The most common approach to estimating conditional quantile curves is to fit a curve, typically linear, pointwise for each quantile. Linear functional forms, coupled with pointwise fitting, are used for a number of reasons including parsimony of the resulting approximations and good computational properties. The resulting fits, however, may not respect a logical monotonicity requirement -- that the quantile curve be increasing as a function of probability. This paper studies the natural monotonization of these empirical curves induced by sampling…

One of Shannon's earliest results was his determination of the capacity of the binary symmetric channel (BSC). Shannon went on to show that, with randomly chosen codes and optimal decoding, the probability of decoding error decreases exponentially for any transmission rate less than capacity. Much of the important early work of Shannon, Elias, Fano and Gallager was devoted to determining bounds on the corresponding "error exponent." A later approach to this problem, pioneered by Csiszar and Korner, and now adopted…

Many performance engineering and operations research modeling formulations lead to Markov models in which the key performance measure is an expectation defined in terms of the stationary distribution of the process. In models of realistic complexity, it is often difficult to compute such expectations in closed form. In this talk, we will discuss a simple class of bounds for such stationary expectations, and describe some of the mathematical subtleties that arise in making rigorous such bounds. We will also discuss…

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.