Limit functionals for superposition of random walk and sequence of independent variables. (Ukrainian, English) Zbl 1164.60396

Ezhov, I. I; Kadankov, V. F.

Teor. Jmovirn. Mat. Stat. 75, 8-20 (2006); translation in Theory Probab. Math. Stat. 75, 9-22 (2007).

Let \(\{\xi_{n}\},\{\eta_{n}\}\in Z,n\in\mathbb N\) be a sequences of i.i.d. integer-valued random variables; \(\xi_{i}\) and \(\eta_{j}\) are independent for all \(i,j\in\mathbb N\). Let us define \(\xi(0)=0, \xi(n)= \xi_1+\dots+\xi_{n}\), \(\eta_{l}(0)=l\), \(\eta_{l}(n)=\eta_{n}\), \(n\in\mathbb N, l\in\mathbb Z\). A random sequence \(X_{l}(n)\in\mathbb Z\) defined by \(X_{l}(0)=l\), \[ X_{l}(n)=\begin{cases} X_{l}(n-1)+\xi_{n}& \text{with probability } \lambda,\\ \eta_{n}&\text{with probability } 1-\lambda,\end{cases} \] is called the superposition of a random walk \(\xi(n)\) and a sequence of independent variables \(\eta_{l}(n)\). The authors derive the generating function for the joint distribution of the moment of first crossing of a level and the magnitude of a jump over this level at this moment. Also, the generating function for the joint distribution of the moment of first exit from interval and the value of the sequence at this moment are derived.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60J10

Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

generating functions; superposition; random walk; sequence of independent variables; moment of first crossing of a level; jump magnitude