想像一個動點在時間 t = 0 從 x-軸的原點開始移動，每增加 1 個單位時間就往右或往左移動一整數單位，其規則如下：動點在每個整數點往右移動的機率為 p，往左移動的機率為 q = 1 − p。對於 p = q = 1/2 的模型我們通常稱它為簡單對稱隨機漫步，又對於 p ≠ q 的模型我們通常稱它為簡單非對稱隨機漫步。
對於一個隨機漫步模型，我們定義覆蓋時間(cover time) 為該動點經過n 個不同的點所需時間。在這篇碩士論文中我們首先研究由 0 出發且在整數點上移動的隨機漫步之覆蓋時間，然後我們研究由 0 出發且具有一個反彈點的隨機漫步並探討其所對應的覆蓋時間的一些性質。

Imagine that a particle starts from the origin of the x-axis and moves at times t = 0, 1, . . . one step to the right with probability p or one step to the left with probability q = 1 − p. If p = q = 1/2, it is usually called a simple symmetric random walk and if p ≠ q, it is usually called a simple asymmetric random walk.
For a simple random walk and a given positive integer n, define the cover time to be the time when the number of points visited has just increased to the given number n. In this thesis, we first review the cover time of a simple random walk on Z starting from 0. Then we study the cover time in a simple random walk starting from 0 with a reflection barrier.

誌謝i
中文摘要ii
Abstract iii
1 Introduction 1
2 Random Walk on Z 2
3 Random Walk on a Half Line 6
4 Random Walk on {−a, −a + 1, . . .} 10
Appendix 19
Reference 19

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