A new approach to Poisson approximation and applications

Transcription

1 A new approach to Poisson approximation and applications Tran Loc Hung and Le Truong Giang October 7, 014 Abstract The main purpose of this article is to introduce a new approach to Poisson approximation Some bounds in Poisson approximation for probability distributions of a wide class of various arrays of row-wise independent discrete random variables are established via a probability distance Some analogous results related to random sums in Poisson approximation are also considered Keywords: Poisson approximation, Random sums, Le Cam s inequality, Trotter s operator, Poisson-binomial random variable, Geometric random variable, Negative-binomial random variable, Integer-valued random variable, Probability distance Mathematics Subject Classification 010: 60F05, 60G50, 41A36 1 Introduction Let X n,1, X n,, be a triangular array of row-wise independent Bernoulli distributed random variables with success probabilities P X n,j = 1 = 1 P X n,j = 0 = p n,j 0, 1, j = 1,,, n; n 1, such that pˆ n := n max p n,j 0 as n Set S n = n X n,j and λ n = 1 j n ES n = p n,j Suppose that lim λ n = λ, 0 < λ < + We will denote by Z λ the Poisson random variable with mean λ It has long been known that the probability distributions of S n, n 1 are usually approximated by the distribution of Z λ More specifically, under above assumptions on pˆ n and λ, the Poisson approximation theorem states that S n d Zλ as n 1 For other surveys, see?,?,?,?,?,? Note that, and from now on, the notation d means the convergence in distribution Moreover, using the method of convolution operators, Le Cam?, 1960 obtained the remarkable inequality P S n = k P Z λ = k n p n,j University of Finance and Marketing, 306 Nguyen Trong Tuyen st, Tan Binh Dist, Ho Chi Minh city, Vietnam tlhung@ufmeduvn University of Finance and Marketing, 306 Nguyen Trong Tuyen st, Tan Binh Dist, Ho Chi Minh city, Vietnam ltgiang@ufmeduvn 1

2 It is to be noticed that the Le Cam s inequality provides information on the quality of the Poisson approximation, and it is usually expressed via the total variation distance d T V S n, Z λ d T V S n, Z λ := 1 P S n = k P Z λ = k n p n,j 3 The inerested reader is referred to?,?,?,?,?,?,?, and? for more details The Poisson approximation has received extensive attention in the literature and many different approaches have been proposed During the last several decades various powerfull mathematical tools for establishing the Le Cam s inequality have been demonstrated such as the coupling technique, the Stein-Chen method, the method of convolution operators, the method of probability metrics, etc Results of this nature may be found in?,?,?,?,? Recently, one of new approaches to Poisson approximation have been used in providing the Le Cam-style inequalities due to? Specially, for f K A Sn f A Zλn f f n p n,j, 4 where K denoted the class of all real-valued bounded functions f on the set of all non-negative integers Z + := {0, 1,,, n} The norm of function f K is defined by f = sup f x x Z + The linear operators associated with S n and Z λn denoted by A Sn and A Zλn, respectively The operator A X associated a random variable X is called the Trotter-Renyi operator and it is defined by A X fx := fx + kp X = k, f K, x Z + 5 See? and? for more details It is to be noticed that the linear operator defined in?? actually is a discrete form of the Trotter s operator we refer the readers to?,? and?, for a more general and detailed discussion of this operator method Assume, furthermore, that N n, n 1 are non-negative, integer-valued random variables independent of all X n,j, j = 1,, ; n 1 Let us denote by S Nn = X n1 + + X nnn and λ Nn = ES Nn Then, an analog bound in?? is established as follows N n A SNn f A ZλNn f f E, 6 for f K See,? for more details During the last several decades the operator method has risen to become one of the important most tools available for studying with certain types of large scale problems as limit theorems for independent random variables The Trotter-operator method have been used in some areas of probability theory and related fields For a deeper discussion of Trotter-operator method we refer the reader to?,?,?,?,?,?,?,?,?, and? We wish to introduce in this paper a new probability distance based on the Trotter-Renyi operator in??, which establishes many new effective bounds of Le Cam s style inequalities for a wide class of discrete random variables such as geometric random variables, negative-binomial random variables, integer-valued random variables The relationship between the Trotter-Renyi distance and total variaion distance will be shown later The received results in this paper are extensions and generalized of known results in?,?,?,?,?,?,?,?,? The present paper is a continuation of earlier results in? p n,j

3 This paper is organized as follows We begin with the notations and definitions of Trotter operator, Renyi operator and Trotter-Renyi distance with some main properties in sections and 3 The sections 4, 5, 6 and 7 are devoted to the discussions of establishing the Le Cam s inequalities, based on Trotter-Renyi distance, for a wide class of variuos discrete random variables such as Poisson-binomial, Geometric, Negative-binomial and integer-valued distributed random variables The analogous results related to random sums in Poisson approximation problems also considered in these sections Concluding remarks are noticed in last section The Trotter-Renyi operator In the sequel we shall need some properties of Trotter-Renyi operator in??, which has been used for a long time in various studies of classical limit theorems for sums of independent random variables see?,?,?,?,? and?, for the complete bibliography It is to be noticed that the linear operator defined in?? actually is a discrete form of the Trotter s operator we refer the readers to?,?,?,?,? and?, for a more general and detailed discussion of this operator method We will denote by A X and A Y two operators associated with two discrete random variables X and Y We will write α for a real number Then, we easily get following main properties of our operator, for all functions f, g K and for α, β R 1 A X αf + βg = αa X f + βa X g A X f f 3 Suppose that A X, A Y are operators associated with two independent random variables X and Y Then, for all f K, In fact, for all x Z + A X+Y fx = A X+Y f = A X A Y f = A Y A X f fx + lp X + Y = l = = A X A Y fx = A X A Y fx r, fx + k + rp Y = kp X = r 4 Suppose that A X1, A X,, A Xn are the operators associated with the independent random variables X 1, X,, X n Then, for all f K, A Sn f = A X1 A X A Xn f is the operator associated with the partial sum S n = X 1 + X + + X n 5 Suppose that A X1, A X,, A Xn and A Y1, A Y,, A Yn are operators associated with independent random variables X 1, X,, X n and Y 1, Y,, Y n Moreover, assume that all X i and Y j are independent for i, j = 1,,, n Then, for every f K, Clearly, n A n k=1 X f k A n k=1 Y f A k Xk f A Yk f 7 A X1 A X A Xn A Y1 A Y A Yn = k=1 n A X1 A X A Xk 1 A Xk A Yk A Yk+1 A Yn k=1 3

4 Accordingly, n A n k=1 X f k A n k=1 Y f A k X1 A Xk 1 A Xk A Yk A Yk+1 A Yn f k=1 n A Yk+1 A Yn A Xk A Yk f k=1 n A Xk f A Yk f k=1 6 A n X f An Y f n A Xf A Y f Lemma 1 The equation A X fx = A Y fx for f K, x Z +, provides that X and Y are identically distributed random variables Let A X1, A X,, A Xn, be a sequence of Trotter-Renyi s operators associated with the independent discrete random variables X 1, X,, X n,, and assume that A X is a Trotter- Renyi operator associated with the discrete random variable X The following lemma states one of the most important properties of the Trotter-Renyi operator Lemma A sufficient condition for a sequence of random variables X 1, X,, X n converging in distribution to a random variable X is that lim A X n f A X f = 0, for all f K Proof Since lim A X n f A X f = 0, for all f K, we conclude that lim f x + k P X n = k P X = k = 0, for all f K and for all x Z + If we take then, we recover that { 1, if 0 x t f x = 0, if x > t, t lim P X n = k P X = k = 0 It follows that, P X n t P X t 0 as n + We infer that X n d X as n + 3 The Trotter-Renyi distance Before stating the definition of the Trotter-Renyi distance we firstly need the definition of a probability metric Let Ω, A, P be a probability space and let ZΩ, A be a space of real-valued A measurable random variables X : Ω R Definition 31 A functional dx, Y : ZΩ, A ZΩ, A 0, is said to be a probability metric in ZΩ, A if it possesses for the random variables X, Y, Z ZΩ, A the following properties see?,?,?,? and? for more details 1 P X = Y = 1 dx, Y = 0; 4

5 dx, Y = dy, X; 3 dx, Y dx, Z + dz, Y We now return to the main tool establishing our results It is a probability distance based on the Trotter-Renyi opertor in?? We need to be recalled the definition of Trotter -Renyi distance with some main properties see?,? and? Definition 3 The Trotter - Renyi distance X, Y ; f of two random variables X and Y with respect to function f K is defined by X, Y ; f := A X f A Y f = sup x Z + EfX + x EfY + x 8 Based on the properties of the Trotter-Renyi operator, the most important properties of the Trotter - Renyi distance are summarized in the following see?,?,?,? and? for more details 1 It is easy to see that X, Y ; f is a probability metric, ie for the random variables X, Y and Z the following properties are possessed a For every f K, the distance X, Y ; f = 0 if P X = Y = 1 b X, Y ; f = Y, X; f for every f K c X, Y ; f X, Z; f + Z, Y ; f for every f K If X, Y ; f = 0 for every f K, then F X F Y 3 Let {X n, n 1} be a sequence of random variables and let X be a random variable The condition lim n + X n, X; f = 0, for all f K, implies that X n d X as n 4 Suppose that X 1, X, X n ; Y 1, Y, Y n are independent random variables in each group Then, for every f K, n X j, n Y j ; f n X j, Y j ; f Moreover, if the random variables are identically in each group, then we have n X j, n Y j ; f n X 1, Y 1 ; f 5 Suppose that X 1, X, X n ; Y 1, Y, Y n are independent random variables in each group Let {N n, n 1} be a sequence of positive integer-valued random variables that independent of X 1, X,, X n and Y 1, Y,, Y n Then, for every f K, Nn N n X j, Y j ; f P N n = k k=1 k X j, Y j ; f 9 5

6 6 Suppose that X 1, X, X n ; Y 1, Y, Y n are independent identically distributed random variables in each group Let {N n, n 1} be a sequence of positive integer-valued random variables that independent of X 1, X,, X n and Y 1, Y,, Y n Moreover, suppose that EN n < +, n 1 Then, for every f K, we have Nn N n X j, Y j ; f EN n X 1, Y 1 ; f 10 It is easily seen that the Trotter-Reny distance in term of?? and the total variation distance in?? have a close relationship if the function f is chosen as an indicator function I A of a set A Z +, namely X, Y, χ A = d T V X, Y if fx = I A x = { 1, if x A 0, if x / A where we denote by d T V X, Y by the total variation distance between two integer-valued random variables X and Y, defined as follows d T V X, Y = sup P X A P Y A A Z + = 1 P X = k P Y = k k Z + For a deeper discussion of total variation distance, we refer the reader to?,?,?,? and? 4 Bounds in Poisson approximation for Poisson-binomial distribution Let X n,1, X n,, ; n = 1,, ldots be a wise-row tringular array of independent Bernoulli distributed random variables with success probabilities P X n,j = 1 = 1 P X n,j = 0 = p n,j 0, 1, j = 1,,, n; n = 1,, Let Z pn,j, j = 1,,, n; n = 1,, be a tringular array of independent Poisson distributed random variables with positive means p n,1, p n,,, p n,n It is d n clear that Z λn = Z pn,j Then, by an argument analogous to that used for the proof of theorems in?, we can establish the bounds in Poisson approximation via Trotter-Renyi distance Theorem 41 Recall Theorem 31,? Let X n,j, 1 j n, n 1 be a tringular array of wise-row independent Bernuolli distributed random variables with success probabilities P X n,j = 1 = 1 P X n,j = 0 = p n,j 0, 1; 1 j n, n 1 Let us denote by Z λn the Poisson distributed random variable with mean λ n := n p n,j Write S n = n X n,j Then, for all functions f K, S n, Z λn, f f n p n,j 6

7 Proof Using the definition of Trotter-Renyi distance in?? and received bounds in Theorem 31 from?, we have immediate proof Corollary 41 Under the above assumptions of Theorem??, for all k {0, 1,, n}, we have P S n = k P Z λn = k Remark 41 On account of Corollary??, if lim max p n,j = 0 1 j n n p n,j and lim λ n = λ 0 < λ < +, then, S n d Zλ when n Throughout the forthcoming, unless otherwise specified, we shall denote by N n, n 1 a sequence of non-negative integer-valued random variables independent of all X n,j, 1 j n, n 1 Set S Nn = N n X n,j, S 0 = 0 by convention Moreover, assume that N n P as n, here and from now on, the notation P means the convergence in probability Then, in the same way as in proofs of theorems in?, the bounds in Poisson approximation for random sum will be established vie Trotter-Renyi distance Theorem 4 Recall Theorem 41,? Under above assumptions on X n,j, 1 j n, n 1 and N n, n 1 Then, for all functions f K, we have N n S Nn, Z λnn, f f E p N n,j Proof Using the definition of Trotter-Renyi distance in?? with the inequality??, based on received bounds in Theorem 31 from?, we can finish the proof Corollary 4 Under the above assumptions of Theorem??, for all k {0, 1,, n}, P S Nn = k P Z λnn = k N n E p n,j Let us denote by Ω Nn the set of outcomes of the random variables N n, n 1 and h n = max {h : h Ω Nn } Then, according to Theorem??, it follows that Remark 4 According to Corollary?? and let lim max p n,j = 0 1 j h n and lim E N n p n,j = λ 0 < λ < + Then, S Nn d Zλ as n 7

8 For the sake of completeness we consider the results in Poisson approximation for the double arrays of independent Bernoulli distributed random variables We shall consider the following theorems Theorem 43 Let X n,m, n 1, m 1 be a doulbe array of independent Bernoulli distributed random variables with probabilities P X k,j = 1 = 1 P X k,j = 0 = p k,j 0, 1 Let us denote by Z δn,m the Poisson distributed random variable with mean δ n,m = ES nm = n m p k,j Then, for all functions f K, we deduce that Snm, Z δn,m, f f n k=1 m p k,j Proof Our proof starts with the observation that, for all f K and for x Z +, A Xk,j f x A Zpk,j f x = = f x + l P X k,j = l P Z λpk,j = l f x + l P X k,j = l e pk,j p l k,j l! = f x 1 p k,j e p k,j + f x + 1 p k,j p k,j e p k,j It follows that X k,j, ZZ pk,j, f l= l= k=1 f x + l e p k,j p l k,j l! f x 1 p k,j e p k,j + f x + 1 p k,j p k,j e p k,j f x + l e p k,jp l k,j l! = f x 1 pk,j e p k,j + f x + 1 pk,j p k,j e p k,j + e p k,j + p k,j 1 sup f x + p k,j p k,j e p k,j sup x Z+ x Z+ + sup f x x Z+ l= e p k,jp l k,j l! l= f x = sup f x e p k,j + p k,j 1 + p k,j p k,j e p k,j + x Z+ l= = f e p k,j + p k,j 1 + p k,j p k,j e p k,j + 1 e p k,j p k,j e p k,j = f p k,j 1 e p k,j f p k,j Using the properties of Trotter-Renyi distance we have for f K, S nm, Z δn,m, f n k=1 e p k,jp l k,j l! m X k,j, Z pk,j, f f x + l e p kjp l k,j l! Then The proof is complete n m S nm, Z δn,m, f f p k,j k=1 8

9 Corollary 43 On account of the assumptions of Theorem??, for all r {0, 1,, n}, we have P S nm = r P Z δnm = r Proof Let us take the function fx such that { f x + l = 1 if m = r, 0 if m r Set y = x + l, since x, l Z +, it follows that y Z + Then, f = sup x According to Theorem?? we conclude that n k=1 f x = sup f y = 1 y S nm, Z δn,m, f n k=1 m p k,j m p k,j 11 On the other hand, by choosing the function fx, we deduce that S nm, Z δn,m, f = sup A Snm f x A Zδn,m f x x = sup f x + l P S x nm = l P Z δn,m = l = P S nm = r P Z δn,m = r Combining 11 with the last equations, we can assert that The proof is streight-forward P S nm = r P Z δn,m = r n k=1 m p k,j Theorem 44 Let X k,j, 1 j n, n 1 be a wise-row double array of independent Bernoulli distributed random variables with probabilities P X k,j = 1 = 1 P X k,j = 0 = p k,j 0, 1, 1 j n, n 1 Suppose that N n, M m are non-negative integer-valued random variables independent of all X k,j, 1 j n, n 1 Let us denote Z δnn,mm the Poisson distributed random variable with mean δ Nn,Mm = ES NnMm = Nn M m p k,j Then, for all functions f K, we obtain k=1 SNnMm, Z δnn,mm, f N n M n f E k=1 p k,j 9

10 Proof Using definition and properties of Trotter-Renyi operator, we have ASNnMm f x := E f S NnMm + x = E E f S NnMm + x N n = P N n = n E f S nmm + x N n = n = P N n = n E f S nmm + x = P N n = n E f S nmm + x M m = P N n = n P M n = me f S nm + x M m = m = P N n = n P M n = me f S nm + x = P N n = n P M n = m A Snm f x In the same way A ZδNnMm f x = E f Z δnnmm + x = P N n = n P M n = m A Zδnm f x Thus, for all f K, and for x Z +, we obtain S NnM m, Z δnn,mm, f P N n = n P M n = m S nm, Z δn,m, f f P N n = n n P M n = m = f n P N n = n E = f E N n M m p k,j k=1 M m p k,j k=1 m p k,j k=1 Then N n M n S NnMm, Z δnn,mm, f f E k=1 p k,j Remark 43 Note that we have used the fact that min{n n, M m } P as min{n, m} Corollary 44 According to the above Theorem??, for all r {0, 1,, n}, we conclude that P S NnMm = r P Z λnn,mm = r N n M m E k=1 p k,j 10

11 Proof Taking the function fx such that f x + l = { 1 if m = r, 0 if m r Write y = x + l, since x, l Z +, it follows that y Z + Then f = sup x f x = sup f y = 1 y Thus, according to results of Theorem?? we show that N n M m S NnMm, Z δn,m, f p k,j 1 k=1 On the other hand, by choosing the function fx, we conclude that S NnMm, Z δn,m, f = sup A SNnMm f x A ZδNn,Mm f x x = sup f x + l P S x NnMm = l P Z δnn,mm = P S NnM m = r P Z δnn,mm = r = l Combining 1 with the last equations, we have This finishes the proof P SNnM m = r P Z δnn,mm = r N n M m p k,j k=1 5 Bounds in Poisson approximation for distribution of geometric random variables The problem to be considered in this section is that of establshing the bounds in Poisson approximation for geometric random variables This problem has attacted much attention We refer to?-?, for a more general and detailed discussion of this problem It is worth pointing out that Teerapabolarn et al in? used the Stein-Chen method to give a non-uniform bounds in Poisson approximation for geometric random variables in case the Poisson mean λ = n 1 p j, here p j 0, 1, 1 j n, n 1 are parameters of geometric distributed random variables In this subsection, based on Trotter-Renyi distance method Theorems??,??,?? and?? are devoted to the discussion of bounds in Poisson approximation for independent geometric distributed random variables with parameters p n,j 0, 1, 1 j n, n 1, from wise-row triangular or double arrays and for Poisson mean λ n = n 1 p n,j Moreover, via Trotter-Renyi distance four theorems??,??,?? and?? give some results on bounds in case of the Poisson means are λ n = E S n = n 1 p n,j p 1 n,j Let A Xn,j, 1 j n, n 1 be the operators associated with the random variables X n,j, 1 j n, n 1, and let A Zpn,j, 1 j n; n 1 be the operators associated with the Poisson random variables with parameters p n,j, 1 j n, n 1 On account of assumption that Z λn is 11

12 a Poisson distributed random variable with positive parameter λ n = n p n,j, we can perform d that Z λn = n Z p n,j, where Z pn,1, Z pn,,, Z pn,n are independent Poisson distributed random variables with positive parameters p n,1, p n,,, p n,n, and the notation d = denotes coincidence of distributions Theorem 51 Let X n,j, 1 j n, n 1 be a triangular array of wise-row independent, non-identically distributed geometric random variables with parameters p n,j, p n,j 0, 1 Set S n = n X n,j and let us denote by Z λn the Poisson distributed random variable with mean λ n = n 1 p n,j Then, for all functions f K, we have S n, Z λn ; f f Proof Applying the inequality in??, we have S n, Z λn, f n n Xn,j, Z pn,j ; f := 1 p n,j 6 pn,j n A Xn,j f A Zpn,j f Moreover, for all functions f K, for all x Z + and k {0, 1,, n}, we conclude that A Zpn,j f x A Xn,j f x = f x + k P Z pn,j = k P X n,j = k = f x + k e 1 p n,j 1 p n,j k p n,j 1 p n,j k Therefore, for all functions f K, and for all x Z +, we have A Zpn,j f x A Xn,j f x = f x + k e 1 p n,j 1 p nj k 1 p n,k k p n,j f x e 1 pn,j p n,j + f x p n,j e 1 pn,j 1 p n,j p n,j + f x + k e 1 p n,j 1 p n,j k 1 p n,k k p n,j k= sup f x e 1 pn,j p n,j + sup f x 1 p n,j e 1 pn,j 1 p n,j p n,j x Z + x Z + + sup f x e 1 p n,j 1 p n,j k + 1 p n,k k p nj x Z + k= k= p n,j 1 p n,j f + 1 e 1 pn,j 1 p n,j e 1 pn,j + 1 p n,j f 1 p n,j 6 pn,j Thus, This completes the proof S n, Z λn ; f f n 1 p n,j 6 pn,j 1

13 Corollary 51 Under the assumptions in Theorem??, let k {0, 1,, n}, we have P S n = k P Z λn = k n 1 p n,j 6 pn,j Remark 51 According to the Corollary?? and based on the following conditions, we have Then, S n i lim ii d Z λ as n max 1 p n,j = 0, 1 j n lim λ n = n 1 p n,j = λ 0 < λ <, Let A Xn,1, A Xn,,, A XNn,Nn be operators associated with the independent triangular array random variables X n,1, X n,,, X Nn,Nn, and let A Zpn,1, A Zpn,,, A ZpNn,Nn be operators associated with the independent Poisson distributed random variables with positive parameters p n,1, p n,,, p Nn,Nn According to the properties of the linear operator in form of??, we have A SNn = A Xn,1 A Xn, A XNn,Nn and A ZλNn = A Zpn,1 A Zpn, A ZpNn,Nn be the respective operators associated with the random sums S Nn = N n k=1 X n,k and Z λnn = N n k=1 Z p n,k Theorem 5 Let X n,j, 1 j n, n 1 be a row-wise triangular array of independent, geometric distributed random variables with parameters p n,j 0, 1 Moreover, we suppose N n are positive integer-valued random variables, independent of all X n,j, 1 j n, n 1 Let S Nn = Nn X n,j be a random sum and we will denote by Z λnn the Poisson random variable with parameters λ Nn = Nn 1 p Nn,j Then, for all functions f K, we have SNn, Z λnn ; f N n f E 1 p Nn,j 6 pnn,j, f K Proof According to the Theorem??, for all functions f K, x Z +, we have SNn, Z λnn, f P N n = m Xnj, Z pnj ; f P N n = m f m 1 p Nnj 6 pnnj = f m P N n = m 1 p Nnj 6 pnnj N n = f E 1 p Nnj 6 pnnj Therefore, SNn, Z λnn ; f N n f E 1 p Nn,j 6 pnn,j, f K The proof is complete 13

14 Corollary 5 According to Theorem??, let k {0, 1,, n}, we have P SNn = k P Z λnn = k N n E 1 p Nn,j 6 pnn,j We will denote by Ω Nn the set of all results of random variable N n and h n = max {h : h Ω Nn } Then, we have remark below Remark 5 According to the Corollary?? and the following conditions are satisfied Then, S Nn i lim max 1 p Nn,j = 0, 1 j h n ii lim λ N n n = E 1 p n,j = λ 0 < λ < + d Z λ as n Theorem 53 Let X n,m, n 1, m 1 be a doulbe array of independent Geometric distributed random variables with probabilities P X k,j = l = p k,j 1 p k,j l 0 < p k,j < 1 Let us denote by Z δn,m the Poisson distributed random varianle with mean δ n,m = n m 1 p k,j Then, for all functions f K, we have Snm, Z δn,m, f f Proof Applying the inequality in??, we have n m k=1 k=1 6 1 p k,j pk,j S nm, Z δnm, f n W km, Z µkm, f k=1 n k=1 m Wkj, Z λk,j, f According to the theorem??, for all functions f K, x Z +, we have Wk,j, Z λk,j, f 6 1 p k,j pk,j Hence Snm, Z δn,m, f f n m k=1 6 1 p k,j pk,j Corollary 53 Under the assumptions in Theorem??, let k {0, 1,, n}, we have P S nm = k P Z δnm = k n m k=1 6 1 p k,j pk,j Theorem 54 Let X n,m, n 1, m 1 be a wise-row double array of independent Geometric distributed random variables with P X k,j = l = p k,j 1 p k,j l 0 < p k,j < 1 Suppose that N n and M m are non-negative integer-valued random variables independent of all X n,m Let us denote 14

15 by Z δnnmm the Poisson variable with means δ NnMm = Nn f K, we have SNnMm, Z δnnmm, f N n M m f E k=1 Proof According to the difinition??, we have ASNnMm f x := E f S NnM m + x M m k=1 1 p k,j Then, for all functions 6 1 p k,j pk,j = P N n = n P M n = m A Snm f x And A ZδNnMm f x := E f Z δnnmm + x = P N n = n P M n = m A Zδnm f x Thus, for all functions f K, and for all x Z +, we have SNnM m, Z δnnmm, f P N n = n P M m = m S nm, Z δnm, f f P N n = n n m 6 P M m = m 1 p k,j pk,j k=1 = f n M m 6 P N n = n E 1 p k,j pk,j k=1 N n M m 6 = f E 1 p k,j pk,j k=1 Hence, SNnMm, Z δnn,mm, f N n M m f E k=1 6 1 p k,j pk,j Corollary 54 Under the assumptions in Theorem??, let k {0, 1,, n}, we have P SNnMm = k P Z δnnmm = k N n M m E 6 1 p k,j pk,j k=1 Theorem 55 Let X n,j, 1 j n, n 1 be a triangular array of wise-row independent, non-identically distributed geometric random variables with parameters p n,j 0, 1 Set S n = n X n,j and let us denote by Z λn the Poisson random variables with means λ n = E S n = n 1 p nj p 1 n,j Then, for all functions f K, we have S n, Z λn ; f f n 1 p n,j + 1 p n,j p n,j 15

16 Proof We have ASn f A Zλn f n A Xn,j f A Zpn,j f Hence, For all functions f K,and for all x Z +, we have A Zpn,j f x A Xn,j f x = f x + k P Z pn,j = k P X n,j = k = f x + k e 1 p n,jp 1 A Zpn,j f x A Xn,j f x = f x + k sup f y y Z+ = f 1 p n,jp 1 n,j n,j k e 1 p n,jp 1 1 p n,jp 1 n,j n,j k e 1 p n,jp 1 1 p n,jp 1 n,j n,j k e 1 p n,jp 1 1 p n,jp 1 n,j n,j k p n,j 1 p n,j k f x + k e 1 p n,jp 1 1 p n,jp 1 n,j n,j k 1 p n,j k p n,j 1 p n,j k p n,j 1 p n,j k p n,j e 1 p n,j p 1 n,j e p n,j + 1 p n,j p 1 n,j 1 p n,j p 1 = f + k= f p 1 n,j p n,j = f = f e 1 p n,jp 1 1 p n,jp 1 n,j + 1 p n,j p 1 n,j + e 1 p n,jp 1 k 1 p n,jp 1 n,j n,j k 1 p n,j k p n,j n,j 1 p n,j p n,j n,j k 1 p n,j k p n,j p 1 n,j p n,j + 1 p n,j k p n,j k p 1 n,j p n,j + 1 p n,j p 1 n,j p 1 n,j p n,j + 1 e 1 p n,jp 1 n,j 1 p n,j p 1 n,j e 1 p n,jp 1 n,j + 1 p n,j p n,j 4p n,j + p n,j e 1 p n,jp 1 n,j 1 p n,j p 1 n,j e 1 p n,jp 1 n,j f p n,j 4p n,j + p n,j + p 1 n,j 1 p n,j p 1 n,j + 1 p n,j p 1 n,j 1 p n,j p 1 f p n,j 4p n,j + p n,j + p 1 n,j 1 p n,j p 1 n,j + 1 p n,j p 1 n,j 1 p n,j p 1 n,j = f p n,j 4p n,j + p n,j p 1 n,j + = f 1 p n,j + 1 p n,j p n,j Hence, A Xn,j f A Zpn,j f = A Zpn,j f A Xnj f f 1 p n,j + 1 p n,j p n,j Thus, This completes proof S n, Z λn ; f f n 1 p n,j + 1 p n,j p n,j n,j e 1 p n,jp 1 n,j 16

17 Corollary 55 Under the assumptions in Theorem??, let k {0, 1,, n}, we have n P S n = k P Z λn = k 1 p n,j + 1 p n,j p n,j Theorem 56 Let X n,j, 1 j n, n 1 be a row-wise triangular array of independent, geometric distributed random variables with parameters p n,j 0, 1 Moreover, we suppose that N n are independent positive integer-valued random variables, independent of all X n,j, 1 j n, n 1 Let S Nn = Nn be a random sum and let us denote by Z λnn the Poisson distributed random variable with mean λ Nn = E S Nn = Nn functions f K, we have SNn, Z λnn ; f N n f E 1 p Nn, j + 1 p N n,j p N n,j 1 p Nn j p 1 N nj 0 < p N nj < 1 Then, for all, f K Proof According to the Theorem??, for all functions f K, and for all x Z +, we have SNn, Z λnn ; f P N n = m S m, Z λm ; f P N n = m f m 1 p Nn,j + 1 p Nn,j p Nn,j = f m P N n = m 1 p Nn,j + 1 p Nn,j p Nn,j N n = f E 1 p Nn,j + 1 p Nn,j p Nn,j Thus, SNn, Z λnn ; f N n f E 1 p Nn, j + 1 p N n,j p N n,j, f K The proof is complete Corollary 56 According to the Theorem??, let k {0, 1,, n}, we have P SNn = k P Z λnn = k N n E 1 p n,j + 1 p n,j p n,j Theorem 57 Let X n,m, n 1, m 1 be a doulbe array of independent geometric distributed random variables with probabilities P X k,j = l = p k,j 1 p k,j l 0 < p k,j < 1 Let us denote by Z δn,m the Poisson distributed random variable with mean δ n,m = E S nm = n m 1 p k,j p 1 k,j Then, for all functions f K, we have k=1 Snm, Z δn,m, f f n m k=1 1 p kj + 1 p kj p kj 17

18 Proof Applying the inequality in??, we have Snm, Z δn,m, f n W km, Z µkm, f k=1 n k=1 m Wkj, Z λk,j, f According to the Theorem??, for all functions f K, and for all x Z +, we have Thus, This proof is complete Skj, Z λk,j, f f Snm, Z δn,m, f f n k=1 1 p k,j + 1 p kj p k,j m 1 p k,j + 1 p k,j p k,j Corollary 57 According to the Theorem??, let r {0, 1,, n}, we have P Snm = r P Z δn,m = r n m 1 p k,j + 1 p k,j p k,j k=1 Theorem 58 Let X n,m, n 1, m 1 be a a wise-row double array of independent geometric distributed random variables with probabilities P X k,j = l = p k,j 1 p k,j l 0 < p k,j < 1 Suppose that N n, M m are non-negative integer-valued random variables independent of all X n,m, n 1, m 1 Let us denote by Z δnnmm the Poisson distributed random variable with mean δ Nn,Mm = E S NnMm = Nn M m k=1 SNnMm, Z δnnmm, f N n M m f E 1 p kj p 1 k,j Then, for all functions f K, we have k=1 Proof According to the difinition??, we have ASNnMm f x := E f S NnM m + x 1 p k,j + 1 p k,j p k,j = P N n = n P M n = m A Snm f x and A ZδNnMm f x := E f Z δnnmm + x = P N n = n P M n = m A Zδnm f x Hence, for all functions f K, and for all x Z +, we have SNnMm, Z δnnmm, f P N n = n P M m = m S nm, Z δnm, f f P N n = n n P M m = m = f n P N n = n E = f E N n M m k=1 M m k=1 1 p k,j + 1 p k,j p k,j m k=1 1 p k,j + 1 p k,j p k,j 1 p k,j + 1 p k,j p k,j 18

19 Thus, This finishes the proof SNnMm, Z δnnmm, f N n M m f E k=1 1 p k,j + 1 p kj p k,j Corollary 58 According to the Theorem??, let r {0, 1,, n}, we have P S NnMm = r P Z δnnmm = r N n M m E 1 p k,j + 1 p kj p k,j k=1 6 Bounds in Poisson approximation for distribution of negativebinomial random variables The received results in this subsection are generalized ones in above part For r = 1 we shall obtain the established bounds in Poisson approximation for geometric random variables Theorem 61 Let X n,j, 1 j n, n 1 be a row-wise triangular array of independent, non-identically negative-binomial random variables with parameters p n,j 0, 1, r Z + Write S n = n X n,j and let us denote by Z λn with means λ n = n r 1 p n,j Then, for all functions f K, we have S n, Z λn ; f f n Proof Applying the inequality in??, we have S n, Z λn, f r 1 r1 p n,j r 3 p n,j n Xn,j, Z pn,j ; f := For all functions f K, and for all x Z +, we have A Zpnj f x A Xnj f x p l n,j n A Xn,j f A Zpn,j f = f x + k P Z pn,j = k P X n,j = k = f x + k e r1 p n,j r1 p nj k Cr+k 1 k pr n,j 1 p n,j k 19

20 Inferred A Zpn,j f x A Xn,j f x = f x + k e r1 p n,j r1 p nj k Cr+k 1 k 1 p n,j k pn,j r f x + k e r1 p n,j r1 p nj k Cr+k 1 k 1 p n,j k p r n,j sup f y e r1 p j r1 p nj k Cr+k 1 k 1 p n,j k p r n,j y Z + = f e r1 p n,j r1 p n,j k Cr+k 1 k 1 p n,j k p r n,j e r1 p = f n,j p 1 j p r n,j + e r1 pn,j r 1 p n,j r 1 p n,j p r n,j + e r1 p n,jp 1 r1 p j n,j k Cr+k 1 k 1 p n,j k p r n,j k= e f r1 p n,j p r n,j + e r1 pn,j r 1 p n,j r 1 p n,j p r n,j + e r1 p n,j r1 p n,j k + Cr+k 1 k 1 p n,j k p r n,j k= k= Moreover, we have e r1 pn,j p r n,j + r 1 p n,j e r1 pn,j r 1 p n,j p r n,j Inferred Thus r 1 p n,j + r 1 p n,j e r1 p n,j r 1 p n,j k r 1 p n,j Cr+k 1 k 1 p n,j k p r n,j r1 p n,j r 1 p l n,j k= k= Xn,j, Z pn,j ; f f r1 p n,j r 3 p n,j S n, Z λn ; f f This completes the proof n r 1 p l n,j r 1 r1 p n,j r 3 p n,j Corollary 61 Under the assumptions of Theorem??, let k {0, 1,, n}, we have P S n = k P Z λn = k 1 n r 1 r1 p nj r 3 p n,j p l n,j Remark 61 According to the Corollary?? and the following conditions are satisfied Then, S n i lim ii d Z λ as n max 1 p nj = 0, 1 j n lim λ n = n r 1 p nj = λ 0 < λ <, 0 p l n,j

21 Theorem 6 Let X n,j, 1 j n, n 1 be a row-wise triangular array of independent, nonidentically negative-binomial random variables with parameters p n,j 0, 1, r Z + Moreover, we suppose that N n are independent positive integer-valued random variables, independent of all X n,j, 1 j n, n 1 Let S Nn = Nn are random sums and let us denote by Z λnn the Poisson random variables with means λ Nn r1 p Nnj 0 < p < 1, r {1,, } Then, for all = Nn functions f K, we have SNn, Z λnn ; f f E N n r1 p Nn,j r n,j r 1 3 p Nn,j p l N Proof According to the Theorem?? and 10, for all functions f K, and for all x Z +, we have SNn, Z λnn, f = f P N n = m Xn,j, Z pn,j ; f P N n = m f m r1 p Nnj r 3 p Nn,j r 1 m P N n = m = f E N n p l N n,j r 1 r1 p Nnj r 3 p Nn,j r 1 r1 p Nn,j r 3 p Nn,j p l N n,j Therefore, SNn, Z λnn ; f f E N n r1 p Nn,j r n,j r 1 3 p Nn,j p l N, f K The proof is complete Corollary 6 According to the Theorem??, let k {0, 1,, n}, we have P S Nn = k P Z λnn = k 1 E N n r1 p Nn,j r nj r 1 3 p Nn,j p l N n,j p l N Let us denote by Ω Nn the set of all results of random variable N n and h n = max {h : h Ω Nn } Then, we have remark below Remark 6 According to the Corollary?? and the following conditions are satisfied Then, S Nn i lim max 1 p Nnj = 0, 1 j h n ii lim λ N n n = E r 1 p nj = λ 0 < λ < + d Z λ as n 1

22 Theorem 63 Let X n,m, n 1, m 1 be a doulbe array of independent negative-binomial distributed random variables with probabilities P X kj = l = Cr+l 1 l 1 p i l p r i, l = 0, 1, ; r = 1,, n m Let us denote by Z δn,m the Poisson random variables with means δ n,m r1 p k,j Then, for all functions f K, we have Snm, Z δn,m ; f f n m k=1 Proof Applying the inequality in??, we have k=1 r 1 r1 p k,j r 3 p k,j S nm, Z δnm, f n W km, Z µkm, f k=1 n k=1 m Wkj, Z λkj, f According to the Theorem??, for all functions f K, x Z +, we obtain Wk,j, Z λk,j, f f r1 p r 1 k,j r 3 p k,j Hence Snm, Z δn,m, f f n m k=1 p l k,j r 1 r1 p k,j r 3 p k,j p l k,j p l k,j Corollary 63 According to the Theorem??, let k {0, 1,, n}, we have P S nm = r P Z λnm = r 1 n m k=1 r 1 r1 p k,j r 3 p k,j Theorem 64 Let X n,m, n 1, m 1 be a wise-row double array of independent negativebinomial distributed random variables with P X kj = l = Cr+l 1 l 1 p i l p r i, l = 0, 1, ; r = 1,, Suppose that N n, M m are non-negative integer-valued random variables independent of all X n,m, n 1, m 1 Let us denote by Z δnnmm the Poisson variable with means δ NnMm = Nn M m r1 p k,j Then, for all functions f K, we have SNnMm, Z δnnmm ; f f E N n M m k=1 Proof According to the difinition??, we have ASNnMm f x := E f S NnM m + x p l k,j k=1 r 1 r1 p k,j r 3 p k,j p l k,j = P N n = n P M m = m A Snm f x And A ZδNnMm f x := E f Z δnnmm + x = P N n = n P M m = m A Zδnm f x

23 Thus, for all functions f K, x Z +, we have SNnM m, Z δnnmm ; f P N n = n P M n = m S nm, Z δnm ; f f P N n = n n m P M m = m = f n M m P N n = n E k=1 = f E N n M m k=1 k=1 r 1 r1 p k,j r 3 p kj r 1 r1 p k,j r 3 p k,j r 1 r1 p k,j r 3 p k,j Hence, SNnMm, Z δnn,mm, f f E N n M n k=1 p l k,j p l k,j r 1 r1 p k,j r 3 p k,j p l k,j p l k,j Corollary 64 According to the Theorem??, let k {0, 1,, n}, we have P SNnMm = r P Z λnnmm = r 1 E N n M n r1 p k,j r k,j r 1 3 p k,j p l k=1 Theorem 65 Let X n,j, 1 j n, n 1 be a tringular array of wise-row independent negative-binomial distributed random variables with parameters p n,j 0, 1, r Z + Set S n = n X n,j, and let us denote by Z λn the Poisson random variables with means λ n = E S n = n r 1 p n,j p 1 n,j Then, for all functions f K, we have S n, Z λn, f f Proof Consider n rp n,j r 1 p r n,j + r r p 1 n,j + r p n,j + r r A Zpn,j f x A Xn,j f x = f x + k P Z pn,j = k P X n,j = k = f x + k e r1 p n,jp 1 r1 p n,jp 1 n,j n,j k C k r+k 1 pr n,j 1 p n,j k 3

24 It follows that A Zpn,j f x A Xn,j f x = f x + k e r1 p n,jp 1 r1 p n,jp 1 n,j n,j k Cr+k 1 k 1 p n,j k pn,j r f x + k e r1 p n,jp 1 r1 p n,jp 1 n,j n,j k Cr+k 1 k 1 p n,j k pn,j r sup f y e r1 p jp 1 r1 p n,jp 1 nj n,j k Cr+k 1 k 1 p n,j k p r n,j y Z + = f e r1 p n,jp 1 r1 p njp 1 n,j n,j k Cr+k 1 k 1 p n,j k p r n,j e r1 p = f n,j p 1 j p r n,j + e r1 p n,jp 1 n,j r 1 p n,j p 1 n,j r 1 p n,j p r n,j + e r1 p n,jp 1 r1 p n,j p 1 j n,j k Cr+k 1 k 1 p n,j k p r n,j k= e r1 p f n,j p 1 n,j p r n,j + e r1 p n,jp 1 n,j r 1 p n,j p 1 n,j r 1 p n,j p r n,j + e r1 p n,jp 1 r1 p n,jp 1 n,j n,j k + Cr+k 1 k 1 p n,j k p r n,j k= k= We first obsevrve that e r1 p n,jp 1 n,j p r n,j rp 1 n,j r + 1 pr n,j, and e r1 p n,jp 1 n,j r 1 p n,j p 1 n,j r 1 p n,j p r n,j rp r+1 n,j rpr n,j+ r r p 1 n,j +r p n,j +r r Moverover, k= e r1 p n,jp 1 n,j r 1 p n,j p 1 n,j k = 1 e e1 p n,jp 1 n,j r 1 p n,j p 1 n,j e r1 p n,jp 1 n,j and k= C k r+k 1 1 p n,j k p r n,j = 1 r 1 p n,j p r n,j p r n,j = 1 rp r n,j + rp r+1 n,j p r n,j Therefore, it follows that Xn,j, Z pn,j, f f rp n,j r 1 p r n,j + r r p 1 n,j + r p n,j + r r + 1 and using the inequality S n, Z λn, f n Xn,j, Z pn,j, f for all f K, x Z + This concludes the proof 4

25 Corollary 65 On account of assumptions of above theorem??, for k {0, 1,, n}, we have P S n = k P Z λn = k n rp n,j r 1 p r n,j + r r p 1 n,j + r p n,j + r r + 1 Theorem 66 Let X n,j, 1 j n, n 1 be a tringular array of wise-row independent negative-binomial distributed random variables with parameters p n,j 0, 1, r 1 Moreover, assume that N n, n 1 are non-negative integer-valued random variables independent of all X n,j, 1 j n; n 1 Set S Nn := Nn X Nn,j, with convention S 0 = 0 We will denote by Z λnn the Poisson random variables with means λ Nn = E S Nn = Nn functions f K, we have N n S Nn, Z λn, f f E Proof We shall begin with showing that and It follows that A SNn f x A ZλNn f x = A SNn f x = A ZλNn f x = r 1 p Nn,j p 1 N n,j Then, for all rp Nnj r 1 p r N n,j + r r p 1 N n,j + r p P N n = m f x + k P S m = k P N n = m f x + k P Z λm = k Then, we conclude that, for every function f K SNn, Z λnn, f = f = f E P N n = m f m N n N n,j + r r + 1 P N n = m f x + k P S m = k f x + k P Z λm = k P N n = m m This completes the proof P N n = m S m, Z λm, f rp m,j r 1 p r m,j + r r p 1 m,j + r p m,j + r r + 1 rp m,j r 1 p r m,j + r r p 1 m,j + r p m,j + r r + 1 rp m,j r 1 p r m,j + r r p 1 m,j, + r p m,j + r r + 1 Corollary 66 On account of assumptions in Theorem??, for allk {0, 1,, n}, we have P S Nn = k P Z λnn = k N n E rpnn,j r 1 p r N n,j + r r p 1 N n,j + r p N n,j + r r + 1 5

26 Theorem 67 Let X nm, n 1, m 1 be a wise-row double array of independent negativebinomial distributed random variables with parameter p k,j 0, 1 Let us denote by Z δn,m the Poisson random variables with means δ m,n = E S nm = n m r 1 p k,j p 1 k,j Then, for all functions f K, we have Smn, Z δm,n, f f n m k=1 Proof It is easily seen that k=1 rp k,j r 1 p r k,j + r r p 1 k,j + r p k,j + r r + 1 Snm, Z δm,n, f n k=1 m Xk,j, Z δk,j, f On account of the above theorem, we have Xk,j, Z δk,j, f f rp k,j r 1 p r k,j + r r p 1 k,j + r p k,j + r r + 1 Thus, Snm, Z δm,n, f f n m k=1 rp k,j r 1 p r k,j + r r p 1 k,j + r p k,j + r r + 1 Corollary 67 Under the assumptions of Theorem??, for all l {0, 1,, n}, we have P S nm = l P Z δn,m = l n m k=1 rp k,j r 1 p r k,j + r r p 1 k,j + r p k,j + r r + 1 Theorem 68 Let X n,m, n 1, m 1 be a double array of independent negative-binomial distributed random variables with parameters p k,j 0, 1 Suppose that N n, n 1 and M m, m 1 are non-negative integer-valued random variables independent of all X n,m, n 1, m 1 Set S NnMm := N n Mm k=1 X k,j Let us denote by Z δnn,mm the Poisson variables with means δ Nn,Mm = E S NnMm = Nn M m k=1 SNnMm, Z δnn,mm, f N n M n f E k=1 r 1 p k,j p 1 k,j Then, for all functions f K, we have rp k,j r 1 p r k,j + r r p 1 k,j + r p k,j + r r + 1 Proof According to definition of Trotter - Renyi distance, for all functions f K, x Z +, we have SNnMm, Z δnn,mm, f P N n = n P M n = m Snm, Z δn,m, f N n M n f E k=1 rp k,j r 1 p r k,j + r r p 1 k,j + r p 6 k,j + r r + 1

27 Then SNnMm, Z δnn,mm, f N n M n f E k=1 rp k,j r 1 p r k,j + r r p 1 k,j + r p k,j + r r + 1 Corollary 68 Under assumptions of Theorem??, for all l {0, 1,, n}, we have P SNnMm = l P Z δnnmm = l N n M n E k=1 rp kj r 1 p r kj + r r p 1 kj + r p kj + r r + 1 Remark 63 When r = 1 we get the results related to Poisson approximation for geometric random variables in subsection 5 7 Bounds in Poisson approximation for distribution of integervalued random variables Let A Xn,j, 1 n, j 1 be the operators associated with the random variables X n,j, j 1, n 1, and let A Zpn,j, j 1, n 1 be the operators associated with the Poisson distributed random variables with parameters p n,j, 1 j n, n 1 On the assumption that Z λn is a Poisson distributed random variable with positive parmeter λ n = n p n,j, we can perform d that Z λn = n Z p n,j, where Z pn,1, Z pn,,, Z pn,n are independent Poisson distributed random variables with positive parameters p n,1, p n,,, p n,n, and the notation d = denotes coincidence of distributions Theorem 71 Let X n,m, n 1, m 1 be a row-wise triangular array of independent, nonidentically integer-valued random variables with probabilities P X n,j = 1 = p n,j, q n,j = 1 p n,j P X n,j = 0 p n,j 0, 1; 1 j n, n 1 Let us write S n = n X n,j and λ n = n p n,j We will denote by Z λn the Poisson distributed random variable with parameter λ n Then, for all functions f K, we have S n, Z λn ; f f Proof Applying the inequality in??, we have S n, Z λn, f n p n,j + q n,j n Xn,j, Z pn,j ; f := n A Xn,j f A Zpn,j f k=1 7

28 Moreover, for all f K, for all x Z + and r {0, 1,, n} we conclude that A Xnj f x A Zpn,j f x = r=0 = r=0 f x + r P X nj = r P Z λpn,j f x + r P X nj = r e pn,j p r n,j r! = f x 1 p n,j q n,j e p n,j +f x + 1 p n,j p n,j e p n,j + Therefore, for all functions f K, and for all x Z +, we have A Xn,j f x A Zpn,j f x r= = f x 1 p n,j q n,j e p n,j + f x + 1 p n,j p n,j e p n,j + r= f x + r P X n,j = r e pn,j p r n,j r! = r f x + r P X n,j = r e pn,j p r n,j r! = f x 1 p n,j q n,j e p n,j + f x + 1 p n,j p n,j e p n,j + r= f x + r P X n,j = r e pn,j p r n,j r! f x 1 p n,j q n,j e p n,j + f x + 1 p n,j p n,j e p n,j + f x + r P X n,j = r + r= e p n,j + p n,j + q n,j 1 sup x Z r= f x + r e p n,j p r n,j r! f x + p n,j p n,j e p n,j sup x Z f x + sup f x P X n,k = r + sup e f x pn,j p r n,j x Z r= x Z r! r= = sup f x e p n,j + p n,j + q n,j 1 + p n,j p n,j e p n,j + q n,j + 1 e p n,j p n,j e p n,j x Z = f p n,j p n,j e p n,j + q n,j f p n,j + q n,j Inferred f K, A Xn,j f A Zpn,j f f p n,j + q n,j Therefore, applying??, we can assert that n S n, Z λn ; f f p n,j + q n,j This completes the proof Corollary 71 Under the assumptions in Theorem??, let r {0, 1,, n}, we have n P S n = r P Z λn = r p n,j + q n,k 8

29 Remark 71 According to the Corollary?? and the following conditions are satisfied Then, S n i lim ii iii d Z λ as n n q n,j = 0, lim max p n,j = 0, 1 k n n p n,j = λ, 0 < λ < +, lim λ n = lim Theorem 7 Let X n,j, 1 j, n 1 be a row-wise triangular array of independent, nonidentically integer-valued random variables with probabilities P X n,j = 1 = p n,j, q n,j = 1 p n,j P X n,j = 0 p n,j 0, 1; 1 j n, n 1 Moreover, we suppose that N n, n 1 are positive integer-valued random variables, independent of all X n,j, 1 j n, n 1 Let us write S Nn = Nn X n,j and λ Nn = Nn p n,j We will denote by Z λnn the Poisson distributed random variable with parameter λ Nn Then, for all functions f K, we have SNn, Z λnn ; f N n f E p Nn,j + q Nn,j Proof According to the Theorem?? and??, for all functions f K, and for all x Z +, we have Therefore, The proof is complete SNn, Z λnn ; f = f = f E P N n = m S m, Z λm ; f P N n = m f m N n P N n = m m p N + q n,j N n,j p N n,j + q N n,j p N + q n,j N n,j SNn, Z λnn ; f N n f E p Nn,j + q Nn,j Corollary 7 According to theorem??, let r {0, 1,, n}, we have P S Nn = r P Z λnn = r N n E p Nn,j + q Nn,j Let us denote by Ω Nn the set of all outcomes of random variable N n and h n = max {h : h Ω Nn } Then, we have 9

30 Remark 7 According to the Corollary?? and the following conditions are satisfied Then, S Nn i lim ii iii d Z λ as n N n q Nn,j = 0, lim max p Nn,j = 0, 1 k h n lim E N n p Nn,j = λ, 0 < λ < + Theorem 73 Let X n,m, n 1, m 1 be a doulbe array of independent integer-valued distributed random variables with probabilities P X n,k = 1 = p n,k, q n,k = 1 p n,k = P X n,k = 0, p n,k 0, 1; 1 k n, n 1 Let us denote by Z δn,m the Poisson distributed random variable with means δ n,m = n m p k,j Then, for all f K, we have k=1 Snm, Z δn,m, f f Proof Applying the inequality in??, we have n k=1 m p k,j + q k,j S nm, Z δnm, f n Skm, Z µk,m, f k=1 n k=1 m Sk,j, Z λk,j, f According to the Theorem??, for all functions f K,and for all x Z +, we conclude that Sk,j, Z λk,j, f f p k,j + q k,j Therefore, n m S nm, Z δnm, f f p k,j + q k,j k=1 Theorem 74 Let X n,m, n 1, m 1 be a wise-row double array of independent integervalued distributed random variables with P X n,k = 1 = p n,k ; q n,k = 1 p n,k = P X n,k = 0; p n,k 0, 1; 1 k, n 1 Suppose that N n, M m are non-negative integer-valued random variables independent of all X n,m, n 1; m 1 Let us denote by Z δnnmm the Poisson distributed random variable with mean δ NnMm = ES NnMm = Nn M m p k,j Then, for all functions f K, we have k=1 SNnMm, Z δnnmm, f N n M n f E p k,j + q k,j Proof According to the definition??, we have ASNnMm f x := E f S NnM m + x k=1 = P N n = n P M n = m A Snm f x 30

31 and A ZδNnMm f x := E f Z δnnmm + x = P N n = n P M n = m A Zδnm f x Therefore, for all functions f K, and for all x Z +, we have A SNnMm f A ZδNnMm f Thus, P N n = n P M n = m A Snm f A Zδn,m f f P N n = n n m P M n = m p k,j + q k,j k=1 = f n M m P N n = n E p k,j + q k,j = f E N n k=1 k=1 M m p k,j + q k,j SNnMm, Z δnn,mm, f N n M n f E p k,j + q k,j k=1 Remark 73 When q k,j = 0 or q n,k = 0 or q Nn,k = 0 we get the received results in section 3 Conclusions The probability distance method based on Trotter-Renyi operator has been presented the simple but efficient computation for establishing the bounds in Poisson approximation for various classes of independent discrete random variables This tool will certainly be more effective in Poisson approximation on vector spaces Acknowledgments The work was done partially while the first author was visiting the Institute for Mathematical Sciences, National University of Singapore in 013 The visit was supported by the Institute References 1 Arratia, R, Goldstein, L and Gordon, L, Poisson approximation and the Chen-Stein method, Statistical Science, Vol 5, , 1990 Barbour, A D, Holst, L and Janson, S, Poisson Approximation, Clarendon Press-Oxford, Barbour, A D and Chen, L H Y,, An introduction to Stein s method, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, Chen, L H Y, On the convergence of Poisson binomial to Poisson distribution, The Annals of Probability, Vol, No 1, , Chen, L H Y, Poisson approximation for dependent trials, The Annals of Probability, Vol 3, ,

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