Weak convergence theorem for the ergodic distribution of a random walk with normal distributed interference of chance

Abstract

In this study, a semi-Markovian random walk process (X (t)) with a discrete interference of chance is investigated. Here, it is assumed that the zeta(n), n = 1; 2; 3, ..., which describe the discrete interference of chance are independent and identically distributed random variables having restricted normal distribution with parameters (a; sigma(2)). Under this assumption, the ergodicity of the process X (t) is proved. Moreover, the exact forms of the ergodic distribution and characteristic function are obtained. Then, weak convergence theorem for the ergodic distribution of the process W-a (t) = X (t) = a is proved under additional condition that sigma/a -> 0 when a -> infinity.

[Hanalioglu, Z.] Karabuk Univ, Dept Actuery & Risk Managment, TR-78050 Karabuk, Turkey; [Khaniyev, T.] TOBB Univ Econ & Technol, Dept Ind Engn, TR-06560 Ankara, Turkey; [Agakishiyev, I.] Azerbaijan Natl Acad Sci, Inst Cybernet, AZ-1141 Baku, Azerbaijan