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# A Coupling, and the Darling-Erdos Conjectures

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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)]

# Excursion Theory Revisited

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# The asymptotic shift for the principal eigenvalue under small obstacles.

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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.

# An anti-classification theorem for measure preserving transformations

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# An anti-classification theorem for measure preserving transformations

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# Valuing real options in fractional market

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# 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.