Download convergence of random variables

4. Convergence of
random variables
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

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Convergence in probability
Convergence in distribution
Convergence in quadratic mean
Properties
The law of large numbers
The central limit theorem
Delta method
1
Convergence in probability
Xn 
 X 
P
   0 P X n  X     0
CONVERGENCE OF RANDOM VARIABLES
2
Convergence in distribution
Xn 
 X  FX n ( x)  FX ( x)
d
at any continuity point of
FX
CONVERGENCE OF RANDOM VARIABLES
3
Convergence in quadratic mean
X n  X 
qm
E( X n  X )  0
2
CONVERGENCE OF RANDOM VARIABLES
4
Properties
qm
P
d
X


X

X


X

X


X
(i)
n
n
n
d
P

c  Xn 

c
(ii) X n 
{ X n : n  }, {Yn : n  }; X , Y . Then:
(i)
P
P
P
X n 
X , Yn 
Y  X n  Yn 
X Y
qm
qm
qm
(ii) X n  X , Yn  Y  X n  Yn  X  Y
d
d
d
(iii) X n 

X , Yn 

c  X n  Yn 

X c
CONVERGENCE OF RANDOM VARIABLES
5
Properties
(iv)
(v)
P
P
P
X n 
 X , Yn 
 Y  X n  Yn 
 XY
d
d
d
Xn 

X , Yn 

c  X n ·Yn 

c X
Let g(•) be a continuous function. Then:
(i)
P
P
X n 
 X  g ( X n ) 
 g(X )
d
d
X


X

g
(
X
)


g( X )
(ii)
n
n
CONVERGENCE OF RANDOM VARIABLES
6
The law of large numbers
X with EX   ; X 1 ,..., X n ,... i. i. d. sample.
Let X n  1n
X
i
. Then:
X n 

P
CONVERGENCE OF RANDOM VARIABLES
7
The central limit theorem
X with EX   ; VX   2 ; X 1 ,..., X n ,... i. i. d.
sample.
Let X n 
n
1
n
X
i 1
i
. Then:
Xn  

n
2

 N (0,1)
d
CONVERGENCE OF RANDOM VARIABLES
8
The central limit theorem
Remark:
n( X n  ) d

 N (0,1)

d
n( X n  ) 

N (0,  2 )
( X n  ) 
 N (0, )
d
2
n
X n  N ( , )
2
n
(good for n 30)
CONVERGENCE OF RANDOM VARIABLES
9
Delta method
Suppose that CLT holds:
n ( X n  ) 
 N (0, )
d
2
g(•) differentiable function. Then:
d
n ( g ( X n )  g ( )) 

g ' ( ) N (0,  2 )
i. e.,
g ( X n )  N ( g (  ), g ' (  )
2 2
n
)
CONVERGENCE OF RANDOM VARIABLES
10