Local times and sample function properties of stationary Gaussian processes

1. Introduction.The theory of local times of a stochastic process was conceived in the work of Paul Levy on linear Brownian motion [9].H. Trotter proved the first major theorem for the Brownian model [11]; and much has been discovered by many other authors, too numerous to list here.A survey of the theory and a bibliography are contained in the monograph of Ito and McKean [8].Local times have apparently been studied and used for Markov processes only.In this paper, local times of another class of stochastic processes are examined-a class of stationary Gaussian processes.Extensions to other processes are also indicated.In their recent monograph, Cramer and Leadbetter [6] summarized the current research for stationary Gaussian processes and their sample functions; their book is mostly about processes whose correlation functions are twice differentiable at the origin.These processes have absolutely continuous sample functions.The present paper is about a very different class: the correlation function r(t) satisfies the following relation for i->0: t2(l -r(t))^-oo.A typical example is: l-r(t) is asymptotic to a constant multiple of \t\", 0<a<2.The sample functions, though continuous, are not only not absolutely continuous, but are nondifferentiable at almost every point; this can be inferred from the work of J. Yeh [12].It is shown that such processes have local times with continuous sample functions, and then the connection between local times and first passage times are revealed.These results are used to establish the following peculiar property of the Gaussian process: The values crossed by the sample function finitely many times in an interval form a set of category 1 in the image ofthat interval.A new inequality for the probability of first passage is incidentally obtained in §6.The well-known dichotomy theorem of Beljaev [1] states that the sample functions of a separable stochastically continuous, stationary Gaussian process are either almost all continuous, or else almost all unbounded; furthermore, a sufiicient condition for continuity is the boundedness of |log \t\ \a(l -r(t)), t -»• 0, for some constant a> 1.In §8, a result for the unbounded case is proven: If liminf |log|i| |(1 -r(t)) t-o