Lecture 12 - 3/12/2010
1
DISCRETE
STOCHASTIC
PROCESSES
Lecture 12
Renewal Processes - Chapter 3
Review:
Stopping Times
Wald's Theorem
The Stopping Time
(N(t) +1)
A Lower Bound on E[N(t)]
The Renewal Equation for N(t)
An Approach to the Renewal Equation Using Laplace Transforms
The Elementary Renewal Theorem
Blackwell's Theorem
Using Renewal Processes to Think about Finite Markov Chains

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
Lecture 12 - 3/12/2010
2
Stopping Times
Stopping time
J
is random, and can depend on observations
up to and including
X
N
.
If experiment has proceeded to
X
n
, the
decision to stop immediately following
n
(i.e., to choose J = n) should be independent of
X
n
+
1
,
X
n
+
2
,...
.
Therefore, for any fixed t, J =N(t) is
not
a valid stopping time, since N(t) = n means that
X
1
+ X
2
+
- - - + X
n
≤
t and X
1
+ X
2
+
- - - + X
n
+ X
n+1
> t, which depends on X
n+1.
But , for any fixed t, J =N(t) + 1
is
a valid stopping time, since N(t) + 1 = n means that
N(t) = n -1, and therefore X
1
+ X
2
+
- - - + X
n-1
≤
t and X
1
+ X
2
+
- - -
+ X
n
> t, which depends
only on X
1,
X
2,
- - - , X
n.