Persistence exponents of Gaussian stationary functions

Persistence exponents of Gaussian stationary functions

Naomi Feldheim, Bar-Ilan University
Fine Hall 214

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A Gaussian stationary function is a stationary process f, indexed on either R or Z, whose finite marginals are mean zero multi-normals. The persistence probability of such a process, denoted by P^x(T), is the probability that f remains above a fixed level x on an interval of length T. This quantity has been extensively studied by Slepian, Rice, Rosenblatt, Majumdar, Schehr, Dembo and others for the last 60 years. 

When the correlations decay fast, it is natural to expect that f would demonstrate an i.i.d. like behavior so that P^x(T) would be roughly exponential. In recent years, a spectral point of view on the problem allowed us to show that indeed, the value of log P^x(T)/T as T tends to infinity is bounded between two constants, for a much broader family of processes, some of which have non-decaying correlations. However, in general, the convergence of log P^x(T)/T to a persistence exponent remained open.

In this talk, I will present our very recent work, in which we establish a nearly exact condition for the existence of the persistence exponent. We show that such an exponent exists if the spectral measure of the process has a continuous  non-vanishing density at the origin. Our methods involve establishing several continuity properties of persistence probabilities, borrowing ideas from harmonic and convex analysis.

Joint work with Ohad Feldheim and Sumit Mukherjee.