Leadership Exponent in the Pursuit Problem for 1-D Random Particles

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 exceedTwhennandTare 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:=-ln p(T,n)/(Tlnn)=1/d(0)+o(1), whered(0)/(2 pi) 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 ofLwas understood as a sequential limit first overTand then overn. For processes that do not necessarily have non-negative covariances, the limit overTmay not exist. To overcome this difficulty, the growing parametersTandnare considered in the domain c lnT<n <= CT where c > 1. The Lamperti transform allows us to transfer the described result to self-similar processes by changing the ln pT,n normalization to the value lnT ln n.