Vietnam J. Math. https://doi.org/10.1007/s10013-018-0294-y

Some Results on Approximation for Distributions of Random Sums by Poisson Distribution Tran Loc Hung1

Received: 23 February 2017 / Accepted: 20 January 2018 © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Abstract The main purpose of this paper is to investigate the bounds in Poisson approximation of the distributions for random sums by Poisson distribution via a probability distance based on linear operator introduced by R´enyi. Some Le Cam type bounds for distributions of random sums of row-wise independent Poisson-binomial distributed and negative-binomial distributed random variables are obtained. Keywords Poisson approximation · Random sum · Le Cam’s bound · R´enyi’s operator · Probability distance Mathematics Subject Classification (2010) Primary 60G50 · Secondary 41A36

1 Introduction For n ≥ 1, let {Xn,j , j = 1, 2, . . . , n} be a triangular array of row-wise independent, Bernoulli distributed random variables with success probabilities P (Xn,j = 1) = 1 − P (Xn,j = 0) = pn,j ∈ (0, 1), j ∈ {1, 2, . . . , n}.   Set Sn = nj=1 Xn,j and write λn := E(Sn ) = nj=1 pn,j . Assume that limn→∞ λn = λ ∈ (0, +∞). We denote by Zλ the Poisson random variable with mean λ (λ > 0), that is e−λ λk , k = 0, 1, 2, . . . k! It has long been known that the probability distribution of sum Sn is approximated by the distribution of a Poisson random variable Zλ (see for instance [17, Theorem 2, p. 523]). A P (Zλ = k) =

 Tran Loc Hung

[email protected] 1

University of Finance and Marketing, 2/4 Tran Xuan Soan street, District 7, Ho Chi Minh City, Vietnam

T. L. Hung

well-known bound in this approximation originated by Le Cam (1960) (see [14] for details) is defined as follows: ∞ 

|P (Sn = k) − P (Zλ = k)| ≤ 2

n 

2 pn,j .

(1)

j =1

k=0

During the last several decades, various powerful mathematical tools for establishing the bound in (1) have been developed, such as the Stein–Chen method, the method of convolution operators, the Kerstan’s method, the Charlier–Parseval approach, and the method of probability metrics, etc. Results of this nature may be found in [1–3, 5, 6, 10, 12, 14, 15, 18, 24–26, 28]. It is worth noticing that the bound in (1) usually presented via distance of total variation in the following form n  2 dTV (Sn , Zλ ) ≤ pn,j , (2) j =1

where the dTV (Sn , Zλ ) is defined by ∞

dTV (Sn , Zλ ) :=

1 |P (Sn = k) − P (Zλ = k)|. 2

(3)

k=0

Up to the present, the study of Le Cam’s bound has attracted much attention. Kerstan in [12] improved the bound in (2) to the following inequality ⎛ ⎞ ⎛ ⎞ n n   ⎝ dTV (Sn , Zλ ) ≤ 1.05 × ⎝ pj2 ⎠ pj ⎠, (4) j =1

j =1

where pj ≤ 1/4. Barbour and Hall in [2] further improved the bound in (2) to n

 pj2 . dTV (Sn , Zλ ) ≤ min 1, λ−1

(5)

j =1

Recently, Chen and R¨ollin in [5] showed the Le Cam type bound in the following form n n  1 − e− j =1 pj  2 n pj . (6) dTV (Sn , Zλ ) ≤ j =1 pj j =1

(The interested reader is referred to [1–5, 7, 14, 15, 18, 27] for more details). Furthermore, let N be a non-negative, integer-valued random variable, which is independent of all Xn,j , j ∈ {1, 2, . . . , n}; n ≥ 1. Then, the Le Cam type bound for random sum of independent Bernoulli random variables is given by Yannaros in [27] as follows:

⎞ ⎛ 



N − N j =1 pj 

N

1 − e dTV (SN , Zλ ) ≤ E

pj − λ

+ E ⎝ N pj2 ⎠ , p

j =1

j j =1 j =1  where SN = N j =1 Xn,j and S0 = 0. Besides, some uniform and non-uniform bounds in Poisson approximation for random sums SN are investigated by Kongudomthrap and Chaidee in [13]. However, as far as we know, the number of results related to the Le Cam’s bound in Poisson approximation for random sums is still little.

Some Results on Approximation for Distributions of Random Sums...

In recent years, using the linear operator method introduced by R´enyi in [17], an analogous bound to Le Cam’s bound in (1) is obtained by Hung and Thao (see [8] for more details). Specifically, for f ∈ K, ASn (f ) − AZλn (f ) ≤ 2f 

n 

2 pn,j ,

j =1

where K denotes the class of all real-valued bounded functions on the set of all nonnegative integers Z+ := {0, 1, 2, . . . }. The norm of function f ∈ K is defined by f  = supx∈Z+ |f (x)|. The linear operators associated with Sn and Zλn are denoted by ASn and AZλn , respectively. Here Sn := Xn,1 + · · · + Xn,j denotes the sum of row-wise independent Bernoulli random variables from a triangular array, λn := E(Sn ) and Zλn denotes the Poisson random variable with mean λn . We recall that the operator AX associated to a random variable X is introduced by R´enyi in [17], defined in the form (AX f )(x) := E[f (x + X)] =

∞ 

f (x + k)P (X = k)

∀f ∈ K, ∀x ∈ Z+ .

(7)

k=0

(See [17] for more details). It is to be noticed that the linear operator defined in (7) actually is a discrete form of a linear operator originated by Trotter in [23]. Additionally, let {Nn , n ≥ 1} be a sequence of integer-valued positive random variables such that for every n ≥ 1 the variables Xn,j , j = 1, 2, . . . n and Nn are independent. For n ≥ 1, set a random sum  SNn = Xn,1 + Xn,2 + · · · + Xn,Nn . Write λ˜ n := E(SNn ) = ∞ k=1 P (Nn = k)E(Sk ). Then, the Le Cam type bound for random sum is obtained in the following form ⎛ ⎞ Nn  2 ⎠ ASNn (f ) − AZλ˜ (f ) ≤ 2f E ⎝ pn,j n

j =1

for f ∈ K (see [8] for more details). Before starting the main results of this paper, we first have the definition and main properties of probability distance d based on linear operator introduced by R´enyi in [17]. Definition 1 The probability distance d(X, Y ; f ) of two discrete random variables X and Y with respect to function f ∈ K is defined by d(X, Y ; f ) := AX (f ) − AY (f ) = sup |Ef (X + x) − Ef (Y + x)| ,

(8)

x∈Z+

where the operators AX and AY are given in (7). Proposition 1 Based on the properties of R´enyi operator defined in (7) (see [17] for more details), the following properties of probability distance d will be used in subsequent sections. 1.

Let {Xn , n ≥ 1} be a sequence of random variables on a probability space (, B , P). Let X be a random variable defined on the same probability space (, B , P). If D

limn→∞ d(Xn , X; f ) = 0 for all functions f ∈ K, then Xn −→ X as n → ∞, where D

2.

−→ denotes the convergence in distribution. Let {Xn , n ≥ 1} and {Yn , n ≥ 1} be two sequences of independent random variables (and they are independent in each group). Let {Nn , n ≥ 1} be a sequence of integervalued positive random variables independent of all Xn and Yn for n ≥ 1. Then, for all

T. L. Hung

functions f ∈ K, N Nn ∞ k n     d Xi , Yi ; f ≤ P (Nn = k) d(Xi , Yi ; f ). i=1

3.

i=1

k=1

i=1

Let {X, Xn , n ≥ 1} and {Y, Yn , n ≥ 1} be two sequences of independent, identically distributed random variables (and they are independent in each group). Let {Nn , n ≥ 1} be a sequence of integer-valued positive random variables independent of all Xn and Yn for n ≥ 1. Moreover, assume that E(Nn ) < +∞. Then, for all functions f ∈ K, N Nn n   d Xi , Yi ; f ≤ E(Nn )d(X, Y ; f ). i=1

i=1

Remark 1 It is worth pointing out that the relation between the probability distance d in (8) and the distance of total variation dTV in (3) is given by sup d(X, Y ; IA ) = dTV (X, Y ), A⊆Z+

here IA (·) is the indicator function of a set A ⊆ Z+ . The main aim of this paper is to investigate the bounds in approximation of the distributions for random sums by Poisson distribution via probability distance d. Some Le Cam type bounds for distributions of random sums of row-wise independent Poisson-binomial and negative-binomial random variables are established. The present paper is a continuation of discussions about the Le Cam type bounds in Poisson approximation in [8, 9, 11]. The received results in this paper also are extensions of known results due to Le Cam (1960) in [14], Steele (1992) in [18], Teerapabolarn et al. (2007, 2009, 2012) in [19–22], and Neammanee (2003) in [15]. It is to be noticed that the operator approach in Poisson approximation has been used by Le Cam (1960) in [18] and Upadhye and Vellaisamy (2012, 2013, 2014) in [24–26]. This paper is organized as follows. Section 2 is devoted to the discussions on Le Cam type bounds, based on probability distance d, for distributions of random sums of a wide class of various discrete independent random variables, such as row-wise independent Poissonbinomial and negative-binomial random variables. The proofs of all theorems presented in Section 2 (Theorems 1, 2, and 3) will be shown in Section 3. Concluding remarks are noticed at the end of the paper. For making the proofs of main theorems in the paper, some propositions are given in the Appendix.

2 Main Results P

Throughout this paper, the notation −→ will mean convergence in probability. The Le Cam type bound for random sums of row-wise independent Bernoulli distributed random variables will be established via a probability distance defined in (8) as follows: Theorem 1 Let {Xk,j , k ≥ 1, j ≥ 1} be a double array of row-wise independent Bernoulli distributed random variables with success probabilities P (Xk,j = 1) = 1− P (Xk,j = 0) = pk,j ∈ (0, 1), k ≥ 1, j ≥ 1. Let {Nn , n ≥ 1} and {Mm , m ≥ 1} be two sequences of integer-valued positive random variables such that for every n ≥ 1

Some Results on Approximation for Distributions of Random Sums...

and m ≥ 1, the random variables Xk,j , (k ≥ 1, j ≥ 1), and Nn , Mm are indepenP

P

dent. Moreover,  assume Nn −→ ∞ as n → ∞, and Mm −→ ∞ as m → ∞. that Mm n Set SNn,Mm := N k=1 j =1 Xk,j . Let us denote by Zλ˜ n,m the Poisson distributed random variable with mean λ˜ n,m := E(SNn ,Mm ) =

∞ 

P (Nn = u)

u=1

P (Mm = v)



⎛

d SNn ,Mm , Zλ˜ n,m ; f ≤ 2f E ⎝

u  v 

pk,j .

k=1 j =1

v=1

Then, for all functions f ∈ K, 

∞ 

Mm Nn  

⎞ 2 ⎠ pk,j .

(9)

k=1 j =1

Corollary 1 According to Theorem 1, from (9), for all l ∈ {0, 1, . . . , n}, the Le Cam type bound in Poisson approximation for distribution of random sums of row-wise independent, Bernoulli distributed random variables in a double array is stated as follows: ⎞ ⎛ Nn  Mm

    

2 ⎠ pk,j .

P SNn ,Mm = l − P Zλ˜ n,m = l ≤ 2E ⎝ k=1 j =1

Remark 2 Theorem 1 is an extension of the results in [8] for the double arrays of row-wise independent, Bernoulli distributed random variables. That is, for f ∈ K, n  m    2 pk,j d Sn,m , Zλn,m ; f ≤ 2f  k=1 j =1

and n  m 

    2

P Sn,m = l − P Zλ = l ≤ 2 pk,j , n,m k=1 j =1

when P (Nn = n) = 1, P (Mm = m) = 1 and λn,m = E(Sn,m ) = E



m n k=1 j =1 Xk,j

 .

For making usefulness of our approach more apparent, illustrative applications in Poisson approximation for distributions of random sums of row-wise independent negative-binomial random variables will be presented as follows: Theorem 2 For n ≥ 1, let {Xn,j , j = 1, 2, . . . , n} be a triangular array of row-wise independent negative-binomial random variables with probabilities   rn,j + k − 1 (1 − pn,j )k (pn,j )rn,j ; pn,j ∈ (0, 1); k ≥ 0; rn,j ∈ N. P (Xn,j = k) = k Moreover, let {Nn , n ≥ 1} be a sequence of integer-valued positive random variables inde n P pendent of all Xn,j . Assume that Nn −→ ∞ as n → ∞. For n ≥ 1, set SNn := N j =1 Xn,j and write ∞ k   −1 P (Nn = k) rn,j (1 − pn,j )pn,j . β˜n := E(SNn ) = k=1

j =1

T. L. Hung

Let us denote by Zβ˜n the Poisson random variable with mean β˜n . Then, for f ∈ K, ⎛  ⎞ 2 Nn      1 rn,j +1 rn,j rn,j + pn,j + 1 ⎠. d SNn , Zβ˜n ; f ≤ f E ⎝ 1− pn,j

(10)

j =1

Corollary 2 On account of Theorem 2, from (10), for all k ∈ {1, 2, . . . , n}, the Le Cam type bound in Poisson approximation for distribution of random sums of row-wise independent, negative-binomial distributed random variables in a triangular array is established as follows: ⎛  ⎞ 2 Nn

       1 r +1

n,j rn,j rn,j + pn,j + 1 ⎠. 1−

P SNn = k − P Zβ˜n = k ≤ E ⎝ pn,j j =1

Theorem 3 Let {Xi,j , i ≥ 1, j ≥ 1} be a double array of row-wise independent negativebinomial distributed random variables with probabilities   ri,j + k − 1 (1 − pi,j )k (pi,j )ri,j ; pi,j ∈ (0, 1); k ≥ 0; ri,j ∈ N. P (Xi,j = k) = k Let {Nn , n ≥ 1} and {Mm , m ≥ 1} be two sequences of integer-valued positive random variables such that for every n ≥ 1 and m ≥ 1 the random variables Xi,j , Nn and Mm P

P

are independent. Assume that Nn −→ ∞ as n → ∞ and Mm −→ ∞ as m → ∞. Set  n Mm SNn ,Mm := N i=1 j =1 Xi,j and write β˜n,m := E(SNn ,Mm ) =

∞  ∞ 

P (Nn = u)P (Mm = v)E(Su,v ).

u=1 v=1

Let us denote by Zβ˜n,m the Poisson random variable with mean β˜n,m . Then, for f ∈ K, ⎛  ⎞ 2 Nn  Mm      1 rn,j +1 rn,j rn,j + pn,j d SNn Mm , Zβ˜n,m ; f ≤ f E ⎝ + 1 ⎠. 1− pn,j i=1 j =1

(11) Corollary 3 Under assumptions of Theorem 3, from (11), for all k ∈ {1, 2, . . . , n}, the Le Cam type bound in Poisson approximation for distribution of random sums of row-wise independent, negative-binomial random variables in a double array is stated in the form ⎛ ⎞  2   Mm Nn     1 rn,j +1 rn,j rn,j + pn,j +1 ⎠ . 1− |P (SNn ,Mm = k)−P (Zβ˜n,m = k)| ≤ E ⎝ pn,j i=1 j =1

3 Proofs Proof of Theorem 1 According to Definition in (7), we have ∞ ∞     ASNn ,Mm f (x) = E f (SNn ,Mm + x) = P (Nn = u) P (Mm = v)ASu,v f (x) u=1

v=1

Some Results on Approximation for Distributions of Random Sums...

and

AZλ˜

Nn ,Mm

 f (x) = E f (Zλ˜ N

∞ ∞    + x) = P (Nn = u) P (Mm = v)AZλ˜

n ,Mm

u,v

u=1

f (x),

v=1

where λ˜ u,v := E(Su,v ). Then, for all f ∈ K, and for x ∈ Z+ , we obtain ∞ ∞       P (Nn = u) P (Mm = v)d Su,v , Zλ˜ u,v ; f . d SNn ,Mm , Zλ˜ n,m ; f ≤ u=1

(12)

v=1

It is easy to check that (see [8] for more details) u  v    2 pk,j . d Su,v , Zλ˜ u,v , f ≤ 2f 

(13)

k=1 j =1

Combining (12) with (13) yields 

d SNn ,Mm , Zλ˜ n,m ; f



≤ 2f 

∞ 

P (Nn = u)

u=1

⎛

= 2f E ⎝

∞ 

⎛ P (Mm = v) ⎝

⎞ 2 ⎠ pk,j

k=1 j =1

v=1 Mm Nn  

v u  

⎞

2 ⎠ pk,j .

k=1 j =1

The proof is complete. Proof of Theorem 2 We shall begin with showing that ∞    ASNn f (x) = E f (SNn + x) = P (Nn = u)ASu f (x) u=1

=

∞ 

P (Nn = u)

u=1

∞ 

f (x + k)P (Su = k),

k=0

and ∞    AZβ˜n f (x) = E f (Zβ˜n + x) = P (Nn = u)AZβ˜ f (x) u u=1

=

∞ 

P (Nn = u)

u=1

∞ 

f (x + k)P (Zβu = k),

k=0

where βu := E(Su ). It follows that

    ASNn f (x) − AZβ˜n f (x) = E f (SNn + x) − E f (Zβ˜n + x) ∞ ∞ ∞    = P (Nn = u) f (x + k)P (Su = k) − f (x + k)P (Zβu = k) . u=1

k=0

k=0

Then, for f ∈ K, it can be verified that ∞    P (Nn = u)d(Su , Zβu , f ). d SNn , Zβ˜n ; f ≤ u=1

(14)

T. L. Hung

Consider −1 f (x) − AXn,j f (x) =

AZ

rn,j (1−pn,j )pn,j

=

∞ 



∞ 

    f (x + k) P Zrn,j (1−pn,j )p−1 = k −P (Xn,j = k) n,j

k=0 −1

f (x + k) e−rn,j (1−pn,j )pn,j

−1 k [rn,j (1 − pn,j )pn,j ]

k!    rn,j + k − 1 (pn,j )rn,j (1 − pn,j )k . − k

k=0

It follows that

f (x) − A f (x) X n,j −1



AZ

rn,j (1−pn,j )pn,j

 k

−1   ∞

r (1 − p )p  n,j nj −1 k n,j rn,j + k − 1 

−rn,j (1−pn,j )pn,j rn,j − ≤ f  1 − pn,j (pn,j )

e

k! k k=0

   −1 −1 rn,j

−r (1−pn,j )pn,j −1 rn,j (1 − pn,j )pn,j = f  e−rn,j (1−pn,j )pn,j − pn,j

+ e n,j

−rn,j (1 − pn,j )(pn,j )rn,j

⎞  k

−1   ∞

rn,j (1 − pn,j )pn,j  −1 rn,j + k − 1

−rn,j (1−pn,j )pn,j k rn,j ⎟ + (1 − pn,j ) (pn,j ) ⎠ −

e

k k! k=2

 −1

≤ f  e−rn,j (1−pn,j )pn,j − (pn,j )rn,j

 −1 

−1 + e−rn,j (1−pn,j )pn,j rn,j (1 − pn,j )pn,j − rn,j (1 − pn,j )(pn,j )rn,j ⎞  k −1  ∞ ∞    −1 rn,j (1−pn,j )pn,j rn,j + k − 1 ⎟ e−rn,j (1−pn,j )pn,j + + (1 − pn,j )k (pn,j )rn,j ⎠ . k! k k=2

k=2

We first observe that, for n ≥ 1, rn,j ∈ N and pn,j ∈ (0, 1),

−rn,j (1−pn,j )(pn,j )−1

− (pn,j )rn,j < (pn,j )rn,j ,

e and

(15)

   

−rn,j (1−pn,j )(pn,j )−1

rn,j 1 − pn,j (pn,j )−1 − rn,j (1 − pn,j )(pn,j )rn,j

e < rn,j (1 − pn,j )(pn,j )rn,j −1 .

(16)

Moreover, ∞  k=2

e

−rn,j (1−pn,j )(pn,j )−1

 k rn,j (1 − pn,j )(pn,j )−1 k! −1

= 1 − e−rn,j (1−pn,j )(pn,j )

−1

− rn,j (1 − pn,j )(pn,j )−1 e−rn,j (1−pn,j )(pn,j )

2 2 2 ≤ rn,j (pn,j )−2 − 2rn,j (pn,j )−1 + rn,j ,

(17)

Some Results on Approximation for Distributions of Random Sums...

and

 ∞   rn,j + k − 1 (1 − pn,j )k (pn,j )rn,j = 1 − rn,j (1 − pn,j )(pn,j )rn,j − (pn,j )rn,j k k=2

= 1 − rn,j (pn,j )rn,j + rn,j (pn,j )rn,j +1 − (pn,j )rn,j .

(18)

Combining (15), (16), and (17) with (18), it follows that    d Xn,j , Zrn,j (1−pn,j )p−1 ; f ≤ f  rn,j (pn,j )rn,j +1 − 2rn,j (pn,j )rn,j + rn,j (pn,j )r−1 n,j  2 2 2 −2rn,j (pn,j )−1 + rn,j (pn,j )−2 + rn,j +1 and according to properties of probability distance d (see [9]), we have   n    d Sn , Zβn ; f ≤ d Xn,j , Zrn,j (1−pn,j )p−1 ; f j =1

n,j

for f ∈ K, x ∈ Z+ . Hence, d(Sn , Zβn ; f ) ≤ f 

n  

rn,j (pn,j )rn,j +1 − 2rn,j (pn,j )rn,j + rn,j (pn,j )r−1

j =1

 2 2 + rn,j (pn,j )−2 + rn,j +1     1 2 rn,j +1 +1 . rn,j + pn,j 1− pn,j

2 −2rn,j (pn,j )−1

= f 

n 



rn,j

j =1

(19)

Combining (14) with (19), it follows that ⎛ Nn     rn,j (pn,j )rn,j +1 − 2rn,j (pn,j )rn,j + rn,j (pn,j )r−1 d SNn , Zβ˜n ; f ≤ f E ⎝ j =1

 2 2 2 −2rn,j (pn,j )−1 + rn,j (pn,j )−2 + rn,j +1 ⎛  ⎞ 2  Nn    1 rn,j +1 = f E ⎝ rn,j rn,j + pn,j + 1 ⎠. 1− pn,j j =1

This completes the proof. Proof of Theorem 3 The proof is similar to that of Theorem 1 so it will be omitted.

4 Concluding Remarks We conclude this paper with the following remarks. 1.

Taking all rn,j = 1 for n ≥ 1, j ∈ {1, 2, . . . , n}, from Theorem 2 and Theorem 3, we will get the Le Cam type bounds for random sums of row-wise independent geometric distributed random variables. The results of this nature also have been studied by various approaches such as Stein–Chen method or method of ω- function (see e.g. [16, 19–22]).

T. L. Hung

2.

We compare the received bounds in this paper with the existing ones, by considering the Le Cam type bounds in Remark 2 with (2), (4), (5), and (6), and using table. Results of Le Cam (1960) Kerstan (1964) Barbour and Hall (1984) Chen and R¨ollin (2013) Hung and Thao (2013) Remark 2 from Theorem 1

3.

with n = 103 , pj = 10−3

Le Cam type bounds  dTV (Sn , Zλ ) ≤ nj=1 pj2   n 2 dTV (Sn , Zλ ) ≤ 1.05 n j =1 pj j =1 pj  dTV (Sn , Zλ ) ≤ min(1, λ−1 ) nj=1 pj2  − n p  n j =1 j 2 dTV (Sn , Zλ ) ≤ 1−en p j =1 pj j =1 j n 2 d(Sn , Zλ ; f ) ≤ 2f  j =1 pj   2 d(Sn,m , Zλn,m ; f ) ≤ 2f  nk=1 m j =1 pk,j

Bound = 1.0 × 10−3 Bound = 1.05 × 10−3 Bound = 1.0 × 10−3 Bound = 6.3212 × 10−4 Bound = 1.0 × 10−3 Bound = m × 10−3

It should be noted that in the last row of the table according to Remark 2 of Theorem 1, the probability distance d(Sn,m , Zλn,m ; f ) can be operated with a function f ∈ K such that f  = 1/2. Then, the bound depends on m. If m = 1, the bound will be the same of Hung and Thao (2013). When m will get a great value, for example m ≥ 103 , then the bound should be worse than another bounds. From the above table, we can conclude that the Le Cam type bound estimated by distance d defined in (8) has the analogous rate like Le Cam’s (1960) and Barbour and Hall’s (1984). They are better than the bound of Kerstan (1964) but are worse than the bound of Chen and R¨ollin (2013). Thus, the results of this paper are not sharper than the results obtained by Stein–Chen method (see e.g. [5]), but the probability distance approach gives a simple and efficient technique for establishing the Le Cam type bounds in Poisson approximation for random sums of row-wise independent discrete random variables. Numerical computation for bounds in Theorem 1 (with Corollary 1) and Theorem 3 (with Corollary 3) shows that the bounds are bad for double arrays with Nn and Mn P

P

such that Nn −→ ∞ and Mm −→ ∞ when n → ∞ and m → ∞. For the sake of simplicity, assume that ri,j = r ∈ N and pi,j = p ∈ (0, 1). Let Nn ∼ Unif{1, 2, . . . , n} and Mm ∼ Unif{1, 2, . . . , m} for n ≥ 1, m ≥ 1. Consider Corollary 1 Mm Nn 

  

p2

P (SNn ,Mm = k) − P Zλ˜ n,m ≤ 2E i=1 j =1

= 2p E(Nn )E(Mm ) = 2p 2

 2

1+n 2



 1+m . 2

It is clear that the left-hand side of the inequality could not be smallness when n and m are great numbers. By an argument analogous for discussion on received bound in Corollary 3 (with bound from Theorem 3), it follows that

   

P SNn ,Mm = k − P Zβ˜n,m = k        m+1 n+1 1 2 r+1 . +1 1− ≤ r r +p p 2 2 Hence, we can remark that the received bounds in our paper via probability distance d are formal, and they are not better than bounds receiving by another method as Stein– Chen method.

Some Results on Approximation for Distributions of Random Sums... Acknowledgements The author is grateful to the referee for carefully reading the manuscript and for offering comments which enabled him to substantially improve the paper. A part of this work was done when the author was working at the Vietnam Institute of Advanced Study in Mathematics (VIASM). Special thanks are due to the VIASM for providing a fruitful research environment and working condition.

Funding Information The research has been supported by the Vietnam’s National Foundation For Science and Technology Development (NAFOSTED) under Grant No. 101.01-2010.02.

Appendix Proposition A.1 For all x ∈ (0, 1) and for n ∈ N, x n > e−n

1−x x

(20)

.

Proof Taking logarithm of both sides of inequality in (20), it follows that n ln(x) > −n

1−x x

⇐⇒

ln(x) +

1−x >0 x

⇐⇒

x ln(x) + 1 − x > 0.

Considering the function f (x) = x ln(x) + 1 − x, we have f (x) = ln(x) < 0, because x ∈ (0, 1). Thus, the function f (x) is a decreasing function on (0, 1), and lim f (x) = 1;

x→0+

lim f (x) = 0.

x→1−

Then, f (x) > 0 ∀x ∈ (0, 1). The proof is complete. It is obvious that with n = 1, we have the following result. Proposition A.2 For all x ∈ (0, 1), x > e−

1−x x

(21)

.

On account of Propositions A.1 and A.2, it follows that Proposition A.3 For pn,j ∈ (0, 1) and for rn,j ∈ N, inequality (15) holds, that is

−rn,j (1−pn,j )(pn,j )−1

− (pn,j )rn,j < (pn,j )rn,j .

e Proof Substituting x = pn,j ∈ (0, 1) and n = rn,j ∈ N, then, on account of Proposition A.1, it follows that −1

(pn,j )rn,j ≥ e−rn,j (1−pn,j )(pn,j ) . Then,

−1

−rn,j (1−pn,j )(pn,j )−1 − (pn,j )rn,j = (pn,j )rn,j − e−rn,j (1−pn,j )(pn,j ) < (pn,j )rn,j .

e Thus, inequality (15) is proved.

T. L. Hung

Proposition A.4 For pn,j ∈ (0, 1) and for rn,j ∈ N, inequality (16) holds, that is



−rn,j (1−pn,j )(pn,j )−1 rn,j (1 − pn,j )(pn,j )−1 − rn,j (1 − pn,j )(pn,j )rn,j

e < rn,j (1 − pn,j )(pn,j )rn,j −1 . Proof We shall begin with showing that

−n(1−x)x −1

n(1 − x)x −1 − n(1 − x)x n < n(1 − x)x n−1 ,

e

(22)

where x ∈ (0, 1) and n ∈ N. For x ∈ (0, 1), we have 1−x x ∈ (0, +∞). Then, inequality (22) should be written as follows:

  

    

−n 1−x

1−x 1−x 1−x n+1 x

e −n x x n+1 n


 

−n 1−x 1−x x ⇐⇒

e − x n+1

< x n ⇐⇒ −x n < e−n( x ) − x n+1 < x n (23) ⇐⇒ −x n < e−n(

1−x x )

− x n+1

and e−n(

1−x x )

− x n+1 < x n .

Because x ∈ (0, 1), hence x n > x n+1 for every n ∈ N. Then, the first inequality (in the last line) in (23) −x n < e−n(

1−x x )

− x n+1

⇐⇒

−(x n − x n+1 ) < e−n(

1−x x )

should be held. According to (21), we have e−n(

1−x x )

< x n < x n (1 + x) = x n + x n+1 .

It means that the second inequality in the last line of (23) is true. According to the inequality in (23), substituting x = pn,j ∈ (0, 1) and n = rn,j ∈ N, we obtain inequality (16) is proved. Proposition A.5 Assume that pn,j ∈ (0, 1) and rn,j ∈ N. Then ∞ 

−1

e−rn,j (1−pn,j )(pn,j )

k=2

rn,j (1 − pn,j )(pn,j )−1 2 2 2 ≤ rn,j (pn,j )−2 − 2rn,j (pn,j )−1 + rn,j . k!

Proof We see at once that inequality (17) is equivalent to the following inequality ∞ 

−1

e−rn,j (1−pn,j )(pn,j )

k=2

rn,j (1 − pn,j )(pn,j )−1 2 [(pn,j )−1 − 1]2 . ≤ rn,j k!

(24)

To prove inequality (24), let us consider the following inequality, for every x ∈ (0, 1) and for all n ∈ N, ∞  −1 [n(1−x)x −1 ]k k! e−n(1−x)x ≤ n2 (x −1 − 1)2 . (25) k=2

Let us first prove that

e−y > 1 − y for y > 0. Consider the function g(y) = − 1 + y. It is easily seen that e−y

g (y) = −e−y + 1

(26)

Some Results on Approximation for Distributions of Random Sums...

and

g (y) = 0

⇐⇒

y = 0.

It is clear that lim g(y) = 0

y→0+

and

lim g(y) = +∞.

y→∞

Thus, g(y) > 0 for y > 0. Inequality (26) is proved. Substituting y = n(1 − x)x −1 with x ∈ (0, 1), we have −1

e−n(1−x)x [1 + n(1 − x)x −1 ] > 1 − [n(1 − x)x −1 ]2 . Then, the left-hand side of inequality (24) should be written as follows: ∞ 

e−n(1−x)x

k=2 ∞ 

=

−1 [n(1−x)x −1 ]k k!

e−n(1−x)x

−1 [n(1−x)x −1 ]k k!

  −1 −1 − e−n(1−x)x + e−n(1−x)x n(1 − x)x −1

k=0 −1

= 1 − e−n(1−x)x [1 + n(1 − x)x −1 ] ≤ 1 − [1 − (n(1 − x)x −1 )2 ] = n2 (x −1 − 1)2 . Substituting n = rn,j and x = pn,j we get ∞ 

−1

e−rn,j (1−pn,j )(pn,j )

k=2

rn,j (1 − pn,j )(pn,j )−1 2 ≤ rn,j [(pn,j )−1 − 1]2 . k!

Thus, inequality in (17) is proved.

References 1. Arratia, R., Goldstein, L., Gordon, L.: Poisson approximation and the Chen–Stein method. Stat. Sci. 5, 403–434 (1990) 2. Barbour, A.D., Hall, P.: On the rate of poisson convergence. Math. Proc. Camb. Philos. Soc. 95, 473– 480 (1984) 3. Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation. Clarendon Press, Oxford (1992) 4. Barbour, A.D., Chen, L.H.Y.: An Introduction to Stein’s Method. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, vol. 4. World Scientific, Singapore (2004) 5. Chen, L.H.Y., R¨ollin, A.: Approximating dependent rare events. Bernoulli 19, 1243–1267 (2013) 6. Deheuvels, P., Karr, A., Pfeifer, D., Serfling, R.: Poisson approximations in selected metrics by coupling and semi-group methods with applications. J. Stat. Plan. Inference 20, 1–22 (1988) 7. Gut, A.: Probability: A Graduate Course. Springer Texts in Statistics. Springer-Verlag, New York (2005) 8. Hung, T.L., Thao, V.T.: Bounds for the approximation of Poisson-binomial distribution by Poisson distribution. J. Inequal. Appl. 2013, 30 (2013) 9. Hung, T.L., Giang, L.T.: On bounds in Poisson approximation for integer-valued independent random variables. J. Inequal. Appl. 2014, 291 (2014) 10. Hung, T.L., Giang, L.T.: On bounds in Poisson approximation for distributions of independent negativebinomial distributed random variables. SpringerPlus 2016(5), 79 (2016) 11. Hung, T.L., Giang, L.T.: On the bounds in Poisson approximation for independent geometric distributed random variables. Bull. Iran. Math. Soc. 42, 1087–1096 (2016) 12. Kerstan, J.: Verallgemeinerung eines Satzes von Prochorow und Le Cam. Z. Wahrscheinlichkeitstheorie Verw. Gebiete. 2, 173–179 (1964) 13. Kongudomthrap, S., Chaidee, N.: Bounds in Poisson approximation for random sums of Bernoulli random variables. J. Math. Res. 4, 29–35 (2012) 14. Le Cam, L.: An approximation theorem for the Poisson binomial distribution. Pac. J. Math. 10, 1181– 1197 (1960) 15. Neammanee, K.: A Nonuniform bound for the approximation of Poisson binomial by Poisson distribution. Int. J. Math. Math. Sci. 2003, 3041–3046 (2003)

T. L. Hung 16. Papadatos, N., Papathanasiou, V.: Distance in variation between two arbitrary distributions via the associated ω-functions. Theory Probab. Appl. 40, 567–575 (1995) 17. R´enyi, A.: Probability Theory. American Elsevier Publishing Co., Inc., New York (1970) 18. Steele, J.M.: Le Cam’s inequality and Poisson approximations. Am. Math. Mon. 101, 48–54 (1994) 19. Teerapabolarn, K., Wongkasem, P.: Poisson approximation for independent geometric random variables. Int. Math. Forum. 2, 3211–3218 (2007) 20. Teerapabolarn, K.: A note on Poisson approximation for independent geometric random variables. Int. Math. Forum. 4, 531–535 (2009) 21. Teerapabolarn, K.: A point-wise approximation for independent geometric random variables. Int. J. Pure Appl. Math. 76, 727–732 (2012) 22. Teerapabolarn, K.: A non-uniform bound on Poisson approximation by ω-functions. Int. J. Pure Appl. Math. 78, 1063–1075 (2012) 23. Trotter, H.F.: An elementary proof of the central limit theorem. Arch. Math. 10, 226–234 (1959) 24. Upadhye, N.S., Vellaisamy, P.: Improved bounds for approximations to compound distributions. Stat. Probab. Lett. 83, 467–473 (2013) 25. Upadhye, N.S., Vellaisamy, P.: Compound Poisson approximation to convolutions of compound negative binomial variables. Methodol. Comput. Appl. Probab. 16, 951–968 (2014) ˇ 26. Vellaisamy, P., Upadhye, N.S., Cekanaviˇ cius, V.: On negative binomial approximation. Theory Probab. Appl. 57, 97–109 (2013) 27. Yannaros, N.: Poisson approximation for random sums of Bernoulli random variables. Stat. Probab. Lett. 11, 161–165 (1991) 28. Zacharovas, V., Hwang, H.-K.: A Charlier-Parseval approach to Poisson approximation and its applications. arXiv:0810.4756v1 (2008)