalis shuffle.
http://front.math.ucdavis.edu/math.PR/0501401
---------------------------------------------------------------
3193. DYNAMICALLY CONSISTENT NONLINEAR EVALUATIONS AND EXPECTATIONS
Shi-Ge Peng
How an economic agent (a firm, an investor or a financial market)
evaluates a
contingent claim, say a European type of derivatives X, with maturity
t? In
this paper we study a mechanism of dynamic expectations and
evaluations. We
give the axiomatic conditions of the time consistency. We prove that,
under a
domination condition, a time consistent nonlinear evaluation is in fact
a
g-expectation, i.e., it is completely determined a BSDE in which the
generator
is a given function g.
http://front.math.ucdavis.edu/math.PR/0501415
---------------------------------------------------------------
3194. TAIL-SENSITIVE GAUSSIAN ASYMPTOTICS FOR MARGINALS OF CONCENTRATED
MEASURES IN HIGH DIMENSION
Sasha Sodin
If the Euclidean norm is strongly concentrated with respect to a
measure, the
average distribution of an average marginal of this measure has Gaussian
asymptotics that captures tail behaviour. If the marginals of the
measure have
exponential moments, Gaussian asymptotics for the distribution of the
average
marginal implies Gaussian asymptotics for the distribution of most
individual
marginals. We show applications to measures of geometric origin.
http://front.math.ucdavis.edu/math.MG/0501382
---------------------------------------------------------------
3195. A FREE ANALOGUE OF THE TRANSPORTATION COST INEQUALITY ON THE
CIRCLE
F. Hiai and D. Petz
We give a new proof of the free transportation cost inequality for
measures
on the circle following M. Ledoux's idea.
http://front.math.ucdavis.edu/math.OA/0501389
---------------------------------------------------------------
3196. ALMOST SURE ESTIMATES FOR THE CONCENTRATION NEIGHBORHOOD OF
SINAI'S WALK
Pierre Andreoletti (LATP)
We consider Sinai's random walk in random environment. We prove that
infinitely often (i.o.) the size of the concentration neighborhood of
this
random walk is almost surely bounded. As an application we get that
i.o. the
maximal distance between two favorite sites is almost surely bounded.
http://front.math.ucdavis.edu/math.PR/0501439
---------------------------------------------------------------
3197. SCALING OF PERCOLATION ON INFINITE PLANAR MAPS, I
Omer Angel
We consider several aspects of the scaling limit of percolation on
random
planar triangulations, both finite and infinite. The equivalents for
random
maps of Cardy's formula for the limit under scaling of various crossing
probabilities are given. The limit probabilities are expressed in terms
of
simple events regarding Airy-Levy processes. Some explicit formulas for
limit
probabilities follow from this relation by applying known results on
stable
processes. Conversely, natural symmetries of the random maps imply
identities
concerning the Airy-Levy processes.
http://front.math.ucdavis.edu/math.PR/0501006
---------------------------------------------------------------
3198. POSITIVE HARMONIC FUNCTIONS FOR SEMI-ISOTROPIC RANDOM WALKS ON
TREES, LAMPLIGHTER GROUPS, AND DL-GRAPHS
Sara Brofferio and Wolfgang Woess
We determine all positive harmonic functions for a large class of
"semi-isotropic" random walks on the lamplighter group, i.e., the wreath
product of the cyclic group of order q with the infinite cyclic group.
This is
possible via the geometric realization of a Cayley graph of that group
as the
Diestel-Leader graph DL(q,q). More generally, DL(q,r) is the horocyclic
product
of two homogeneous trees with respective degrees $q+1$ and $r+1$, and
our
result applies to all DL-graphs. This is based on a careful study of the
minimal harmonic functions for semi-isotropic walks on trees.
http://front.math.ucdavis.edu/math.PR/0501440
---------------------------------------------------------------
3199. AN EQUIVALENT REPRESENTATION OF THE JACOBI FIELD OF A L\'EVY
PROCESS
Eugene Lytvynov
In [Yu.M. Berezansky, E. Lytvynov, D. A. Mierzejewski, Ukrainian Math.
J. 55
(2003), 853--858 ], the Jacobi field of a L\'evy process was derived.
This
field consists of commuting self-adjoint operators acting in an extended
(interacting) Fock space. However, these operators have a quite
complicated
structure. In this note, using ideas from [L. Accardi. U. Franz, M.
Skeide,
Comm. Math. Phys. 228 (2002), 123--150] and [E. Lytvynov, Infin. Dimen.
Anal.
Quant. Prob. Rel. Top. 7 (2004), 619--629], we obtain a unitary
equivalent
representation of the Jacobi field of a L\'evy process. In this
representation,
the operators act in a usual symmetric Fock space and have a much
simpler
structure.
http://front.math.ucdavis.edu/math.PR/0501450
---------------------------------------------------------------
3200. MIN-MAX VARIATIONAL PRINCIPLE AND FRONT SPEEDS IN RANDOM SHEAR
FLOWS
James Nolen and Jack Xin
Speed ensemble of bistable (combustion) fronts in mean zero stationary
Gaussian shear flows inside two and three dimensional channels is
studied with
a min-max variational principle. In the small root mean square regime
of shear
flows, a new class of multi-scale test functions are found to yield
speed
asymptotics. The quadratic speed enhancement law holds with probability
arbitrarily close to one under the almost sure continuity (dimension
two) and
mean square H\"older regularity (dimension three) of the shear flows.
Remarks
are made on the conditions for the linear growth of front speed
expectation in
the large root mean square regime.
http://front.math.ucdavis.edu/math.AP/0501445
---------------------------------------------------------------
3201. ON THE CONCENTRATION OF SINAI'S WALK
Pierre Andreoletti (CPT)
We consider Sinai's random walk in random environment. We prove that
for an
interval of time [1,n] Sinai's walk sojourns in a small neighborhood of
the
point of localization for the quasi totality of this amount of time.
Moreover
the local time at the point of localization normalized by $n$ converges
in
probability to a well defined random variable of the environment.
http://front.math.ucdavis.edu/math.PR/0501466
---------------------------------------------------------------
3202. ALTERNATIVE PROOF FOR THE LOCALIZATION OF SINAI'S WALK
Pierre Andreoletti
We give an alternative proof of the localization of Sinai's random walk
in
random environment under weaker hypothesis than the ones used by Sinai.
Moreover we give estimates that are stronger than the one of Sinai on
the
localization neighborhood and on the probability for the random walk to
stay
inside this neighborhood.
http://front.math.ucdavis.edu/math.PR/0501467
---------------------------------------------------------------
3203. ON THE CONCENTRATION OF SINAI'S WALK
Pierre Andreoletti (CPT)
We consider Sinai's random walk in random environment. We prove that
for an
interval of time [1,n] Sinai's walk sojourns in a small neighborhood of
the
point of localization for the quasi totality of this amount of time.
Moreover
the local time at the point of localization normalized by $n$ converges
in
probability to a well defined random variable of the environment.
http://front.math.ucdavis.edu/math.PR/0501466
---------------------------------------------------------------
3204. ALTERNATIVE PROOF FOR THE LOCALIZATION OF SINAI'S WALK
Pierre Andreoletti
We give an alternative proof of the localization of Sinai's random walk
in
random environment under weaker hypothesis than the ones used by Sinai.
Moreover we give estimates that are stronger than the one of Sinai on
the
localization neighborhood and on the probability for the random walk to
stay
inside this neighborhood.
http://front.math.ucdavis.edu/math.PR/0501467
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