Two-boundary problems for random walk with geometrically distributed negative jumps. (Ukrainian, English) Zbl 1050.60049

Kadankova, T. V.

Teor. Jmovirn. Mat. Stat. 68, 49-60 (2003); translation in Theory Probab. Math. Stat. 68, 55-66 (2004).

Let \(\alpha\in\{0,1,\ldots\}\) be a nonnegative integer-valued random variable, let \(\beta\in\{1,2,\ldots\}\) be a geometrically distributed random variable, and let \(\xi=\alpha-\beta\), \(\xi\in\{0,\pm1,\ldots\}\). This paper deals with the random walk \(\xi_0=0\), \(\xi_{n}=\xi_1+\ldots+\xi_{n}\), \(n>0\), where \(\{\xi,\xi_{i}\}\), \(i\geq0\), is a sequence of i.i.d. random variables. Let \(P\{\xi_0=k\}=1\), \(k\in\{1,\ldots,N-1\}\) and let \(\tau_{k}=\inf\{n>0: \xi_{n}\not\in\{1,\ldots,N-1\}\}\), \(\tau_{k}^{+}=\inf\{n>0: \xi_{n}\not\in\{1,\ldots,N-1\},\xi_{\tau_{k}}\geq N\}\), \(\tau_{k}^{-}=\inf\{n>0: \xi_{n}\not\in\{1,\ldots,N-1\}, \xi_{\tau_{k}}\leq0\}\), \(\mu_{n}^{+}=\sup\{\xi_0,\ldots,\xi_{n}\}\), \(\mu_{n}^{-}=\inf\{\xi_0,\ldots,\xi_{n}\}\). The author obtains distributions \(P_{k}^{-}=P\{\xi_{\tau_{k}}\leq0\}\), \(P_{k}^{+}=P\{\xi_{\tau_{k}}\geq N\}\) and \(M[\tau_{k}]\). The joint distribution of \(\mu_{n}^{+}\), \(\mu _{n}^{-}\) and \(\xi_{n}\) is derived and the transition probability and ergodic distribution of the stochastic process, describing the evolution of random walk with two barriers, are obtained.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60G50

Sums of independent random variables; random walks

Keywords:

two-boundary problem; random walk; geometric distribution; negative jumps; transition probability; ergodic distribution