Probability Abstracts 45

1003. IMPROVED INCLUSION-EXCLUSION IDENTITIES AND INEQUALITIES BASED ON A PARTICULAR CLASS OF ABSTRACT TUBES

Klaus Dohmen

Recently, Naiman and Wynn [Ann. Statist. 25 (1997), 1954-1983]
introduced the concept of an abstract tube in order to obtain improved
inclusion-exclusion identities and inequalities that involve much fewer
terms than the classical ones. In this paper, we focus on a particular
class of abstract tubes which seem to play an important role in the
context of combinatorial enumeration and reliability computations. In
particular, we establish a link between the Naiman-Wynn theory of
abstract tubes and Brylawski's theory of broken circuit complexes.

dohmen@informatik.hu-berlin.de

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1004. ON A RELATION BETWEEN INTRINSIC AND EXTRINSIC DIRICHLET FORMS FOR INTERACTING PARTICLE SYSTEMS

Jose Luis Da Silva, Yuri G. Kondratiev and Michael Roeckner

In this paper we extend the result obtained in \cite{AKR97} (see also \cite
{AKR96}) on the representation of the intrinsic pre-Dirichlet form $\mathcal{%
E}_{\pi _{\sigma }}^{\Gamma }$ of the Poisson measure $\pi _{\sigma }$ in
terms of the extrinsic one $\mathcal{E}_{\pi _{\sigma },H_{\sigma
}^{X}}^{\Gamma }$. More precisely, replacing $\pi _{\sigma }$ by a Gibbs
measure $\mu $ on the configuration space $\Gamma _{X}$ we derive a relation
between the intrinsic pre-Dirichlet form $\mathcal{E}_{\mu }^{\Gamma }$ of
the measure $\mu $ and the extrinsic one $\mathcal{E}_{\mu ,H_{\sigma
}^{X}}^{P}$. As a consequence we prove the closability of $\mathcal{E}_{\mu
}^{\Gamma }$ on $L^{2}(\Gamma _{X},\mu )$ under very general assumptions on
the interaction potential of the Gibbs measures $\mu $.

roeckner@mathematik.uni-bielefeld.de

1005. A SUPPORT PROPERTY FOR INFINITE DIMENSIONAL INTERACTING DIFFUSION PROCESSES

Michael Roeckner and  Byron Schmuland:

The Dirichlet form associated with the intrinsic gradient on Poisson space is
known to be quasi-regular on the complete metric space $\ddot\Gamma=$
$\{Z_+$-valued Radon measures on $\IR^d\}$. We show that under mild conditions
, the set $\ddot\Gamma\setminus\Gamma$ is $\e$-exceptional, where $\Gamma$ is
the space of locally finite  configurations in $\IR^d$, that is,
measures $\gamma\in\ddot\Gamma$ satisfying $\sup_{x\in\IR^d}\gamma(\{x\})
\leq 1$.Thus, the associated diffusion lives on the smaller space $\Gamma$. 
This result also holds for Gibbs measures with superstable interactions.

roeckner@mathematik.uni-bielefeld.de

1006. RADEMACHER'S THEOREM ON CONFIGURATION SPACES AND APPLICATIONS

Michael Roeckner and Alexander Schied

We consider an $L^2$-Wasserstein  type distance $\rho$ on the configuration
space $\Gamma_X$ over a   Riemannian manifold $X$, and we prove that 
$\rho$-Lipschitz functions 
are contained in a   Dirichlet space associated with a measure on $\Gamma_X$ 
satisfying  some general assumptions. These assumptions are in particular 
fulfilled by a large  class of tempered  grandcanonical Gibbs measures with 
respect to a superstable lower regular pair potential. As an application we 
prove a criterion in terms of $\rho$ for a set to be exceptional. This result 
immediately implies, for instance, a quasi-sure version of the spatial ergodic
theorem. We also show that $\rho$ isoptimal in the sense that it is the
intrinsic metric of our Dirichlet form. 

roeckner@mathematik.uni-bielefeld.de

1007. STOCHASTIC ANALYSIS ON CONFIGURATION SPACES: BASIC IDEAS AND RECENT RESULTS

Michael Roeckner

The purpose of this paper is to provide a both comprehensive and summarizing
account on recent results about analysis and geometry on configuration spaces
$\Gamma_X$ over Riemannian manifolds $X$. Particular emphasis is given to  a
complete description of the so--called ``lifting--procedure'', Markov
resp. strong resp. $L^1$--uniqueness results, the non--conservative case, the
interpretation of the constructed diffusions as solutions of the respective
classical ``heuristic'' stochastic differential equations, and a 
self--containedpresentation of a general closability result for the
corresponding pre--Dirichlet forms. The latter is presented in the general 
case of arbitrary (not necessarily pair) potentials describing the singular 
interactions. A support property for the  diffusions, the intrinsic metric, 
and a Rademacher theorem on $\Gamma_X$, recently proved, are also discussed.

roeckner@mathematik.uni-bielefeld.de

1008. STOCHASTIC DYNAMICS OF COMPACT SPINS: ERGODICITY AND IRREDUCIBILITY

Sergio Albeverio, Alexei Daletskii, Yuri Kondratiev and  Michael Roeckner

Stochastic dynamics associated with Gibbs measures on $M^{Z^d}$, where $M$ is a
compact Riemannian manifold and $Z^d$ is an integer lattice, is
considered. Equivalence of its $L^2$--ergodicity and the extremality of the
corresponding Gibbs measures is proved.

roeckner@mathematik.uni-bielefeld.de

1009. ON THE EXISTENCE OF POSITIVE SOLUTIONS FOR CERTAIN QUASILINEAR ELLIPTIC EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS

Z.-Q. Chen, R. J. Williams and Z. Zhao

We give sufficient conditions for the existence of positive solutions 
to some quasilinear elliptic equations in bounded domains with Dirichlet 
boundary conditions, in dimensions three and higher. 
We impose mild conditions on the domains and
lower order coefficients of the equations in that the bounded domains 
are only required to satisfy an exterior cone condition and we allow 
the coefficients to have singularities controlled by Kato class functions.
Our approach uses an implicit probabilistic representation, 
Schauder's fixed point theorem, and new a priori estimates 
for solutions of the corresponding linear elliptic equations.
In the course of deriving these a priori estimates we show that
the Green functions for operators of the form 
${1\over 2} \Delta + b\cdot \nabla$ on $D$
are comparable when one modifies the 
drift term $b$ on a compact subset of $D$.
This generalizes a previous result of Ancona, obtained
under an $L^p$ condition, to a Kato condition on  $|b|^2$.

williams@math.ucsd.edu (hardcopy available)

1010. ASYMPTOTICS FOR THE RANDOM COUPON COLLECTOR PROBLEM

Vassilis G. Papanicolaou, George E. Kokolakis, and Shahar Boneh

We develop techniques of computing the asymptotics of the expected number of
items that one has to check in order to detect all $N$ existing kinds, as 
$N\rightarrow \infty $. The occurring frequencies of the differend kinds are
random variables.

papanico@cs.twsu.edu

1011. LARGE SCALE DEGREES AND THE NUMBER OF SPANNING CLUSTERS FOR THE UNIFORM SPANNING TREE

Itai Benjamini

We study large scale properties of the uniform spanning tree in $\Z^d$.
It is shown that inside the $n$-cube of $\Z^d$, there are $o(n^d)$ vertices 
with large scale degree 
$2$. In 2D the number of vertices with large 
scale degree $3$
is uniformly bounded and there are no vertices with large scale degree 
bigger than $3$.  
Also, it is shown that the number of spanning clusters for the uniform 
spanning tree in a square is tight as the square grows, and
that this is not the case in dimensions $4$ and higher. A similar 
transition is a known conjecture for critical Bernoulli percolation. 

itai@wisdom.weizmann.ac.il

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1012. LINEAR AND NEAR-LINEAR BOUNDS FOR STOCHASTIC DISPERSION

Mike Cranston, Michael Scheutzow, David Steinsaltz

It has been suggested that stochastic flows might be used to model the
spread of a passive substance on the surface of a body of water.  We
define a stochastic flow by the equations

\phi_0(x) = x 
d\phi_t(x)& = F(dt, \phi_t(x)),

where $F(t,x)$ is a field of semimartingales in d-dimensional Euclidean
space, for $d\geq 2$, whose local characteristics are bounded and
Lipschitz.  The particles are points in a bounded set X, and we ask how
far the substance has spread in a time T.  That is, we define 

\Phi_T^*= \sup_{x\in\X} \sup_{0\leq t\leq T} |\phi_t(x)|,

and seek to bound the probability that \Phi_T^*>z. 

If the field $F$ has a directed drift, of course, $|\phi_t(x)|$ will
grow linearly with time.  Taking the maximum does not increase the growth
rate very much, though: we show that for any $\alpha>1$,

\limsup_{T\to\infty} \Phi_T^* / T(\log T)^\alpha =0  a.s.

Without drift, when $F(\cdot,x)$ are required to
be martingales, although single points move on the order of $\sqrt{T}$, it
is easy to construct examples in which the supremum $\Phi_T^*$ still grows 
linearly in time --- that is, $\liminf_{T\to\infty} \Phi_T^*/T>0$ almost
surely.  We show that this is an upper bound for the growth; that is, we
compute a finite constant $K_0$, depending on the local characteristics,
such that 

\limsup_{T\to\infty} T^{-1}\Phi_T^* \leq K_0 a.s.

cran@math.rochester.edu ms@math.tu-berlin.de dstein@math.tu-berlin.de

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1013. SEQUENTIAL SELECTION OF AN INCREASING SUBSEQUENCE FROM A SAMPLE OF RANDOM SIZE

Alexander V. Gnedin

A random number of independent identically distributed random
variables is inspected in strict succession. As a variable is inspected,
it can be either selected or rejected and this decision
becomes at once final. The selected sequence must increase.
The problem is to maximize the expected length of selected sequence.

We demonstrate decision policies which approach
optimality when the number of observations becomes in a sense large and show
that the maximum expected length is close to an easily computable
value.  

gnedin@math.uni-goettingen.de

1014. SPATIALIZING RANDOM MEASURES: DOUBLY INDEXED PROCESSES AND THE LARGE DEVIATION PRINCIPLE

Christopher Boucher, Richard S. Ellis, and Bruce Turkington

The main theorem is the large deviation principle for the 
doubly indexed sequence of random measures

    W_{r,q}(dx\times dy)\doteq 
    \theta(dx)\otimes\sum_{k=1}^{2^r} 1_{D_{r,k}}(x) L_{q,k}(dy).   

Here $\theta$ is a probability measure on a Polish space ${\cal X}$,
$\{D_{r,k}, k=1,\ldots,2^r\}$ is a dyadic partition of ${\cal X}$ 
(hence the use of $2^r$ summands) satisfying $\theta\{D_{r,k}\} 
= 1/2^r$, and $L_{q,1}, L_{q,2},\ldots, L_{q,2^r}$ is an independent, 
identically distributed sequence of random probability measures 
on a Polish space ${\cal Y}$ such that ${L_{q,k}, q \in \N\}$
satisfies the large deviation principle with a convex rate function.  
A number of related asymptotic results are also derived.  

The random measures $W_{r,q}$ have important applications to the 
statistical mechanics of turbulence.  In a companion paper, 
the large deviation principle presented here is used to give 
a rigorous derivation of maximum entropy principles arising in 
the well known Miller-Robert theory of two-dimensional turbulence as well
as in a modification of that theory recently proposed by Turkington (1998).

boucher@math.umass.edu rsellis@math.umass.edu turk@math.umass.edu

1015. A NOTE ON SEQUENTIAL SELECTION FROM PERMUTATIONS

Alexander V. Gnedin

The maximum expected length of an increasing subsequence which can be
sequentially (i.e. on-line) selected from an unknown random permutation
of $n$ integers is known to be asymptotic to $\sqrt{2n}.$ 
We give a new direct proof of this fact and demonstrate a policy based
solely on relative ranks which achieves this value.

gnedin@math.uni-goettingen.de

1016. EFFICIENT MARKOVIAN COUPLINGS: EXAMPLES AND COUNTEREXAMPLES

Krzysztof Burdzy and Wilfrid S. Kendall

In this paper we study the notion of an efficient coupling of
Markov processes. Informally, an efficient coupling is one which
couples at the maximum possible exponential rate, as given by the
spectral gap. This notion
is of interest not only for its own sake, but also of growing
importance arising from the very recent advent of methods of ``perfect
simulation'': it helps to establish the ``price of perfection''
for such methods.
In general one can always achieve efficient coupling if the coupling
is allowed to ``cheat'' (if each component's behaviour is affected by
future behaviour of the other component), but the situation is
more interesting if the coupling is required to be {\em co-adapted}.
We present an informal heuristic for the existence of an efficient
coupling, and justify the heuristic by proving rigorous results
and examples in the
contexts of finite reversible Markov chains and of reflecting Brownian
motion in planar domains.
 
burdzy@math.washington.edu or W.S.Kendall@csv.warwick.ac.uk
      (postscript file or hard-copy available)

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• Or here.

1017. THE AVERAGE DENSITY OF THE PATH OF PLANAR BROWNIAN MOTION

Peter Moerters

We answer a question by Falconer and Xiao in (Stoch.Proc.Appl.55(1995), 
271-283) about the average densities of the image set of a planar 
Brownian motion by showing that the occupation measure $\mu$ on the path of
a Brownian motion run for an arbitrary finite time interval has an average
density of order three with respect to the gauge function 
$\varphi(t)=t^2\cdot \log(1/t)$. In other words, almost surely,
$$ \lim_{\eps\downarrow 0}\, \frac{1}{\log |\log \eps|} \int_\eps^{1/e} 
\frac{\mu(B(x,t))}{\varphi(t)} \; \frac{dt}{|t\, \log t|} = 2 
\mbox{\ \ at $\mu$-almost every $x$.} $$
We also prove a refinement of this statement: Almost surely, at $\mu$-almost
every $x$, the distribution of the $\varphi$-density function under the
logaritmic laws of order three converges to a gamma distribution with 
parameter two. The main tools of the proof are a Palm distribtuion associated
with the distribution of the occupation measure and an approximation of the
occupation measure of small balls by means of crossing numbers, which is due 
to Ray.

peter@mathematik.uni-kl.de

1018. PATHWISE KALLIANPUR ROBBINS LAWS FOR BROWNIAN MOTION IN THE PLANE

Peter Moerters

The Kallianpur Robbins law describes the long term asymptotic behaviour of
the distribution of the occupation measure of a Brownian motion in the plane 
In this paper we show that this behaviour can be seen at every typical 
Brownian path by choosing either a random time or a random scale according
to the logarithmic laws of order three. We also prove a ratio ergodic theorem
for small scales outside an exceptional set of vanishing logarithmic density 
of order three.  

peter@mathematik.uni-kl.de

1019. SELFDECOMPOSABILITY, PERPETUITY LAWS AND STOPPING TIMES

Zbigniew J. Jurek

In classical probability theory the class L distributions ( or
selfdecomposable distributions is the other name) are introduced as limit
laws of affinely modified partial sums of independent random variales,but
not neccessarily identicaly distributed. Class L includes stable laws but
also gamma,t-Student, F-Fisher, log-normal, log F, inverse gaussian, etc.
The basic characterization of X, as class L distribution, is as follows:

 for each t>0 there exists X(t) such that X = X(t) + exp(-t)X,  (in law),

with X(t) and X being independent. ( This justifies the term "selfdecomposability".)
In the paper we extend the selfdecomposability property to a certain class
of stopping tmes. This allows us to conclude that all selfdecomposable
distributions are the perpetuity laws  ( notion from
actuarial mathematics). As a by-product we get  gamma random variable
written as sum of products of independent  uniform distributions.
Finally, our new approach  applies to random affine mappings in Euclidean
or Banach spaces.

zjjurek@math.uni.wroc.pl (Tex-file or hard copy available)

1020. SAMPLE CORRELATION BEHAVIOR FOR THE HEAVY TAILED GENERAL BILINEAR PROCESS

Eric van den Berg and  Sidney Resnick

Consider the class of general bilinear models given by
$$
X_t = Z_t +\sum_{i=1}^p \phi_i X_{t-i} + \sum_{j=1}^q \theta_j Z_{t-j} +
\sum_{i=1}^m \sum_{j=1}^l b_{ij}X_{t-i} Z_{t-j}, \quad t \in \mathbb{Z},
$$
where $\{ Z_t \}$ is an i.i.d. sequence of heavy tailed noise
variables, and where $\prod_{j=1}^l b_{1j} \neq 0$. By means of a
point process analysis, we show that the sample correlation function
converges in distribution to a nondegenerate random variable.  Thus
standard model selection and fitting tools when applied to nonlinear
heavy tailed models will be grossly misleading.  
Also, consistency of the Hill
estimator as an estimate of the tail index for this class of bilinear
models is proved.

ericvdb@cam.cornell.edu,  sid@orie.cornell.edu

1021. ON A MEASURE CONCENTRATION INEQUALITY OF TALAGRAND FOR DEPENDENT RANDOM VARIABLES

Katalin Marton

We prove measure concentration for Talagrand's "convex hull" distance,
in the case of dependent random variables $(X_1,X_2,...,X_n)$, under
some condition on the dependence. We assume that for any k, $0<k<n$, and any 
two k-length initial sequences $x^k=(x_1,x_2,...x_k)$
and $y^k=(y_1,y_2,...y_k)$, differing only in the last symbol,
there exists a coupling of the "future distributions" 
$$
q(|x^k)=dist(X_{k+1},...X_n|x^k),\qquad  q(|y^k)=dist(X_{k+1},...X_n|y^k),
$$
that yields a small quadratic average for the error probability
$$
Pr(i'th \quad coordinates \quad differ|x^k, x_k^n, y^k).
$$
Thus we only control the influence of individual $X_k$'s on the later
random variables.

If the quadratic averages of the above error probabilities are
bounded by numbers $v_{i,k}, (i>k)$ then
we obtain a measure concentration inequality in terms of the sums
$\max_{k}\sum_i v_{i,k}$ and $\max_{i}\sum_k v_{i,k}$.

As an application we prove measure concentration for random fields
in a large square of the two dimensional lattice, under the so called
"strong mixing" condition, both for the Hamming and the "convex hull"
distance. In these examples it is essential that we only have to
control the influence of individual $X_k$'s on the later random variables.

marton@math-inst.hu

1022. THE INCIPIENT INFINITE CLUSTER IN HIGH-DIMENSIONAL PERCOLATION

Takashi Hara and Gordon Slade

We announce our recent proof that, for independent bond percolation in high 
dimensions, the scaling limits of the incipient infinite cluster's two-point
and three-point functions are those of integrated super-Brownian excursion
(ISE). The proof uses an extension of the lace expansion for percolation.
To appear in Electronic Research Announcements of the AMS, Volume 4, (1998).

hara@ap.titech.ac.jp   slade@mcmaster.ca 

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1023. SEQUENTIAL SELECTION UNDER A SUM CONSTRAINT WHEN THE NUMBER OF OBSERVATIONS IS RANDOM

Alexander V. Gnedin

A random number of non-negative i.i.d. random variables is inspected in
strict sequence. As a variable is inspected, it can be either selected
or rejected and this decision becomes at once final. The problem is to
maximize the expected length of selected sequence subject to the
constraint that the selected variables sum to no more than unity. 
We demonstrate a decision policy which is close to optimality when the
number of observations is in a sense large and give an approximate value
of the maximum expected length.

gnedin@math.uni-goettingen.de

1024. PERCOLATION PERTURBATIONS IN POTENTIAL THEORY AND RANDOM WALKS

Itai Benjamini, Russell Lyons, Oded Schramm

  We show that on a Cayley graph of a nonamenable group, almost surely the
infinite clusters of Bernoulli percolation are transient for simple random
walk, that simple random walk on these clusters has positive speed, and that
these clusters admit bounded harmonic functions. A principal new finding on
which these results are based is that such clusters admit invariant random
subgraphs with positive isoperimetric constant.
  We also show that percolation clusters in any amenable Cayley graph almost
surely admit no nonconstant harmonic Dirichlet functions. Conversely, on a
Cayley graph admitting nonconstant harmonic Dirichlet functions, almost surely
the infinite clusters of $p$-Bernoulli percolation also have nonconstant
harmonic Dirichlet functions when $p$ is sufficiently close to 1. Many
conjectures and questions are presented.

schramm@wisdom.weizmann.ac.il

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1025. MEASURES ON CONTOUR, POLYMER OR ANIMAL MODELS. A PROBABILISTIC APPROACH

Roberto Fern\'andez, Pablo A. Ferrari, Nancy L. Garcia 

  We present a new approach to study measures on ensembles of contours,
polymers or other objects interacting by some sort of exclusion condition. For
concreteness we develop it here for the case of Peierls contours. Unlike
existing methods, which are based on cluster-expansion formalisms and/or
complex analysis, our method is strictly probabilistic and hence can be applied
even in the absence of analyticity properties. It involves a Harris graphical
construction of a loss network for which the measure of interest is invariant.
The existence of the process and its mixing properties depend on the absence of
infinite clusters for a dual (backwards) oriented percolation process which we
dominate by a multitype branching process. Within the region of subcriticality
of this branching process the approach yields: (i) exponential convergence to
the equilibrium (=contour) measures, (ii) standard clustering and finite-effect
properties of the contour measure, (iii) a particularly strong form of the
central limit theorem, and (iv) a Poisson approximation for the distribution of
contours at low temperature.

pablo@ime.usp.br

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1026. A NOTE ON THE EIGENVALUE DENSITY OF RANDOM MATRICES

Michael K.-H. Kiessling and Herbert Spohn

  The distribution of eigenvalues of N times N random matrices in the limit N
to infinity is the solution to a variational principle that determines the
ground state energy of a confined fluid of classical unit charges. This fact is
a consequence of a more general theorem, proven here, in the statistical
mechanics of unstable interactions. Our result establishes the eigenvalue
density of some ensembles of random matrices which were not covered by previous
theorems.

miki@math.rutgers.edu

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1027. BROWNIAN SHEET IMAGES AND BESSEL-RIESZ CAPACITY

Davar Khoshnevisan

  We show that the image of a 2-dimensional set under d-dimensional,
2-parameter Brownian sheet can have positive Lebesgue measure, if and only if
the set in question has positive (d/2)-dimensional Bessel-Riesz capacity. Our
methods solve a problem of J.-P. Kahane.

davar@math.utah.edu

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1028. A NOTE ON SUMS OF INDEPENDENT RANDOM VARIABLES

Pawe{\l} Hitczenko and Stephen Montgomery-Smith

  In this note a two sided bound on the tail probability of sums of
independent, and either symmetric or nonnegative, random variables is obtained.
We utilize a recent result by Lata{\l}a on bounds on moments of such sums. We
also give a new proof of Lata{\l}a's result for nonnegative random variables,
and improve one of the constants in his inequality.

stephen@math.missouri.edu

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1029. POINT PROCESSES AND THE INFINITE SYMMETRIC GROUP. PART I: THE GENERAL FORMALISM AND THE DENSITY FUNCTION

Grigori Olshanski

  We study a 2-parametric family of probability measures on an
infinite-dimensional simplex (the Thoma simplex). These measures originate in
harmonic analysis on the infinite symmetric group (S.Kerov, G.Olshanski and
A.Vershik, Comptes Rendus Acad. Sci. Paris I 316 (1993), 773-778). Our approach
is to interpret them as probability distributions on a space of point
configurations, i.e., as certain point stochastic processes, and to find the
correlation functions of these processes.
  In the present paper we relate the correlation functions to the solutions of
certain multidimensional moment problems. Then we calculate the first
correlation function which leads to a conclusion about the support of the
initial measures. In the appendix, we discuss a parallel but more elementary
theory related to the well-known Poisson-Dirichlet distribution.
  The higher correlation functions are explicitly calculated in the subsequent
paper (A.Borodin, math.RT/9804087). In the third part (A.Borodin and
G.Olshanski, math.RT/9804088) we discuss some applications and relationships
with the random matrix theory.
  The goal of our work is to understand new phenomena in noncommutative
harmonic analysis which arise when the irreducible representations depend on
countably many continuous parameters.

borodine@math.upenn.edu

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1030. POINT PROCESSES AND THE INFINITE SYMMETRIC GROUP. PART II: HIGHER CORRELATION FUNCTIONS

Alexei Borodin

  We continue the study of the correlation functions for the point stochastic
processes introduced in Part I (G.Olshanski, math.RT/9804086). We find an integral
representation of all the correlation functions and their explicit expression
in terms of multivariate hypergeometric functions. Then we define a
modification (``lifting'') of the processes which results in a substantial
simplification of the structure of the correlation functions. It turns out that
the ``lifted'' correlation functions are given by a determinantal formula
involving a kernel. The latter has the form (A(x)B(y)-B(x)A(y))/(x-y), where A
and B are certain Whittaker functions. Such a form for correlation functions is
well known in the random matrix theory and mathematical physics. Finally, we
get some asymptotic formulas for the correlation functions which are employed
in Part III (A.Borodin and G.Olshanski, math.RT/9804088).

borodine@math.upenn.edu

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1031. POINT PROCESSES AND THE INFINITE SYMMETRIC GROUP. PART III: FERMION POINT PROCESSES

Alexei Borodin and Grigori Olshanski

  In Part I (G.Olshanski, math.RT/9804086) and Part II (A.Borodin,
math.RT/9804087) we developed an approach to certain probability distributions
on the Thoma simplex. The latter has infinite dimension and is a kind of dual
object for the infinite symmetric group. Our approach is based on studying the
correlation functions of certain related point stochastic processes.
  In the present paper we consider the so-called tail point processes which
describe the limit behavior of the Thoma parameters (coordinates on the Thoma
simplex) with large numbers. The tail processes turn out to be stationary
processes on the real line. Their correlation functions have determinantal form
with a kernel which generalizes the well-known sine kernel arising in random
matrix theory. Our second result is a law of large numbers for the Thoma
parameters. We also produce Sturm-Liouville operators commuting with the
Whittaker kernel introduced in Part II and with the generalized sine kernel.

borodine@math.upenn.edu

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1032. TIME DYNAMICS OF PROBABILITY MEASURE AND HEDGING OF DERIVATIVES

S. Esipov, I. Vaysburd 

  We analyse derivative securities whose value is a deterministic function of
an underlying which means presence of a basis risk at any time. The key object
of our analysis is conditional probability distribution at a given underlying
value and moment of time. We consider time evolution of this probability
distribution for an arbitrary hedging strategy (dynamically changing position
in the underlying asset). We assume log-brownian walk of the underlying and use
convolution formula to relate conditional probability distribution at any two
successive time moments. It leads to the simple PDE on the probability measure
parametrized by a hedging strategy. For delta-like distributions and
risk-neutral hedging this equation reduces to the Black-Scholes one. We further
analyse the PDE and derive formulae for hedging strategies targeting various
objectives, such as minimizing variance or optimizing quantile position.

iv3@nyu.edu

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1033. QUANTUM FIELD THEORY FOR DISCREPANCIES

A. van Hameren and R. Kleiss

  The concept of discrepancy plays an important role in the study of uniformity
properties of point sets. For sets of random points, the discrepancy is a
random variable. We apply techniques from quantum field theory to translate the
problem of calculating the probability density of (quadratic) discrepancies
into that of evaluating certain path integrals. Both their perturbative and
non-perturbative properties are discussed.

andrevh@sci.kun.nl

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