On extremes and streams of upcrossing. (Q1888779)

Let \(\{\xi (t)\}_{t\in [0,h]}\) be a real stationary process. Assume that the process is separable and continuous in probability. Define \(M(h)=\sup _{t\in [0,h]} \xi (t)\), \(J(q,u)=q^{-1} P\{\xi (0) >u \geq \xi (q)\}\), \(J(u)=\underline {\lim }_{n\to \infty } J(2^{-n}, u)\). The author studies relations between \(P\{M(h)>u\}\) and \(P\{\xi (0)>u\}+h J(2^n,u)\). It is proved that under some general conditions the inequality \(P\{M(h)>u\} \leq P\{\xi (0)>u\}+h J(u)\) holds. Applications of results include argmax process of Brownian motion minus parabolic drift, \(P\)-differentiable \(\alpha \)-stable processes, Markov jump processes, differentiable \(n\)-dimensional Gaussian processes, and moving \(L^2\)-norms of differentiable Gaussian processes.