We present a coupling of the 1-dimensional Ornstein-Uhlenbeck process with an i.i.d. sequence.
We then apply this coupling to resolve two conjectures of Darling and Erd\H{o}s (1956).
Interestingly enough, we prove one and disprove the other conjecture. [This is joint work with David Levin.]

Time-permitting, we may use the ideas of this talk to describe precisely the rate of convergence in the
classical law of the iterated logarithm of Khintchine for Brownian motion (1933).
[This portion is joint work with David Levin and Zhan Shi, and has recently appeared in
the Electr. Comm. of Probab. (2005)]

We study the asymptotic shift for principal eigenvalue for a
large class of second order elliptic operators on bounded domains subject
to perturbations known as obstacles. The results extend the well-studied
self-adjoint case. The approach is probabilistic.

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.