Summer 2007
STAT333
Summary of the Random walk
Random walk is a simple stochastic process, which can be regarded as an example for the
renewal process, and can also be regarded as an example for the discrete Markov process.
The following are some formulas you need to memorize. You should understand them before
memorizing them.
♣
DeFnition of random walk process
(a) We assume that the walk starts at the origin 0 (that is,
X
0
= 0).
(b) On each step the process either jumps one unit to the right or one unit to the
left. Therefore, we have
X
n
+1
=
X
n
+
I
n
+1
, where
I
n
=
b
1
if the process jumps to the right by one unit at step
n
+ 1
−
1
if the process jumps to the left by one unit at step
n
+ 1
.
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 ().
(c) Let
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 Fall '10
 MenZhongxian
 Probability, Probability theory, Stochastic process, Random walk

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