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For a semicontinuous homogeneous process ξ(t) with independent increments the distribution of the its total duration of stay in an interval is obtained. In the case E ξ(1) = 0, E ξ(1) 2 < ∞, the limit theorem on a weak convergence of the time of duration of stay in an interval of the process to distribution of the time of duration of stay of Wiener process in the interval(0, 1) is obtained. ForWiener process the distribution of the total duration of stay in an interval is found.

For a process ξ(t) with independent increments an integral transformations for the joint distribution of the moment of the first exittime of process from an interval and the value of overjump of process through border at the moment of this exittime are obtained. 1.Let ξ(t) ∈ R, t ≥ 0, ξ(0) = 0 be homogeneous process with independent increments [1], with the cumulantand continuous from the right sample trajectories. Note that this process is strictly Markov process.([2]) Let B > 0 be fixed, x ∈ (0, B), y = B − x, and introduce random variableis the moment of the first exittime from the interval (−y, x) of the process ξ(t), t ≥ 0. The moment of the first exittime of this process from an interval is a Markov moment.which are a value of exittime of process through upper bound at the moment of the first exittime of the process from an interval through upper bound and a value of overjump of process through lower bound at the moment of the first exittime of the process from an interval through lower bound. The goal of this paper is the finding of the integral transformations1 Translated by A. I. Vladimirova

A two-sided exit problem is solved for a difference of a compound Poisson process and a compound renewal process. The Laplace transforms of the joint distribution of the first exit time, the value of the overshoot, and the value of a linear component at this instant are determined. The results obtained are applied to solve the two-sided exit problem for a particular case of this process, namely, the difference of the compound Poisson process and the renewal process whose jumps are geometrically distributed. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process.

519.21We solve main two-boundary problems for a random walk. The generating function of the joint distribution of the first exit time of a random walk from an interval and the value of the overshoot of the random walk over the boundary at exit time is determined. We also determine the generating function of the joint distribution of the first entrance time of a random walk to an interval and the value of the random walk at this time. The distributions of the supremum, infimum, and value of a random walk and the number of upward and downward crossings of an interval by a random walk are determined on a geometrically distributed time interval. We give examples of application of obtained results to a random walk with one-sided exponentially distributed jumps.