Ofer Zeitouni

Random walks in random environments

Random walks in random environments and their diffusion analogues have been a source of surprising phenomena and challenging problems, especially in the non-reversible situation, since they began to be studied in the 1970s. We review the model, available results and techniques, and point out several gaps in the understanding of these processes.

When is the generalized likelihood ratio test optimal

The generalized likelihood ratio test (GLRT), which is commonly used in composite hypothesis testing problems, is investigated. Conditions for asymptotic optimality of the GLRT in the Neyman-Pearson sense are studied and discussed. First, a general necessary and sufficient condition is established, and then based on this, a sufficient condition, which is easier to verify, is derived. A counterexample where the GLRT is not optimal, is provided as well. A conjecture is stated concerning the optimality of the GLRT for the class of finite-state sources. >

Exponential stability for nonlinear filtering

Abstract We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoffs contraction coefficient. Criteria for exponential stability and explicit bounds on the rate are given in the specific cases of a diffusion process on a compact manifold, and discrete time Markov chains on both continuous and discrete-countable state spaces. A similar question regarding the optimal smoother is investigated and a stability criterion is provided.

Searching for a Trail of Evidence in a Maze

Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following hypothesis testing problem: under the null, the random variables have common distribution N(0,1) while under the alternative, there is an unknown path along which random variables have distribution N(�, 1), � > 0, and distribution N(0,1) away from it. For which values of the mean shiftcan one reliably detect and for which values is this impossible? This paper develops detection thresholds for two types of common graphs which exhibit a different behavior. The first is the usual regular lattice with vertices of the form {(i, j) : 0 ≤ i, −i ≤ j ≤ i and j has the parity of i} and oriented edges (i, j) → (i+1, j+s) where s = ±1. We show that for paths of length m start- ing at the origin, the hypotheses become distinguishable (in a minimax sense) ifm ≫ √ log m, while they are not ifm ≪ log m. We derive equivalent results in a Bayesian setting where one assumes that all paths are equally likely; there the asymptotic threshold ism ≈ m 1/4 . We obtain corresponding results for trees (where the threshold is of order 1 and independent of the size of the tree), for distributions other than the Gaussian, and for other graphs. The concept of predictability profile, first introduced by Benjamini, Pemantle and Peres, plays a crucial role in our analysis.

Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk

and conjectured that the limit exists and equals 1/Tr a.s. The importance of determining the value of this limit is clarified in (1.3) below, where this value appears in the power laws governing the local time of the walk. The Erd6s-Taylor conjecture was quoted in the book by R~v~sz [19, w but to the best of our knowledge, the bounds in (1.1) were not improved prior to the present paper. As it turns out, an important step towards our solution of the Erd6s-Taylor conjecture was the formulation by Perkins and Taylor [17] of an analogous problem on the maximal occupation measure that planar Brownian motion (run for unit time) can

Lyapunov Exponents for Finite State Nonlinear Filtering

Consider the Wonham optimal filtering problem for a finite state ergodic Markov process in both discrete and continuous time, and let

On roots of random polynomials

\sigma

On universal hypotheses testing via large deviations

be the noise intensity for the observation. We examine the sensitivity of the solution with respect to the filters initial conditions in terms of the gap between the first two Lyapunov exponents of the Zakai equation for the unnormalized conditional probability. This gap is studied in the limit as

Tail estimates for one-dimensional random walk in random environment

\sigma\to 0

Entropic repulsion of the lattice free field

by techniques involving considerations of nonlinear filtering and the stochastic Feynman--Kac formula. Conditions are given for the limit to be either negative or