W. H. LI ET AL.

Copyright © 2012 SciRes. OJDM

172

1

0

1

lim inf,

0,a s

n

kk

nk

r

Xm

nt

tt

(43)

Proof. Here, we only proof (42) and (43) is omitted.

The moment generating function of binomial distri-

bution is defined by. Using inequa-

lity and

em

t

k

Mt qp

t

ln1 0tt t

ee 1tt

, we have

ln Mt

lnee 1

k

tt

m

tt

mq

pmp x

x

ee

k

t

x

mp t

mp xmpmptxmp

tt

Hence (44) follows directly from (46).

Corollary 3. Under the conditions of the Theorem

and Theorem 2, if are independen

identically distribution andom variables with

parameters

mp

tt

(44)

mp mp

1

1

n

Xn

of Poisson rt and

, then

1

lim supkk

nk

Xm

n

0

1n

1

e

01

as

rm

pt

mp t

tt

(45)

1

0

1

lim inf

0,a s

n

kk

nk

r

Xm

nt

tt

(46)

Proof. The moment generating function of distribution

of Poisson is defined by Using ine-

quality we have

e1

e

t

k

Mt

,

ee1

ttt,

2

e1

ln

e

t

kk

Mt

x

m

tttt

t

(45) follows directly from (47). At the sa

time, by using inequality we have

(47)

Hence me

e10

ttt,

2

e1

ln t

kk

Mt

xx

tt t

m

tt

.

3. Acknowledgements

The authors would like to thank the National Science

41172299) and the Funding

ES

[1] W. Liu, “Relative Entropy Densities and a Class of Limit

Theorems of Sandom Variables,”

Annals of Prob990, pp. 829-839.

Funding of China (Grant No.

Project Education Department Science Project of Hebei

Province (Grant No. Z2010297).

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