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On joint distribution of supremum, infimum and values of semi-continuous process with independent increments. (Ukrainian, English) Zbl 1075.60093

Teor. Jmovirn. Mat. Stat. 70, 54-62 (2004); translation in Theory Probab. Math. Stat. 70, 61-70 (2005).
A homogeneous lower semi-continuous process \(\xi(t)\) with independent increments is considered. The Laplace transform for the probability \(\Pr\{-y\leq\inf_{u<t}\xi(u),\xi(t)\in[\alpha,\beta)\), \(\sup_{u\leq t}\xi(u)\leq x\}\) is derived. It is shown that if \(E\xi(1)=0\), \(E\xi^2(1)=\sigma^2<\infty\) and \(x=x'B\), then, as \(B\to\infty\), this probability tends to the analogous probability for a Wiener process. This result is applied to derive the asymptotic distribution of \(\chi=\inf\{t>0:\;\xi(t)\not\in(-yB,xB)\}\).

MSC:

60J25 Continuous-time Markov processes on general state spaces
60J75 Jump processes (MSC2010)
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