V(c„Aj/n) = anw2yn, n= l , . . . , m - 1,
which are equivalent to (1.1.6) and (1.1.7), respectively.
1.3 Random Walk With Discrete Time Process
At time t = 0, a particle is at one of m places 0 , 1 , . . . , m — 1. At successive
instants t = 1,2,..., it can move one place to the right or to the left or can
remain fixed. Suppose that the particle is in position n at some t = to. There is
a probability Q„ that it will be in position n + 1 at o -f 1. There is a probability
/?„ that it will be in position n — 1 at f o +1, and therefore a probability 1 - a
that it will remain in position n.
At the endpoints, the particle is considered lost if it moves to the left of 0 or
to the right of m — 1.
ffo *0 Pn Xnv ftm-1 0 * m - l
« 1 1
0 n m - 1
Figure 1.3: Random Walk
If a particle starting initially at r is at s when = n + l, then, when t = n, it
must have been at s or s — 1 or s -f 1 with respective probabilities 1 — as — fi9,
a,_i, /?,+i of moving then from these positions to s. Hence, if prs(n) represents
the probability of the particle being in position s at time n, starting in position
r at t = 0, then