On Weak Limiting Distributions for Random Walks on a Spider

Youngsoo Seol

doi:10.3390/sym12122000

Abstract

In this article, we study random walks on a spider that can be established from the classical case of simple symmetric random walks. The primary purpose of this article is to establish a functional central limit theorem for random walks on a spider and to define Brownian spider as the resulting weak limit. In special case, random walks on a spider can be characterized as skew random walks. It is well known for skew Brownian motion as the resulting weak limit of skew random walks. We first will study the tightness and then it will be shown for the convergence of finite dimensional distribution for random walks on a spider.

Highlights

Introduction and the Statement of theMain ResultsItô and Mckean [1] first proposed an elementary but interesting stochastic process which is named for Skew Brownian motion to provide a construction of some random processes related to Feller’s classification of differential operators of the second order associated with diffusion processes
We first will study the tightness and it will be shown for the convergence of finite dimensional distribution for random walks on a spider
The following theorem states the convergence of finite dimensional distribution for random walks on a spider

Summary

Introduction

Introduction and the Statement of theMain ResultsItô and Mckean [1] first proposed an elementary but interesting stochastic process which is named for Skew Brownian motion to provide a construction of some random processes related to Feller’s classification of differential operators of the second order associated with diffusion processes (see Section 4.2 in [2]). From Evans and Sowers [5] and Barlow et al [6], we propose a kind of Walsh’s Brownian motion that occupies on N partial axes on the plane which is the so-called Walsh’s spider, or Brownian spider. This motion behaves as a standard Brownian motion over each of the legs. Someone can make the Brownian spider by independently setting the excursions from zero of a usual Brownian motion over the η-th leg of the spider with probability pη , η = 1, .

Objectives

Results

Conclusion