Leadership exponent in the pursuit problem for 1-D random particles

Abstract

For n + 1 particles moving independently on a straight line, we study the question of how long the leading position of one of them can last. Our focus is the asymptotics of the probability p(T,n) that the leader time will exceed T when n and T are large. It is assumed that the dynamics of particles are described by independent, either stationary or self-similar, Gaussian processes, not necessarily identically distributed. Roughly, the result for particles with stationary dynamics of unit variance is as follows: L= -log p(T,n) /(Tlog n)=1/d+o(1), where d/(2pi) is the power of the zero frequency in the spectrum of the leading particle, and this value is the largest in the spectrum. Previously, in some particular models, the asymptotics of L was understood as a sequential limit first over T and then over n. For processes that do not necessarily have non-negative correlations, the limit over T may not exist. To overcome this difficulty, the growing parameters T and n are considered in the domain clog T 1 . The Lamperti transform allows us to transfer the described result to self-similar processes with the normalizer of log p(T,n) becoming log T log n.