Stat 1301B
Probability & Statistics I
Fall 2009-2010
Chapter IV
Limit Theorems
The probabilistic behaviour of the sample mean when the sample size
n
is large
(say, tends to infinity) is called the
limiting distribution
of the sample mean.
Law
of large number
(
LLN
) and the
central limit theorem
(
CLT
) are two of the most
important theorems in statistics concerning the limiting distribution of the sample
mean. These two theorems suggest the “nice” properties of the sample mean and
justify its advantages.
Before proceeding, we need to define what ‘convergence’ means in the context of
random variables.
§ 4.1
Modes of Convergence
Let
be a sequence of random variables (not necessarily independent),
X
be another random variable. Let
,...
,
2
1
X
X
( )
x
F
n
X
( )
x
F
X
be the distribution function of
,
n
X
be the distribution function of
X
.
Converges in Distribution / Converges in Law / Weak Convergence
n
X
is said to converge in distribution to
X
if
( )
( )
x
F
x
F
X
X
n
n
=
∞
→
lim
for all points
x
at which
is continuous. It is denoted as
.
( )
x
F
X
X
X
L
n
⎯→
⎯
Example 4.1
Let
. Define
as the maximum of
. Then the
distribution function of
is given by
(
1
,
0
~
,...
,
2
1
U
U
U
iid
)
n
X
n
U
U
U
,...,
,
2
1
n
X
( )
0
=
x
F
n
X
for
0
≤
x
;
( )
1
=
x
F
n
X
for
1
≥
x
;
( )
(
)
(
)
x
U
x
U
x
U
P
x
X
P
x
F
n
n
X
n
≤
≤
≤
=
≤
=
,...,
,
2
1
(
) (
)
(
)
x
U
P
x
U
P
x
U
P
n
≤
≤
≤
=
L
2
1
n
x
=
,
for
1
0
<
<
x
.
P.152

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