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.

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.

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.

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.