Sojourns Extremes And Self Intersections Of Stochastic Processes

The subjects of this research are certain probabilistic properties of real and vector valued random functions X(t), where t is a generalized time parameter taking values in a subset T of Euclidean space. The distributions of three functionals of the random function X are studied. For any Borel set A in the range, the sojourn time of X in A is defined as the measure of the subset of the points t in T such that X(t) belongs to A. The self-intersection set of X is the subset of the set of points (s, t) in the product space of T such that s = t and X(s) = X(t). Finally, for a real valued random function X, the extreme value is the functional equal to the maximum value of X on the domain T. The research is concerned with the determination of the distributions of these functionals under various hypotheses about the probabilistic structure of X. It is assumed here that the random function is Gaussian or Markovian. Stochastic process, Extreme value, Sojourn time, Local time, Limiting distribution, Self- intersections of paths, Gaussian process, Markov process.