Miklos Schweitzer 1952_7
Source:
October 12, 2008
probabilitylimitprobability and stats
Problem Statement
A point is performing a random walk on the -axis. At the instant t\equal{}0, is at a point (, where and denote integers, ). If at an instant ( being a nonnegative integer), is at a point of integer abscissa and , then by the instant t\plus{}1 it reaches either the point x\plus{}1 or the point x\minus{}1, each with probability . If at the instant , is at the point x\equal{}N [ x\equal{}\minus{}N], then by the instant t\plus{}1 it is certain to reach the point N\minus{}1 [ \minus{}N\plus{}1]. Denote by the probability of being at x\equal{}k at instant ( is an integer). Find and \lim_{t\to \infty}P_k(2t\plus{}1) for every fixed .