文档介绍：C H A P T E R 5 L ARG E— S AM PLE PRO PERTI ES OF THE LSE 1
C h a p t e r 5 L arg e— s am pl e pro perti es of the LS E
5 . 1 S t o c h a s ti c con v e r g ence
S u p po s e t h a t { X n } i s a seq uen c e of r and om v ariab l es w ith a corresponding sequence of
distribution functions { F n } .
I f F n ( x ) → F ( x ) at every continuity point x of F, F n is said to converge weak ly to
F, written F n ⇒ F. In this case, { X n } is said to converge in distribution to X where X
d
is a random variable with distribution function F, written X n → X.
If X is a random variable, and for all ε> 0
l i m P ( | X n − X | < ε) = 1 ,
n →∞
P
X n is said to converge in probability to X, written X n → X. X is known as the probability
limit of X n , written X = plim X n .
If
2
lim E ( X n − X ) = 0 ,
m . s .
X n is said to converge in mean square to X, written X n → X.
Some useful results regarding stochastic convergence are:
P
1. X n → X and g (
) is a continuous function
P
⇒ g ( X n ) → g ( X ) .
E x a m p l e 1 L e t
1 w i th p r o b a bil ity 1
X = n .
n 0 with probability 1 − 1
n
P P
O bv iou s ly, X n → 0 . Let g ( x ) = x + 1 . T hen , g ( X n ) → g (0) = 1 .
d P
2 . Suppose that Y n → Y and X n → c ( a constant) . T hen
d
(a) X n + Y n → c + Y
d
(b) X n Y n → cY
d
(c) Y n → Y when c
= 0 .
X n c
d
3 . X n → X and g (
) is continuous
d
⇒ g ( X n ) → g ( X ) .
(This is called continuous mapping theorem)
C H A P T E R 5 L ARG E— S AM PLE PRO PERTI ES OF THE LSE 2
d 2 d 2
E x a m p l e 2 I f X n → N ( 0 , 1 ) , X n →χ(1) .
P d
4 . X n − Y n → 0 a n d X n → X.
d
⇒ Y n → X.
P d
5 . X n → X i m p l ie s X n → X.
( T h e c o nv er se is not necessarily tru e.)
d
6 . X n → c (a constant)
P
⇒ X n → c.
m . s .
7 . X n → X
P
⇒ X n → X.
I f for any ε> 0 , there ex ists B ε< ∞ such that

| X |
P n > B < ε
n r ε

for all n ≥ 1 , w rite X = O ( n r )