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Theory of Probability and Mathematical Statistics



On a feature of distributions of the overshoot functionals for upper semicontinuous processes on a Markov chain

D. V. Husak, Ie. V. Karnaukh

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Abstract: We examine the question whether the memoryless property associated with the geometric and exponential distributions remains true for the overshoot functionals of almost upper semicontinuous integer- and real-valued processes on a Markov chain. It is established that under the condition of a level attainability and knowing the environment state at the moment of reaching the level this property only holds for overshoot through the level x≥0 and its distribution depends neither on the overshoot moment nor on x. The undershoot distribution of a level x is determined in terms of the zero-level undershoot distribution. A similar dependence is established for a jump that covers the level x.

Keywords: Almost semicontinuous processes on a Markov chain, memoryless property, functionals of a positive level achievement, infinitely divisible factorization.

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