Stimulated by applications to internet traffic modeling and insurance mathematics, distributions with heavy tails have been widely studied over the past decades. This talk addresses a fundamental large-deviation problem for random walks with heavy-tailed step-size distributions. We consider so-called subexponential step-size distributions, which constitute the most widely-used class of heavy-tailed distributions. I will present a theory to study sequences {x_n} for which P{S_n>x_n} behaves asymptotically like n P {S_1>x_n} for large n. (joint work with D. Denisov and V. Shneer)

The quantum adiabatic algorithm is a general approach to solving combinatorial search problems using a quantum computer. It will succeed if the run time is long enough. I will discuss how this algorithm works and our current understanding of the required run time.

We consider the Gaussian random walk (one-dimensional random walk with normally distributed increments), and in particular the moments of its maximum M. Explicit expressions for all moments of M are derived in terms of Taylor series with coefficients that involve the Riemann zeta function. We build upon the work of Chang and Peres (1997) on P(M=0) and Bateman's formulas on Lerch's transcendent. Our result for E(M) completes earlier work of Kingman (1965), Siegmund (1985), and Chang and Peres (1997). The…

We introduce the unitary Schwinger-Dyson equation associated to a selfadjoint polynomial potential V. The V=0 case corresponds to the free product state, so the Schwinger-Dyson equation can be considered as a deformation of free probability. We show that the solutions of this equation are unique for small V's and correspond to a large N limit of a multi-matrix model. This technique allows to show that a wide class of unitary and orthogonal multi-matrix models converge asymptotically. We also give a…

Large-scale complex networks have been the objects of study for the past two decades, and one central problem have been constructing or designing realistic models for such networks. This problem appears in a variety of applications including coding theory and systems biology. Unfortunately, the existing algorithms for this problem have large running times, making them impractical to use for networks with millions of links. We will present a technique to design an almost linear time algorithm for generating random graphs…

We develop an information-theoretic foundation for compound Poisson approximation and limit theorems (analogous to the corresponding developments for the central limit theorem and for simple Poisson approximation). First, su?cient conditions are given under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. In particular, it is shown that a maximum entropy property is valid if the measures under consideration are log-concave, but that it fails in general. Second, approximation bounds…

Fractional Brownian motions are the most natural generalizations of ordinary (one-dimensional) Brownian motions that allow for some amount of long range dependence (a.k.a. "momentum effects"). They are sometimes used in mathematical finance as models for logarithmic asset prices. We describe some natural random simple walks on the integers that have fractional Brownian motion as a scaling limit. In a sense, these walks are the most natural discrete analogs of fractional Brownian motion.

We present a novel linear program for the approximation of the dynamic programming value function in high-dimensional stochastic control problems. LP approaches to approximate DP naturally restrict attention to approximations that, depending on the context, are upper or lower bounds to the optimal value function. Our program -- the `smoothed LP' -- relaxes this restriction in an appropriate fashion while remaining computationally tractable. Doing so appears to have several advantages: o We demonstrate superior bounds on the quality of approximation…

Resolution of a data acquisition system is usually thought to be determined completely by sensing properties, such as density and accuracy of measurements. This talk advocates the view that resolution is, in addition, dependent on signal modeling and the complexity of computations allowed. This view is developed concretely for the acquisition of sparse signals, where the asymptotic relationship between resolution and the number of measurements is studied for algorithms with various complexities. The sparse signal recovery problem corresponds perfectly to…

Preferential attachment random graphs evolve in time by sequentially adding vertices and edges randomly so that connections to vertices having high degree are favored. There has been an explosion of research on these models since Barabasi and Albert popularized them in 1999 to explain the so-called power law behavior observed in real networks such as the graph of the Internet where Web pages correspond to vertices and hyperlinks between them correspond to edges. In this talk we will show how…