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STAT 333
Random Walk Summary
The random walk is a Markov chain taking values in the integers
S
=
{
...,

2
,

1
,
0
,
1
,
2
,...
}
.
We assume that the walk starts at the origin 0 (that is,
X
0
= 0) although we may relabel the
axis to make any chosen point the origin. On each step the process either jumps one unit to
the right or one unit to the left. Therefore, we either have
X
n
+1
=
X
n
+1or
X
n
+1
=
X
n

1
where
X
n
= position of the walk after
n
steps. The jumps themselves (+ or ) can be viewed
as a sequence of Bernoulli trials (or coin tosses) where
p
is the probability of a jump to the
right (+) and
q
=1

p
is the probability of a jump to the left (). We assume that 0
<p<
1.
•
The event
λ
00
= “return to 0” is a renewal event of period
d
=2.
•
Let
T
= number of steps to Frst return to 0. Then the probability of at least one
return to 0 is
f
=
P
(
T
<
∞
)=1

p

q

.
Therefore
λ
is
transient
if
p
±
/
2 and
recurrent
if
p
/
2.
We further calculate
E
(
T
)=
∞
when
p
/
2 so “return to 0” is
null recurrent
for
the balanced walk.
•
Let
V
= number of returns to 0. Then
E
(
V
f
1

f
=
1
p

q


p

q

•
Let
k
be a positive integer and let
T
0
k
= number of steps to Frst visit
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 Spring '08
 Chisholm

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