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Applied Mathematical Sciences, Vol. 8, 2014, no. 172, 8577 - 8579
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.410812
A Pointwise Approximation for Random Sums
of Independent Discrete Random Variables
K. Teerapabolarn
Department of Mathematics, Faculty of Science
Burapha University, Chonburi 20131, Thailand
c 2014 K. Teerapabolarn. This is an open access article distributed under
Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Abstract
We determine a new result of pointwise Poisson approximation for random sums of independent integer-valued random variables. The bound
in this study is sharper than that reported in [1].
Mathematics Subject Classification: 60F05, 60G05
Keywords: Integer-valued random variable, Pointwise Poisson approximation, Random sums
1
Introduction
Let {Xn,i , i = 1, ..., n; n ∈ N} be a sequence of independent integer-valued
random variables with success probabilities P (Xn,i = 1) = pn,i and P (Xn,i =
0) = 1 − pn,i − qn,i for every i ∈ {1, ..., n} and n ∈ N, where pn,i , qn,i ∈ (0, 1)
and pn,i + qn,i ∈ (0, 1). Suppose that Nn for n ∈ N are positive integer-valued
PNn
random variables and independent of the Xn,i ’s. Let SNn =
i=1 Xn,i be
random sums ofPa sequence of independent integer-valued random variables,
n
and let λNn = N
i=1 pn,i . In this case, Hung and Giang [1] gave a pointwise
bound for approximating the probability function of the random sums SNn by
the Poisson probability function with mean λNn as follows:
(N
)
n
X
dP (SNn , ℘λNn ) ≤ 2E
(p2Nn ,i + qNn ,i ) ,
(1.1)
i=1
8578
K. Teerapabolarn
where dP (SNn , ℘λNn ) = |P (SNn = x) − P (℘λNn = x)| for x ∈ N ∪ {0} and ℘λNn
is the Poisson random variable with mean λNn .
In this paper, we determine a new bound for dP (SNn , ℘λNn ) by using the
method in [3], which is in Section 2. In Section 3, we determine a new bound
for this approximation and the conclusion of this study is presented in the last
section.
2
Method
We will prove our main result by using the method in [3]. For arbitrary nonnegative integer-valued X, Y and Z and A ⊆ N ∪ {0},
|P (X ∈ A) − P (Y ∈ A)| ≤ |P (X ∈ A) − P (Z ∈ A)|
+ |P (Z ∈ A) − P (Y ∈ A)|.
(2.1)
The following lemmas are directly obtained from [3] and [2], respectively.
0
0
Lemma 2.1. Let Xn,i = I(Xn,i = 1) for every i ∈ {1, ..., n} and Sn =
Pn
0
i=1 Xn,i . Then, for x ∈ N ∪ {0},
0
dP (Sn , Sn ) ≤
n
X
qn,i
(2.2)
i=1
and
0
−λn
dP (Sn , ℘λn ) ≤ λ−1
)
n (1 − e
n
X
p2n,i .
(2.3)
i=1
3
Result
This Section presents a new bound for dP (SNn , ℘λNn ).
Theorem 3.1. With the above definitions, the following inequality holds:
(N
)
n
X
dP (SNn , ℘λNn ) ≤ E
(p2Nn ,i + qNn ,i )
i=1
for x ∈ N ∪ {0}.
(3.1)
8579
A pointwise approximation for random sums
Proof. From (2.1), (2.2) and (2.3), we obtain
dP (SNn , ℘λNn ) ≤
≤
≤
≤
∞
X
m=1
∞
X
m=1
∞
X
m=1
∞
X
m=1
P (Nn = m)dP (Sm , ℘λm )
0
0
P (Nn = m)[dP (Sm , ℘λm ) + dP (Sm , Sm )]
"
−λm
)
P (Nn = m) λ−1
m (1 − e
m
X
p2m,i +
i=1
P (Nn = m)
m
X
#
qm,i
i=1
m
X
(p2m,i + qm,i ),
i=1
which implies that
)
(N
n
X
(p2Nn ,i + qNn ,i ) .
dP (SNn , ℘λNn ) ≤ E
i=1
Hence, (3.1) is obtained.
o
o
nP
nP
Nn
Nn
2
2
< 2E
Remark. Because E
i=1 (pNn ,i + qNn ,i ) , the
i=1 (pNn ,i + qNn ,i )
bound in (3.1) is sharper than the bound in (1.1).
4
Conclusion
A new result for approximating the probability function of random sums of
independent integer-valued
PNn random variables and the Poisson probability function with mean λNn = i=1 pn,i was obtained. By comparing between two such
bounds, the bound in this study is sharper than that reported in [1].
References
[1] T.L. Hung, L.T. Giang, On bounds in Poisson approximation for integervalued independent random variables, J. Inequal. Appl., 2014(2014):291.
http://dx.doi.org/10.1186/1029-242x-2014-291
[2] R. Kun, K. Teerapabolarn, A piontwise Poisson approximation by wfunctions, Appl. Math. Sci., 6(2012), 5029-5037.
[3] R.J. Serfling, A general Poisson approximation theorem, Ann. Prob.,
3(1975), 726–731. http://dx.doi.org/10.1214/aop/1176996313
Received: October 11, 2014; Published: December 1, 2014