# Quasi-cumulants and limit theorems for stable laws.

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We will present several new results about global theorem and asymptotic expansions for the distributions of iid random variables in the domain of attraction of stable laws. Particular attention will be paid to the Cuachy case which exhibits especially interesting features.

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# Moment problems, exchangeability, and the Curie-Weiss model

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The Curie-Weiss model is an exchangeable probability measure $\mu$ on $\{0,1\}^n.$

It has two parameters -- the external magnetic field $h$ and the interaction $J$.

A natural problem is to determine when this measure extends to an exchangeable measure

on $\{0,1\}^{\infty}$. We will discuss two approaches to the following result:

$\mu$ can be (infinitely) extended if and only if $J\geq 0$. One of these

approaches relies on the classical Hausdorff moment problem. When $Jn$ can $\mu$ be extended to an exchangeable measure on $\{0,1\}^l$. Our approach

to this question involves an apparently new type of moment problem, which we will

solve. We then take $J=-c/l$, and determine the values of $c$ for which $l$-extendibility

is possible for all large $l$. This is joint work with Jeff Steif and Balint Toth.

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# Local martingale functions of Brownian motion

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ABSTRACT: A harmonic function of the Brownian path is a local martingale. Is the converse true? We show that the class of local martingale functions of Brownian motion is co-extensive with the class of finely harmonic functions, and then use a results of Fuglede and Gardiner to answer this question in the negative, in dimensions bigger than 2.

# `Volatility perturbations in financial markets'.

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# Results and Challenges in the study of Motion In Random Environments

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# Statistics of processes with applications to Mathematical Finance.

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# Renormalized self-intersection local time and the range of random walks.

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Self-intersection local time $\beta_t$ is a measure of how often

a Brownian motion (or other process) crosses itself. Since Brownian

motion in the plane intersects itself so often, a renormalization

is needed in order to get something finite. LeGall proved that

$E e^{\gamma \beta_1}$ is finite for small $\gamma$ and infinite

for large $\gamma$. It turns out that the critical value is related

to the best constant in a Gagliardo-Nirenberg inequality. I will discuss

this result (joint work with Xia Chen) as well as large deviations

for $\beta_1$ and $-\beta_1$ and LILs for $\beta_t$ and $-\beta_t$.

The range of random walks is closely related to self-intersection

local times, and I will also discuss joint work with Jay Rosen

making this idea precise.