Probability Abstracts 52

1324. A PROOF OF THE RIEMANN HYPOTHESIS

Andrzej Madrecki

The starting point for the proof is the consideration of the abstact Hodge
decompositions and measures of functors. Involving the technique of
Gaussian measures in Banach spaces (specially important is possibility of
using of Fernique theorem) we linearize Green's function \mid z \mid^{-2}
through the functional Laplace transform which is the key to the proof
of the Riemann Hypothesis. Next we explain how the above results imply the
first of the two main technical results of the proof : the so called
Casteulnovo-Weil-Serre type inequality on the strong positivity of the
trace associated with the Riemann zeta function. The abstract Hodge type
decomposition and measure of \mid z \mid^{-2} (z \in C^{*}) is next used
to prove the main result of the paper- so called Riemann hypothesis
functional eqquation (RHFE in short) of the form:

Im(\zeta^{*}(s)) = Im(s)(2Re(s)-1)Tr(M_{G}M_{s})   s\in C, 

which immediately implies the Riemann Hypothesis.

madrecki@im.pwr.wroc.pl

1325. WAVELET ANALYSIS OF CONSERVATIVE CASCADES

Sidney Resnick, Anna Gilbert and Walter Willinger

A {\em conservative cascade} is an iterative process that fragments a
given set into smaller and smaller pieces according to a rule
which  
preserves the total mass
of the initial set at each stage of the construction almost surely and
not just in expectation. Motivated by the importance of conservative cascades
in analyzing multifractal behavior of measured Internet traffic traces,
we consider wavelet based statistical techniques for inference about
the cascade {\it generator\/}, the random mechanism determining the
re-distribution of the set's mass at each iteration. We provide two
estimators of the structure function, one asymptotically biased and one
not, prove consistency and asymptotic normality in a range of values of
the argument of the structure function less than a critical value
$q_*$. Simulation experiments illustrate the asymptotic properties of these
estimators and provide also  interesting conjectures for the
uninformative behavior of the estimators beyond the critical value
$q_*$. 

sid@orie.cornell.edu agilbert@research.att.com walter@research.att.com

• To see a preprint or other information provided by the author click here.

1326. EXTENDING THE WONG-ZAKAI THEOREM TO REVERSIBLE MARKOV PROCESSES

Richard Bass, Ben Hambly, and Terry Lyons

We show how to construct a canonical choice of stochastic area for paths of
reversible Markov processes satisfying a weak Holder condition, and
hence demonstrate that the sample paths of such processes are rough paths in
the sense of Lyons. We further prove that certain polygonal approximations
to these paths and their areas converge in $p$-variation norm. As a
corollary of this result and standard properties of rough paths, we are ableto p
rovide a significant generalization of the classical result of
Wong-Zakai on the approximation of solutions to stochastic differential
equations. Our results allow us to construct solutions to differential
equations driven by reversible Markov processes of finite $p$-variation with
$p<4$.

bass@math.uconn.edu  benham@hplb.hpl.hp.com   t.lyons@ic.ac.uk

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1327. ASYMPTOTIC ANALYSIS OF THE SIEVE ESTIMATOR FOR A CLASS OF PARABOLIC SPDES

Marianne Huebner and Sergey Lototsky

In this paper we consider the problem of estimating
a coefficient of a strongly elliptic partial 
differential operator in 
stochastic parabolic  equations. The coefficient is a
bounded function of time. We compute the maximum likelihood estimate of
the function on an approximating space (sieve)  using a finite number of 
the spatial Fourier coefficients 
 of the solution  and establish  conditions  that guarantee  
consistency and asymptotic normality of the resulting estimate
as the number of the coefficients increases. The equation is assumed 
diagonalizable in the sense that all the operators have a common 
system of eigenfunction. 

huebner@stt.msu.edu  lototsky@math.mit.edu

1328. A SPATIAL MODEL OF RANGE-DEPENDENT SUCCESSION

Stephen M. Krone and Claudia Neuhauser

We consider an interacting particle system in which each site of the
d-dimensional integer lattice can be in state 0, 1, or 2.  Our aim is
to model the spread of disease in certain plant populations, so think of
0=vacant, 1=(healthy) plant, 2=infected plant.  A vacant site becomes
occupied by a plant at a rate which increases linearly with the number
of plants within range R, up to some saturation level, F, above which
the rate is constant.  Similarly, a plant becomes infected at a rate which
increases linearly with the number of infected plants within some range
M, up to some saturation level G.  An infected plant dies (and the site
becomes vacant) at constant rate $\delta$.  We discuss coexistence results
in one and two dimensions.  These results depend on the relative dispersal
ranges for plants and disease.

krone@uidaho.edu

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1329. ON THE SURVIVING PROBABILITY OF AN ANNIHILATING BRANCHING PROCESS AND APPLICATION TO A NONLINEAR VOTER MODEL

Smail Alili and Irina Ignatiouk

We consider a discrete time interacting particle system which can be considered as 
an annihilating branching process on $\Z$ where at each time each particle either 
performs a jump as a nearest neighboor random walk, or splits (with probability $\ep$) 
into two particles which will occupy the nearest neighboor sites. Furthermore, if 
two particles come to the same site, then they are removed from the system. We show 
that if the branching probability $\ep >0$ is small enough, and the number of 
particles at initial time is finite, then the surviving probability 
$$p(t)=\P\bigl( \mbox{there is at least one particle at time } \; t\Bigr)$$
decays to zero exponentialy fast. This result is applied to a nonlinear discrete time 
voter model (in a random and non-random environment) obtained as a small 
perturbation with parameter $\ep$ of the classical voter model. For this 
class of models, we show that if $\ep >0$ is small enough, then the process 
converges to a unique invariant probability measure independently on the initial 
distribution. It is known that the classical one-dimensional voter model (in a  
random environment as well as without environment) is not ergodic, that is there
exist at least two extremal invariant probability measures. Our results prove therefore the 
phase transition in $\ep =0$.

alili@u-cergy.fr   ignatiouk@u-cergy.fr

1330. ZERO-TEMPERATURE ISING SPIN DYNAMICS ON THE HOMOGENEOUS TREE OF DEGREE THREE

C. Douglas Howard

We investigate zero-temperature dynamics for a homogeneous 
ferromagnetic Ising model on the homogeneous tree of degree
three with random (i.i.d. Bernoulli) spin configuration 
at time 0.  Letting $\theta$ denote the probability that 
any particular vertex has a $+1$ initial spin, for 
$\theta=1/2$, infinite spin clusters do not exist at 
time 0 but we show that infinite ``spin chains'' (doubly 
infinite paths of vertices with a common spin) exist in 
abundance at any time $\epsilon>0$.  We study the 
structure of the subgraph of the tree generated by the 
vertices in time-$\epsilon$ spin chains. We show the 
existence of a phase transition in the sense that, for 
some critical $\theta_c$ with $0<\theta_c<1/2$, 
$+1$ spin chains almost surely never form for 
$\theta < \theta_c$ but almost surely do form in 
finite time for $\theta > \theta_c$. We relate 
these results to certain quantities of physical interest, 
such as the $t\to\infty$ asymptotics of the probability  
that any particular vertex changes spin after time $t$. 

dhoward@math.baruch.cuny.edu

1331. A CLASS OF VELOCITY FIELDS WITH KNOWN LAGRANGIAN LAW

Curtis D. Bennett and Craig L. Zirbel

We introduce a large class of random velocity fields on
the periodic lattice and in discrete time having a certain
hidden Markov structure.  The generalized Lagrangian
velocity (the velocity field as viewed from the location
of a single moving particle) has similar hidden Markov
structure, and its law is found exactly.  As a result, the
law of the trajectory of an individual particle is known
in principle.  The rate of convergence to equilibrium of
the generalized Lagrangian velocity is studied in small
numerical examples and in rigorous results giving absolute
and relative bounds on the size of the spectral gap.  The
effect of molecular diffusion on the rate of convergence
is also investigated.

zirbel@aurora.bgsu.edu

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• Or here.

1332. MIXED PERCOLATION AS A BRIDGE BETWEEN SITE AND BOND PERCOLATION

L. Chayes and Roberto H. Schonmann

By using mixed percolation as a bridge between site and
bond percolation, we derive a new inequality between the
critical points of these processes, which is optimal in a
certain sense, and we also extend a result on the crossover
exponent of bond diluted Potts models to site diluted
Potts models. Some new results about the critical line in
mixed percolation are also proved.

lchayes@math.ucla.edu  rhs@math.ucla.edu

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1333. MARTINGALES ON RANDOM SETS AND THE STRONG MARTINGALE PROPERTY

Michael J. Sharpe

Suppose given be a process defined on an optional random set. 
The paper develops two different conditions  guaranteeing
under secondary hypotheses that the process is the
restriction of a uniformly integrable martingale. 
It is supposed in each case that the process is known to be
the restriction  of some special semimartingale. The first
condition, which is both necessary and sufficient, is an
absolute continuity condition on the predictable part of the
semimartingale. Under additional hypotheses, the existence
of a martingale extension can also be characterized by  a
strong martingale property. Uniqueness of the extension is
also considered. 

msharpe@ucsd.edu

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1334. WEIGHTED APPROXIMATIONS OF TAIL PROCESSES UNDER MIXING CONDITIONS

Holger Drees

While the extreme value statistics for i.i.d. data is well developed, much
less is known about the asymptotic behavior of statistical procedures in the
presence of dependence. We establish convergence of tail empirical processes
to Gaussian limits for $\beta$--mixing stationary time series. As a consequence
one obtains weighted approximations of the tail empirical quantile function
that is based on a random sequence with marginal distribution belonging to the
domain of attraction of an extreme value distribution. Moreover, the asymptotic
normality is concluded for a large class of estimators of the extreme value
index. These results are applied to stationary solutions of a general
stochastic difference equation.

hdrees@mi.uni-koeln.de

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1335. OVERCOMING PRIORS ANXIETY

G. D'Agostini

The choice of priors may become an insoluble problem if priors and Bayes'
rule are not seen and accepted in the framework of subjectivism. Therefore, the
meaning and the role of subjectivity in science is considered and defended from
the pragmatic point of view of an ``experienced scientist''. The case for the
use of subjective priors is then supported and some recommendations for routine
and frontier measurement applications are given. The issue of reference priors
is also considered from the practical point of view and in the general context
of ``Bayesian dogmatism''.

Giulio.Dagostini@roma1.infn.it

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1336. A VARIATIONAL COUPLING FOR A TOTALLY ASYMMETRIC EXCLUSION PROCESS WITH LONG JUMPS BUT NO PASSING

Timo Seppalainen

We prove a weak law of large numbers for a tagged particle in a totally
asymmetric exclusion process on the one-dimensional lattice. The particles are
allowed to take long jumps but not pass each other. The object of the paper is
to illustrate a special technique for proving such theorems. The method uses a
coupling that mimics the Hopf-Lax formula from the theory of viscosity
solutions of Hamilton-Jacobi equations.

seppalai@iastate.edu

• This article is available from the xxx mathematics archive as math.PR/9906192.

1337. STRONG LAW OF LARGE NUMBERS FOR THE INTERFACE IN BALLISTIC DEPOSITION

Timo Seppalainen

We prove a hydrodynamic limit for ballistic deposition on a multidimensional
lattice. In this growth model particles rain down at random and stick to the
growing cluster at the first point of contact. The theorem is that if the
initial random interface converges to a deterministic macroscopic function,
then at later times the height of the scaled interface converges to the
viscosity solution of a Hamilton-Jacobi equation. The proof idea is to
decompose the interface into the shapes that grow from individual seeds of the
initial interface. This decomposition converges to a variational formula that
defines viscosity solutions of the macrosopic equation. The technical side of
the proof involves subadditive methods and large deviation bounds for related
first-passage percolation processes.

seppalai@iastate.edu

• This article is available from the xxx mathematics archive as math.PR/9906193.

1338. RECENT RESULTS AND OPEN PROBLEMS ON THE HYDRODYNAMICS OF DISORDERED ASYMMETRIC EXCLUSION AND ZERO-RANGE PROCESSES

Timo Seppalainen

This paper summarizes results and some open problems about the large-scale
and long-time behavior of asymmetric, disordered exclusion and zero-range
processes. These processes have randomly chosen jump rates at the sites of the
underlying lattice. The interesting feature is that for suitably distributed
random rates there is a phase transition where the process behaves differently
at high and low densities. Some of this distinction is visible on the
hydrodynamic scale.

seppalai@iastate.edu

• This article is available from the xxx mathematics archive as math.PR/9906194.

1339. NOISE SENSITIVITY ON CONTINUOUS PRODUCTS: AN ANSWER TO AN OLD QUESTION OF J. FELDMAN

Boris Tsirelson

A relation between sigma-additivity and linearizability, conjectured by Jacob
Feldman in 1971 for continuous products of probability spaces, is established
by relating both notions to a recent idea of noise stability/sensitivity.

tsirel@math.tau.ac.il

• This article is available from the xxx mathematics archive as math.PR/9907011.

1340. L\'EVY PROCESSES ON $U_Q(G)$ AS INFINITELY DIVISIBLE REPRESENTATIONS

V.K. Dobrev, H.-D. Doebner, U. Franz, and R. Schott

L\'evy processes on bialgebras are families of infinitely divisible
representations. We classify the generators of L\'evy processes on the compact
forms of the quantum algebras $U_q(g)$, where $g$ is a simple Lie algebra. Then
we show how the processes themselves can be reconstructed from their generators
and study several classical stochastic processes that can be associated to
these processes.

ptvd@pt.tu-clausthal.de

• This article is available from the xxx mathematics archive as math.PR/9907016.

1341. MEAN-FIELD THEORY FOR SCALE-FREE RANDOM NETWORKS

Albert-Laszlo Barabasi, Reka Albert, Hawoong Jeong

Random networks with complex topology are common in Nature, describing
systems as diverse as the world wide web or social and business networks.
Recently, it has been demonstrated that most large networks for which
topological information is available display scale-free features. Here we study
the scaling properties of the recently introduced scale-free model, that can
account for the observed power-law distribution of the connectivities. We
develop a mean-field method to predict the growth dynamics of the individual
vertices, and use this to calculate analytically the connectivity distribution
and the scaling exponents. The mean-field method can be used to address the
properties of two variants of the scale-free model, that do not display
power-law scaling.

ralbert@iron.helios.nd.edu

• To see a preprint or other information provided by the author click here.

1342. GAUSSIAN FLUCTUATION FOR THE NUMBER OF PARTICLES IN AIRY, BESSEL, SINE AND OTHER DETERMINANTAL RANDOM POINT FIELDS

Alexander B. Soshnikov

We prove the Central Limit Theorem for the number of eigenvalues near the
spectrum edge for hermitian ensembles of random matrices. To derive our
results, we use a general theorem, essentially due to Costin and Lebowitz,
concerning the Gaussian fluctuation of the number of particles in random point
fields with determinantal correlation functions. As another corollary of
Costin-Lebowitz Theorem we prove CLT for the empirical distribution function of
the eigenvalues of random matrices from classical compact groups.

sashas@cco.caltech.edu

• To see a preprint or other information provided by the author click here.

1343. UNIVERSALITY AT THE EDGE OF THE SPECTRUM IN WIGNER RANDOM MATRICES

Alexander Soshnikov

We prove universality at the edge for rescaled correlation functions of
Wigner random matrices in the limit $n\rightarrow +\infty$. As a corollary, we
show that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of
Wigner random hermitian (resp. real symmetric) matrix weakly converge to the
distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases.

sashas@cco.caltech.edu

• To see a preprint or other information provided by the author click here.

1344. INFINITE WEDGE AND MEASURES ON PARTITIONS

Andrei Okounkov

We use elementary commutation relations computations in the infinite wedge
space to evaluate: 1) correlation functions of what we call the Schur measure
on partitions, 2) polynomial averages with respect to the uniform measure on
partitions. The first result is a generalization of the formula due to Borodin
and Olshanski for the so called z-measure, which is a specialization of the
Schur measure (see math.RT/9904010 and also math.RT/9905032 for applications).
The second result provides a new proof of a formula due to Bloch and the
author, see alg-geom/9712009. We also comment on the local structure of a
typical partition with respect to the uniform measure.

okounkov@math.uchicago.edu

• To see a preprint or other information provided by the author click here.

1345. EQUITY ALLOCATION AND PORTFOLIO SELECTION IN INSURANCE: A SIMPLIFIED PORTFOLIO MODEL

Erik Taflin 

A quadratic discrete time probabilistic model, for optimal portfolio
selection in (re-)insurance is studied. For positive values of underwriting
levels, the expected value of the accumulated result is optimized, under
constraints on its variance and on annual ROE's. Existence of a unique solution
is proved and a Lagrangian formalism is given. An effective method for solving
the Euler-Lagrange equations is developed. The approximate determination of the
multipliers is discussed. This basic model is an important building block for
more complete models.

etaflin@u-bourgogne.fr

• To see a preprint or other information provided by the author click here.

1346. LIMIT THEOREMS FOR MOTIONS IN A FLOW WITH A NONZERO DRIFT

Albert Fannjiang and Tomasz Komorowski 

We establish diffusion and fractional Brownian motion approximations for
motions in a Markovian Gaussian random field with a nonzero mean.

fannjian@math.ucdavis.edu

• This article is available from the xxx mathematics archive as math.PR/9907159.

1347. EQUITY ALLOCATION AND PORTFOLIO SELECTION IN INSURANCE

Erik Taflin 

A discrete time probabilistic model, for optimal equity allocation and
portfolio selection, is formulated so as to apply to (at least) reinsurance. In
the context of a company with several portfolios (or subsidiaries),
representing both liabilities and assets, it is proved that the model has
solutions respecting constraints on ROE's, ruin probabilities and market shares
currently in practical use. Solutions define global and optimal risk management
strategies of the company. Mathematical existence results and tools, such as
the inversion of the linear part of the Euler-Lagrange equations, developed in
a preceding paper in the context of a simplified model are essential for the
mathematical and numerical construction of solutions of the model.

etaflin@u-bourgogne.fr

• To see a preprint or other information provided by the author click here.

1348. PERCOLATION AND NUMBER OF PHASES IN THE 2D ISING MODEL

H.-O. Georgii, Y. Higuchi

We reconsider the percolation approach of Russo, Aizenman and Higuchi for
showing that there exist only two phases in the Ising model on the square
lattice. We give a fairly short alternative proof which is only based on FKG
monotonicity and avoids the use of GKS-type inequalities originally needed for
some background results. Our proof extends to the Ising model on other planar
lattices such as the triangular and honeycomb lattice. We can also treat the
Ising antiferromagnet in an external field and the hard-core lattice gas model
on $Z^2$.

georgii@rz.mathematik.uni-muenchen.de

• This article is available from the xxx mathematics archive as math.PR/9907186.

1349. DISCRETE GROWTH MODELS

Dorothea M. Eberz-Wagner

The work is the author's Ph.D. Thesis.
Part I of the thesis gives a complete analysis of gaps in a one-dimensional
creation-annihilation model. Part II contains a proof of the existence of
infinitely many holes in the two-dimensional DLA cluster.

burdzy@math.washington.edu

• This article is available from the xxx mathematics archive as math.PR/9908030.

1350. TEACHING STATISTICS IN THE PHYSICS CURRICULUM: UNIFYING AND CLARIFYING ROLE OF SUBJECTIVE PROBABILITY

G. D'Agostini

Subjective probability is based on the intuitive idea that probability
quantifies the degree of belief that an event will occur. A probability theory
based on this idea represents the most general framework for handling
uncertainty. A brief introduction to subjective probability and Bayesian
inference is given, with comments on typical misconceptions which tend to
discredit it and comparisons to other approaches.

Giulio.Dagostini@roma1.infn.it

• To see a preprint or other information provided by the author click here.

1351. QUANTITATIVE ESTIMATES OF DISCRETE HARMONIC MEASURES

E. Bolthausen and K. Muench-Berndl

A theorem of Bourgain states that the harmonic measure of a domain in \R^d is
supported on a set of Hausdorff dimension strictly less than d [J. Bourgain, On
the Hausdorff dimension of harmonic measure in higher dimension. Invent. Math.
87 (1987), 477-483]. We apply Bourgain's method to the discrete case, i.e., to
the distribution of the first entrance point of a random walk into a subset of
\Z^d, d \geq 2. By refining the argument, we prove that for all \b > 0 there
exists \rho(d,\b) < d and N(d,\b), such that for any n > N(d,\b), any x \in
\Z^d, and any A \subset {1,...,n}^d |{y \in \Z^d: \nu_{A,x}(y) \geq n^{-\b}}|
\leq n^{\rho(d,\b)}, where \nu_{A,x}(y) denotes the probability that y is the
first entrance point of the simple random walk starting at x into A.

kmb@amath.unizh.ch

• This article is available from the xxx mathematics archive as math.PR/9908047.

1352. CENTRAL LIMIT THEOREM FOR LOCAL LINEAR STATISTICS IN CLASSICAL COMPACT GROUPS AND RELATED COMBINATORIAL IDENTITIES

Alexander Soshnikov

We discuss CLT for the global and local linear statistics of random matrices
from classical compact groups. The main part of our proofs are certain
combinatorial identities much in the spirit of works by Kac and Spohn.

sashas@cco.caltech.edu

• This article is available from the xxx mathematics archive as math.PR/9908063.

1353. THE PROPAGATION OF MOLECULAR CHAOS BY MARKOV TRANSITIONS

Alexander David Gottlieb

We establish a necessary and sufficient condition for the propagation of
chaos by a family of many-particle Markov processes, if the particles live in a
Polish space $S$: A sequence of $n$-particle Markov transition functions
$\{K_n\}$ propagates molecular chaos if and only if the sequence
$\{K_n(\bs_n,\cdot)\}$ is chaotic whenever $\bs^n = (s^n_1,s^n_2,...,s^n_n) \in
S^n$ is such that $\oon \sum_i \delta(s^n_i)$ converges to a law on S as $n
\longrightarrow \infty$.

gottlieb@indus.lbl.gov

• This article is available from the xxx mathematics archive as math.PR/9908084.