Weak convergence of the integrated number of level crossings to the local time for Wiener processes

Abstract

Ключевые слова и фразы: винеровский процесс, локальное вре мя, пересечения уровня, приращения.1. Introduction. Let X t = {X(t,cj), t £ [0,1], w e ft} be a standard Wiener process. For each t and e > 0 define A e (i) = e~1^2(X t+£ -X t ), the normalized increments of the process. If we fix a trajectory and consider A e (tf) as a random variable (r.v.) on t with Lebesgue's measure A then, as M. Wschebor showed [10], for almost every trajectory, this variable converges in distribution, as e goes to zero, to a Gaussian distribution: