Sojourns of Stationary Processes in Rare Sets

Let X(t), t ≥ 0, be a stationary process assuming values in a measure space B. The family of measurable subsets Aᵤ, u > 0 is called "rare" if P(X(0) ∈ Aᵤ) → 0 for u → ∞. Put Lₜ(u) = mes: 0 ≤ s ≤ t, X(s) ∈ Aᵤ\. Under specified conditions it is shown that there exists a function $v = v(u)$ and a nonincreasing function -Γ'(x) such that P(v(u)Lₜ(u) > x)/E(v(u)Lₜ(u)) → - Γ'(x), x > 0, for u → ∞ and fixed $t > 0$. If $u = u(t)$ varies appropriately with t, then, under suitable conditions, the random variable v(u)Lₜ(u) has, for t → ∞, a limiting distribution of the form of a compound Poisson distribution. The results are applied to Markov processes and Gaussian processes.