Limit distribution for a semi-Markovian random walk with Weibull distributed interference of chance

Tülay Kesemen,Rovshan Aliyev,Tahir Khaniyev

doi:10.1186/1029-242x-2013-134

Abstract

In this paper, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered. In this study, it is assumed that the sequence of random variables {ζn}, n =1 , 2, ... , which describes the discrete interference of chance, forms an ergodic Markov chain with the Weibull stationary distribution. Under this assumption, the ergodic theorem for the process X(t) is discussed. Then the weak convergence theorem is proved for the ergodic distribution of the process X(t) and the limit form of the ergodic distribution is derived. MSC: 60G50; 60K15; 60F99

Highlights

Many interesting problems of stochastic finance, mathematical biology, reliability, queuing, stochastic inventory and mathematical insurance can be expressed by means of random walk processes
Unlike Aliyev et al [ – ] and Khaniyev et al [, ], we assume that the random variables ζn, n =, . . . , which describe the discrete interference of chance, are independent and identically distributed random variables with the Weibull distribution, and the weak convergence theorem is proved for the ergodic distribution of a semi-Markovian random walk process, and the limit distribution is derived for the ergodic distribution of the considered process
The main purpose of this study is to prove the weak convergence theorem for the ergodic distribution of the process X(t), as λ →

Summary

Introduction

Many interesting problems of stochastic finance, mathematical biology, reliability, queuing, stochastic inventory and mathematical insurance can be expressed by means of random walk processes. Which describe the discrete interference of chance, have exponential, gamma and triangular distribution, respectively, and stationary moments of the ergodic distribution of a semi-Markovian random walk process have been investigated. Which describe the discrete interference of chance, are independent and identically distributed random variables with the Weibull distribution, and the weak convergence theorem is proved for the ergodic distribution of a semi-Markovian random walk process, and the limit distribution is derived for the ergodic distribution of the considered process. This process might be useful in the following situation

Objectives

Conclusion