- Tài liệu text

Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (133.7 KB, 5 trang )

(1)<div class='page_container' data-page=1>

DOI: 10.22144/ctu.jen.2018.050

New bounds on poisson approximation for random sums of independent

negative-binomial randomvariables

Le Truong Giang

and Trinh Huu Nghiem

1University of Finance - Marketing, Vietnam 
2Nam Can Tho University, Vietnam

*Correspondence: Le Truong Giang (email: ) 
Article info. ABSTRACT

Received 11 Feb 2018 
Revised 04 Jun 2018 
Accepted 30 Nov 2018

The aim of this paper is to establish new bounds on Poisson

approxima-tion for random sums of independent negative-binomial random 
varia-bles. The bounds showed in current paper are a uniform bound and a 
non-uniform bound. The received results in this paper are extensions and 
generalizations of known results.

Keywords

Negative - binomial variable, 
Poisson approximation, 
ran-dom sums, total variation 
distance, uniform and 
non-uniform bound

Cited as: Giang, L.T. and Nghiem, T.H., 2018. New bounds on poisson approximation for random sums of
independent negative-binomial randomvariables. Can Tho University Journal of Science. 54(8):
149-153.

1 INTRODUCTION

In recent times, Poisson approximation problem for
random sums of discrete random variables has
at-tracted the attention of mathematicians. Several
interesting results can be found in Yannaros
(1991), Vellaisamy and Upadhye (2009),
Kongudomthrap and Chaidee (2012),
Teerapabo-larn (2013a), TeerapaboTeerapabo-larn (2014), Tran Loc
Hung and Le Truong Giang (2014), Tran Loc Hung
and Le Truong Giang (2016a, 2016b), and Le
Tru-ong Giang and Trinh Huu Nghiem (2017).

Let X X 1, 2, be a sequence of independent
nega-tive-binomial random variables with probabilities

(

)

(

)

r
k

k i

P Xi k C pi pi

r k

= = + − −

where

pi(0,1);ri=1,2, ; 1,2, ; =i  =k 0,1, .

Let

n

Wn Xi

i

= 

= and U be a Poisson random vari-n

able with mean

( )

(

1

)

1.

n

E W r p p

n n i i i

i

 = = − −

In addition, throughout this paper, dTV is denoted
a probability distance of total variation, defined by

(

)

sup

(

) (

)

dTV X Y P X A P Y A

A

=  − 

</div>
(2)<div class='page_container' data-page=2>

A uniform bound and a non-uniform bound for the
distance between the distribution functions of

W

n

and
n

U

 were presented in Tran Loc Hung and Le
Truong Giang (2016a) as follows:

(

,

)

min



(

1

)

(

1

)

1,1



1
1

n r p

i
i
n

dTV W Un n e ri p pi i pi

n pi

i




  − −− − − − −

= (0.1)

and

(

)

(

(

)

)

( 0) 0 min ,1 ,

1
0

n

n r p

e i i ri pi

P Wn w P U n w pi

w p p

n i i i



 −  −  −

 −     + − 

 

 

= (0.2)

where w 0 +: {0,1,2, }.= 

Consider the sum

N

WN Xi

i

= 

= , where N is a

non-negative integer valued random variable and
inde-pendent of the Xi's. The sum is called random
sums of independent negative -binomial random

variables. Let U be a Poisson random variable

with =E

( )

N , where

(

)

N

r p

N i i

i

 = −

= .

Teerapabolarn (2014) gave a uniform bound for the
distance between the distribution functions of WN
and U as follows:

(

)

(

)

(

)

2
, min 1,

2
1
2

1 1

min , .

2
1

d WN U E N

TV e

N ri pi

N r p pi

i i i

E E

pi e

i N

 

 



 

   −

 

  

−

   

 

 −  

   = 

+      

  =   

  

  

 

(0.3)

In this paper, some of the bounds on Poisson
ap-proximation for random sums of independent
nega-tive-binomial random variables with mean

( )

E N

=  , where

(

)

N

r p p

N i i i

i

 = − −

= , are

present-ed in Section 2.

2 MAIN RESULTS

The following lemma is necessary to prove the
main result, which is directly obtained from
Bar-bour et al. (1992).

Lemma 2.1. Let U and N U denote a Poisson

random variable with mean  and N  ,
respec-tively. Then, for A +, the total variation distance

between the distributions of U
N

 and U

satis-fies the following inequality:

(

)

, min 1, .

dTV U U E N

N e  

     −

 

  (0.4)

The following theorems present non-uniform and
uniform bounds for the distance between the
distri-bution functions of WN and U , which are the

expected results.

2.1 A uniform bound on Poisson

approximation for random sums of independent 
negative-binomial random variables

Theorem 2.1. For A +,

(

,

)

min 1

(

1

)

1,1

(

1

)

N e N ri

d WNU E ri p pi i pi p pi i

TV

N
i



 

  − − −  − 

 

   − −  − 

 

=  

</div>
(3)<div class='page_container' data-page=3>

(

)



1

(

)

(

)

1



, min 1 1 ,1 .

n r p

i
i
n

dTV W Un n n e ri p pi i pi

pi
i




  − − − − − − −

= (0.6)

From the triangular inequality, combining (1.4) and
(1.6), it follows the fact that

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

, ,
1
, ,

1
, ,
1
1 1
1
min ,1
1 1
2
min 1,
1 1

min 1 1 ,1

d WN U P N n dTV W Un

TV

n

P N n dTV W Un dTV U U

n n

n

P N n dTV W Un dTV U U

n N

n

e r p

n i i r p

i
i

P N n pi

p p

n i i

n i

E N
e

r
N

E N e ri p pi i pi

 
  
  



 





= =
=

 
  =  + 
=

=  = +
=
−
 − − 
   −
  =   − 
 
= =
 
 
 
+   −
 
 
− − −





− − −



min 1, .

N pi

i
pi
i

E N

e  


 − 
 
 = 
 
 
+   −
 

This finishes the proof.

Remark 2.1. The result of (1.5) is interesting

be-cause of considering

(

)

1
1

N

r p p

N i i i

i

 = − −

= instead of

(

)

N

r p

N i i

i

 = −

= as in Teerapabolarn (2014). It is

easily seen that the (1.1) is a special case of the
(1.5) when N n=  + is fixed.

Corollary 2.1. For r r1 2= = = =... rn 1, then

( , ) min



(

1

)

1,1 1



( )2 1
1

min 1, .

N

d WNU E N e pi pi pi

TV
i
E N
e



 


 − − − − 
 
  − − 
=
 
 
+   −
 
(0.7)

Remark 2.2. The result (1.7) is a Poisson

approx-imation for the random sums of independent
geo-metric random variables, which is introduced in
Teerapabolarn (2013a).

2.2 A non-uniform bound on Poisson

approximation for random sums of independent 
negative-binomial random variables

Theorem 2.2. For w 0 +, we have

(

)

(

)

(

)

(

)

(

)

2 2

( 0) 0 min ,min 1,

1
0

1 1 min ,1 1 1 .

1
0
1

P WN w P U w E N

w e

N ri pi r

i
N

E N e pi p pi i

p wi
i
  
 



   
   
 −    +   − 
 
   
 
 −  −  − 
 
+  −   + −  − 
 
 
=
 
(0.8)

Proof. Applying the corresponding results in Tran

Loc Hung and Le Truong Giang (2016a) and
Teerapabolarn (2013a) yields

(

)

(

)

1 1

( 0) min ,1

! 1 0 1

ke n e n n ri pi r p

n i i

P Wn w pi

k n p wi pi

k w i

 


− −  −  −
 −     + − 
 
 

 = (0.9)

and

(

)

(

)

2 2

min ,min 1, .

0 0

1
0

P U w P U w E N

N w e

  

  −    +   − 

 

   

  (0.10)

</div>
(4)<div class='page_container' data-page=4>

(

) (

)

(

) (

)

(

) (

)

(

) (

)

(

)

(

) (

)

(

)

(

)

(

)

(

)

(

(

)

)

0 0 0 0

0 0

1 1

min ,1

1
0

0 1

2 2

min ,min 1,
1
0

P WN w P U w P N n P Wn w P U w

n

P N n P Wn w P U n w P U w P U w

n

P N n P Wn w P U w

n
n

P U N w P U w

n

n r p

e i i ri pi

P N n pi

p w p

n i i

n

i

w e

 

  



 









 −   =  − 

  

  =   −  +  −  

=


  =  − 

+  − 

 −  −  −

  =   + − 

 

 

= =


+

(

)

(

)

(

)

(

)

1 1 min ,1 1 1

1
0
1

2 2

min ,min 1, .

1
0

E N

N ri pi r

i
N

E N e pi p pi i

p wi
i

E N

w e

 



  



  

   − 

   

 

   

 

 −  −  − 

  −   + −  − 

 

 

 

   

   

+  +   − 

 

   

 

The proof is completed.

Remark 2.3. It is easily to check that the (1.2) is a

special case of the (1.8) when N n=  + is fixed.

Corollary 2.2. For r r1 2= = = =... rn 1, then

(

)

(

)

(

)

(

)

2 2

( 0) 0 min ,min 1,

1
0

2
1
1

1 1 min ,1 .

1
0
1

P WN w P U w E N

w e

N pi

N

E N e

p wi pi

i

  

 



   

   

 −    +   − 

 

   

 

   − 

−

 

+  −   +  

 

 

 

(0.11)

Remark 2.4. The result (1.11) is a non-uniform

bound on Poisson approximation for the random
sums of independent geometric random variables.

3 CONCLUSIONS

Bounds for the distance between the distribution
function of random sums of independent
negative-binomial random variables and an appropriate
Poisson distribution function were obtained. The
results in this paper are extensions and
generaliza-tions of results in Teerapabolarn (2013a), and
Teerapabolarn (2014), Tran Loc Hung and Le
Tru-ong Giang (2016a, 2016b). The results will be
more interesting and valuable if Poisson
approxi-mation for random sums of dependent negative -

random variables. Journal of Mathematics Research.
4(3): 29-35.

Le Truong Giang and Trinh Huu Nghiem, 2017.
Charac-teristic functions method for some of the limit
theo-rems in probability. Can Tho University Journal of
Science. 53: 88-95.

Teerapabolarn, K., 2013a. Improved bounds on Poisson
approximation for independent binomial random

summands. International Journal of Pure and
Ap-plied Mathematics. 89(1): 29-33.

Teerapabolarn, K., 2013b. Poisson approximation for
random sums of geometric random variables.
Inter-national Journal of Pure and Applied Mathematics.
89(1): 35-39.

</div>
(5)<div class='page_container' data-page=5>

ran-Tran Loc Hung and Le Truong Giang, 2016a. On bounds
in Poisson approximation for distributions of
inde-pendent negative-binomial distributed random
varia-bles. SpringerPlus,

Tran Loc Hung and Le Truong Giang, 2016b. On the
bounds in Poisson approximation for independent
ge-ometric distributed random variables. Bulletin of the
Irannian Mathematical Society. 42(5): 1087-1096.

Vellaisamy, P. and Upadhye, S., 2009. Compound
nega-tive binomial approximations for sums of random
variables. Probability and Mathematical
Statis-tics, 29(2): 205 - 226.

</div>

<b>New bounds on poisson approximation for random sums of independent </b>

<b>negative-binomial randomvariables </b>

Le Truong Giang

<sub> and Trinh Huu Nghiem</sub>

(

)

(

)

where

( )

(

)

(

)

(

) (

)

<i>W</i>

<i>U</i>

(

)



(

)

(

)



(

)

<sub>(</sub>

(

<sub>)</sub>

)

( )

(

)

(

)

(

)

(

)

( )

(

)

(

)

(

)

(

)

(

)

(

)



(

)

(

)



(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(