On oscillating schemes of a random walk. I. (Russian) Zbl 0669.60066

Gusak, D. V.

Teor. Veroyatn. Mat. Stat. 39, 33-39 (1988).

A modification of oscillating random walks is defined by two Poisson processes with cumulants \((r=1,2)\), \[ \psi_ r(\alpha)=i\alpha a_ r+\int^{\infty}_{-\infty}(e^{i\alpha x}-1)d\Pi_ r(x),\quad a_ 1\geq 0,\quad a_ 2\leq 0, \]
\[ d\Pi_ 1(x)=K_ 1e^{\lambda_ 1x}dx\quad (x<0),\quad d\Pi_ 2(x)=K_ 2e^{-\lambda_ 2x}dx\quad (x>0). \] A relation for the integral transformation of its distribution is established. Under some restrictions the characteristic function of the ergodic (limit) distribution for the walk is obtained.

Reviewer: D.V.Gusak

MSC:

60G50	Sums of independent random variables; random walks
60G99	Stochastic processes
60E99	Distribution theory

Keywords:

factorization identities; random walks; Poisson processes; characteristic function