Transcript Chapter 4 Limit Theorem

Chapter 4 Limit Theorem
§ 4.1 Law of large number
1. Convergence in probability
Suppose that {Xn} is a sequence of r.v.s, if
for any >0, we have
lim P{| X n  X | }  1
n 
it is said that {Xn} convergence to X in
probability and denoted it by
P
Xn 

X.
P
Remark X n  a Means when
n 
the probability that the value of Xn
fall in interval (a   , a   ) is increased to 1
Xn
a 
X n  a means   0, n0
a
when
a
n  n0
| X n  a | 
2. Law of Large Numbers (LLN)
1. Chebyshev’s LLN
Suppose that {Xk,k=1,2,...} are i.i.d r.v.s with
mean  and variance 2>0，then
1 n
P
Yn   X k 


n k 1
i.e. for any give >0, we have
lim P {| Yn   |  }  1
n 
Proof Chebyshev’s inequality, we have
D(Yn )
P{| Yn  E (Yn ) |  }  1 
.
2

where
thus
1 n
E (Yn )   E ( X k )  
n k 1
1
D(Yn )  2
n
n
 D( X
k 1
k
)
2
n
2
P{| Yn   |  }  1  2 .
n
lim P {| Yn   |  }  1
n 
2.Bernoulli’s LLN
Set f n records the numbers of outcomes of A in Bernoulli
we have
n experiment, P( A)  p  0 , then for any ，   0


lim P  n  p     1
 
 n

3. Khinchine’s LLN
Suppose that {Xk,k=1.2,...} are i.i.d sequence with EXk= <, k=1,
2, … then
1 n

lim P   X k  a     1
n
 n k 1

Remark Suppose that {Xi,i=1.2,...} are i.i.d. r.v.s
with E(X1k) <=, then
1 n k P
k
X


E
(
X

i
1 )
n i 1
This remark is very important for moment estimation
for parameters to be discussed in Chapter 6.
§ 4.2. Central Limit Theorems
1. Convergence in distribution
Suppose that {Xn} are i.i.d. r.v.s with d.f. Fn(x), X is a
r.v. with F(x), if for all continuous points of F(x) we have
lim Fn ( x )  F( x ),
n 
It is said that {Xn} convergence to X in distribution
and denoted it by
w
Xn 

X.
n
Set Yn  X k , denoted the standardized Yn by Yn* ,
k 1
w
then Yn* 
  ~ N (0, 1),
2. Central Limit Theorems (CLT)
Levy-Lindeberg’s CLT
Suppose that {Xn} are i.i.d. r.v.s wIth mean
< and variance 2 <，k=1, 2, …, then {Xn}
follows the CLT, which also means that
x  n
p{ X i  x}  (
)
n
i 1
n
De Moivre-Laplace’s CLT
Suppose that n(n=1, 2, ...) follow binomial distribution with
parameters n, p(0<p<1), then
n  np w
  ~ N (0, 1).
npq
Example 2 A life risk company has received 10000 policies, assume each
policy with premium 12 dollars and mortality rate 0.6%，the company
has to paid 1000 dollars when a claim arrived, try to determine：
(1) the probability that the company could be deficit？
(2)to make sure that the profit of the company is not less than 60000
dollars with probability 0.9, try to determine the most payment of each
claim.
Let X denote the death of one year, then, X~B(n, p),
where n= 10000，p=0.6%，Let Y represent the profit of the
company, then, Y=1000012-1000X. By CLT, we have
(1)P{Y<0}=P{1000012-1000X<0}=1P{X120}
1  (7.75)=0.
(2) Assume that the payment is a dollars, then
P{Y>60000}=P{1000012-X>60000}=P{X60000/a}0.9.
By CLT, it is equal to
60000
 10000  0.006
( a
)  0.9
10000  0.006  0.994
 a  3017
Convergence in probability
Convergence in distribution
Chebyshev’s LLN
Levy-Lindeberg
CLT
Bernoulli LLN
De Moivre-Laplace’s CLT
Khinchine’s LLN