EE378 Statistical Signal Processing
Lecture 3  04/11/2007
MeanSquare Ergodicity, WideSense Ergodicity, Martingales
Lecturer: Tsachy Weissman
Scribe: Sudeepto Chakraborty, Lei Zhao, Kamakshi S
1
MeanSquare Ergodicity
Definition 1.
A WSS process
{
X
(
n
)
}
(with E
[
X
(
n
)] =
μ
and
r
(
k
) =
Cov
(
X
(
n
+
k
)
, X
(
n
))
) is
1. Meansquare ergodic in the
1
st
moment if
lim
N
→∞
E
1
2
N
+ 1
N
n
=

N
X
(
n
)

μ
2
= 0
,
(1)
2. Meansquare ergodic in the
2
nd
moment if
∀
k,
lim
N
→∞
E
1
2
N
+ 1
N
n
=

N
(
X
(
n
)

μ
)(
X
(
n

k
)

μ
)

r
(
k
)
2
= 0
(2)
3. A WSS process is widesense ergodic if it is meansquare ergodic in the
1
st
and the
2
nd
moments.
Example 1
{
X
(
n
)
}
is an iid process with the following distribution:
X
(
n
)
∼
1
w.p.
1
/
2

1
w.p.
1
/
2
Verify that this process is widesense ergodic.
Example 2
{
X
(
n
)
}
is a special case of a DC process:
X
(
n
)
∼
1
∀
n w.p.
1
/
2

1
∀
n w.p.
1
/
2
E
1
2
N
+ 1
N
n
=

N
X
(
n
)

0
2
= 1
→
0
Therefore
X
(
n
) is not ergodic in 1
st
moment
⇒
not widesense ergodic. In reference to Theorem 9 in lecture
notes 2, this is an example of a mixture of ergodic processes (namely, a mixture of degenerate processes for
which all the components are equal to the same deterministic constant).
These rather extreme examples make the point that, if process “memory” is sufficiently short you will
have Mean Square Ergodicity. On the other hand, if the “memory” is too long, the process will not be mean
square ergodic. The rate of decay of the covariance sequence can be thought of as a measure of memory.
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 Spring '07
 Weissman,I
 Normal Distribution, Signal Processing, Probability theory, martingales, meansquare ergodicity, covariance sequence

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