A New Approach to Poisson Approximations

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1 A New Approach to Poisson Approximations Tran Loc Hung and Le Truong Giang March 29, 2014 Abstract The main purpose of this note is to present a new approach to Poisson Approximations. Some bounds in Poisson Approximations in term of classical Le Cam s inequalities for various row-wise triangular arrays of independent Poisson-binomial distributed random variables are established via probability distances based on Trotter-Renyi operators. Some analogous results related to random sums in Poisson Approximations are also considered. Keywords: Poisson approximation, Random sums, Le Cam s inequality, Trotter s operator, Renyi s operator, Poisson-binomial random variables, Probability distance Mathematics Subject Classification 2010: 60F05, 60G50, 41A36. 1 Introduction Let X n,j, 1 j n, n 1 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, 1 j n, n 1, such that ˆp n := max p n,j 0 as n. Set 1 j n S n = n X n,j. The random variables S n, n 1 are often called the Poisson-binomial random variables. Write λ n = ES n = n p n,j, and suppose that lim λ n = λ, 0 < n 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, used to being approximated by the distribution of Z λ. More specifically, under above assumptions on ˆp n and λ, the Poisson approximation theorem states that S n d Z λ as n. 1.1 This work is supported by the Vietnam s National Foundation For Science and Technology Development NAFOSTED, Vietnam, code University of Finance and Marketing, 306 Nguyen Trong Tuyen str., Tan Binh Dist., Ho Chi Minh city, Vietnam. tlhungvn@gmail.com truonggiang.gsct@gmail.com 1

2 For other surveys, see [2], [10], [13], [28], [26]. Note that, and from now on, the notation d means the convergence in distribution. Moreover, Le Cam [23], 1960 established the remarkable inequality n P S n = k P Z λ = k 2 p 2 n,j. 1.2 k=0 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 through the total variation distance d T V S n, Z λ d T V S n, Z λ := 1 n P S n = k P Z λ = k p 2 2 n,j. 1.3 k=0 The inerested reader is referred to Barbour, Host and Janson [2], Le Cam [23], Chen [6], Gut [13], Neammanee [24], Steele [30], and Ross [26] for more details. The Poisson approximation for the Poisson-binomial distribution has received extensive attention in the literature and many different approaches have been proposed. During the last several decades various powerful mathematical tools for establishing the Le Cam s inequality have been demonstrated such as the coupling technique, the Stein-Chen method, the semi-group method, the operator method,... However, it should be noted that among various powerful methods, the Stein-Chen method with or without couplings is undoubtedly the most widely used and the most fruitful one. In recent years, finding simpler methods and more efficient is still one of the main task related to the Poisson approximation problems. Results of this nature may be found in [2], [24], [30], [8]. Recently, one of new approaches to Poisson approximation have been used in providing the Le Cam-style inequalities due to Hung and Thao in [17]. Specifically, for f K n A Sn f A Zλn f 2 f p 2 n,j, 1.4 where K is noted the class of all real-valued bounded functions f on the set of all nonnegative integers Z + := {0, 1, 2,..., n}. The norm of function f K is defined by f = sup f x. The linear operators associated with S n and Z λn denoted by A Sn and x Z + 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 = E fx + x = fx + kp X = k, f K, x Z k=0 Recall the definition in [28] and [17]. It is to be noticed that the linear operator defined in 1.5 is actually a discrete form of the Trotter s operator we refer the readers to Trotter [31], Renyi [28] and Hung [17], 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, 1 j n, n 1, the bounds in 1.4 is established as follows Nn A SNn f A ZλNn f 2 f E, p 2 n,j

3 for f K. See, [17] for more details. It is to be noticed that 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. And Trotter 1959, [31] was one of the first mathematicians who succeeded in using the linear operator in order to get elementary proofs in central limit theorem for sums of independent random variables. The Trotter s idea have been used in many areas of probability theory and related fields. For a deeper discussion of Trotter s operator we refer the reader to Trotter in [31], Feller in [10], Renyi in [28], Rychlick and Szynal in [27], Cioczek and Szynal in [7], Kirschfink in [21], Hung in [14], [15], Hung and Thao in [17], Hung and Thanh in [18]. We wish to provide in this paper a new probability distance based on the Trotter- Renyi operator in 1.5, which establishes many new effective bounds of Le Cam s style inequalities for independent Poisson-binomial distributed random variables. The received results in this paper are extensions and generalized of known results in [23], [30]. The present paper is a continuation of an earlier one see [17], too. This paper is organized as follows. We begin with the notations and definitions of Trotter operator, Renyi operator and Trotter-Renyi probability distance with some main properties in the next section. The sections 3 is devoted to the discussions of establishing the Le Cam s inequalities, based on Trotter-Renyi distance, for independent Poissonbinomial distributed random variables. The analogous results related to random sums in Poisson approximation problems have also considered in this sections. Several comments are called for in connection with the new approach to Poisson approximation problems in last section. 2 Preliminaries In the sequel we shall need some properties of Trotter-Renyi operator in 1.5, which has been used for a long time in various studies of classical limit theorems for sums of independent random variables see Trotter [31], Renyi [28], Cioczek [7], Kirschfink [21], Hung [14] and [15], for the complete bibliography. It is to be noticed that the linear operator defined in 1.5 actually is a discrete form of the Trotter s operator we refer the readers to Trotter [31], Renyi [28], Cioczek [7], Kirschfink [21], Hung [14] and [15], 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 R 1. A X f + g = A X f + A X g. 2. A X αf = αa X f. 3. A X f f. 4. A X f + A Y f A X f + A Y f. 3

4 5. Suppose that A X, A Y are operators associated with two independent random variables X and Y. Then, for all f K, A X+Y f = A X A Y f = A Y A X f. In fact, for all x Z + A X+Y fx = fx + lp X + Y = l = l=0 = A X A Y fx = A X A Y fx. fx + k + rp Y = kp X = r 6. Suppose that A X1, A X2,..., A Xn are the operators associated with the independent random variables X 1, X 2,..., X n. Then, for all f K, A Sn f = A X1 A X2... A Xn f is the operator associated with the partial sum S n = X 1 + X X n. 7. Suppose that A X1, A X2,..., A Xn and A Y1, A Y2,..., A Yn are operators associated with independent random variables X 1, X 2,..., X n and Y 1, Y 2,..., Y n. Moreover, assume that all X i and Y j are independent for i, j = 1, 2,..., n. Then, for every f K, Clearly, r,k=0 n A n k=1 X f k A n k=1 Y f A k Xk f A Yk f. 2.7 k=1 A X1 A X2... A Xn A Y1 A Y2... A Yn = n A X1 A X2... A Xk 1 A Xk A Yk A Yk+1... A Yn. k=1 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 8. A n X f An Y f n A Xf A Y f. Lemma 2.1. 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 X2,..., A Xn,... be a sequence of Trotter-Renyi s operators associated with the independent discrete random variables X 1, X 2,..., 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 4

5 Lemma 2.2. A sufficient condition for a sequence of random variables X 1, X 2,..., 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. n Proof. Since lim A Xn f A X f = 0, for all f K, we conclude that n lim n f x + k P X n = k P X = k = 0, for all f K and for all x k=0 Z +. If we take then, we recover that f x = { 1, if 0 x t 0, if x > t, t lim P X n = k P X = k = 0. n k=0 It follows that, P X n t P X t 0 as n +. We infer that X n n +. d X as 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 2.1. 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 [2], [19], [32], [33] and [21] for more details 1. P X = Y = 1 dx, Y = 0; 2. 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 1.5. We need to be recalled the definition of Trotter -Renyi distance with some main properties see [21], [14] and [15]. Definition 2.2. The Trotter - Renyi distance d T R X, Y ; f of two random variables X and Y with respect to function f K is defined by d T R X, Y ; f := A X f A Y f = sup x Z + EfX + x EfY + x. 2.8 Based on the properties of the Trotter-Renyi operator, the most important properties of the Trotter - Renyi distance are summarized in the following see [31], [28], [21], [14] and [15] for more details. 5

6 1. It is easy to see that d T R X, Y ; f is a probability metric, i.e. for the random variables X, Y and Z the following properties are possessed a For every f K, the distance d T R X, Y ; f = 0 if P X = Y = 1. b d T R X, Y ; f = d T R Y, X; f for every f K. c d T R X, Y ; f d T R X, Z; f + d T R Z, Y ; f for every f K. 2. If d T R 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 + d T RX n, X; f = 0, for all f K, implies that X n d X as n. 4. Suppose that X 1, X 2,... X n ; Y 1, Y 2,... Y n are independent random variables in each group. Then, for every f K, n n n d T R X j, Y j ; f d T R X j, Y j ; f. Moreover, if the random variables are identically in each group, then we have n n d T R X j, Y j ; f nd T R X 1, Y 1 ; f. 5. Suppose that X 1, X 2,... X n ; Y 1, Y 2,... 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 2,..., X n and Y 1, Y 2,..., Y n. Then, for every f K, Nn N n k d T R X j, Y j ; f P N n = k d T R X j, Y j ; f. 2.9 k=1 6. Suppose that X 1, X 2,... X n ; Y 1, Y 2,... Y n are independent identically distributed random variables in each group. Let {N n, n 1} be a sequence of positive integervalued random variables that independent of X 1, X 2,..., X n and Y 1, Y 2,..., Y n. Moreover, suppose that EN n < +, n 1. Then, for every f K, we have Nn N n d T R X j, Y j ; f EN n.d T R X 1, Y 1 ; f It is easily seen that the Trotter-Reny distance in term of 2.8 and the total variation distance in 1.3 have a close relationship if the function f is chosen as an indicator function I A of a set A Z +, namely d T R X, Y, χ A = d T V X, Y. if fx = I A x = { 1, if x A 0, if x / A. 6

7 3 Main Results Let X n,j ; 1 j n, n 1 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, 1 j n, n 1. Let Z pn,j, 1 j n, n 1 be a tringular array of independent Poisson distributed random variables with positive means p n,1, p n,2,..., p n,n. It is clear that Z d λn = n Z pn,j. Then, by an argument analogous to that used for the proof of theorems in [17], we can establish the bounds in Poisson approximation via Trotter-Renyi distance. Theorem 3.1. Recall Theorem 3.1, [17] 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, d T R S n, Z λn, f 2 f Proof. Using the definition of Trotter-Renyi distance in 2.8 and received bounds in Theorem 3.1 from [17], we have immediate proof. Corollary 3.1. Under the above assumptions of Theorem 3.1, for all k {0, 1,..., n}, we have n P S n = k P Z λn = k 2 p 2 n,j. Remark 3.1. On account of Corollary 3.1, if lim max p n,j = 0 n 1 j n n p 2 n,j. and lim λ n = λ 0 < λ < +, n then, S n d Z λ 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 P n 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 [17], the bounds in Poisson approximation for random sum will be established vie Trotter-Renyi distance. 7

8 Theorem 3.2. Recall Theorem 4.1, [17] 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 Nn d T R S Nn, Z λnn, f 2 f E. p 2 N n,j Proof. Using the definition of Trotter-Renyi distance in 2.8 with the inequality 2.9, based on received bounds in Theorem 3.1 from [17], we can finish the proof. Corollary 3.2. Under the above assumptions of Theorem 3.2, for all k {0, 1,..., n}, P SNn = k P Z λnn = k Nn 2E. 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 3.2, it follows that Remark 3.2. According to Corollary 3.2 and let p 2 n,j and lim E n Nn lim n max p n,j = 0 1 j h n p n,j = λ 0 < λ < +. Then, S Nn d Z λ as n. 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 3.3. 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 k=1 d T R Snm, Z δn,m, f 2 f n m k=1 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 = l=0 = l=0 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! p 2 k,j. = f x 1 p k,j e p k,j + f x + 1 pk,j p k,j e p k,j 8 l=2 f x + l e p k,j p l k,j l!.

9 It follows that d T R X k,j, ZZ pk,j, f f x 1 p k,j e p k,j + f x + 1 pk,j p k,j e k,j p f x + l e p k,j p l k,j l! = f x 1 pk,j e k,j p + f x + 1 pk,j p k,j e p k,j + f x + l e p kj p l k,j l! e p k,j + p k,j 1 sup f x + p k,j p k,j e k,j p sup x Z+ x Z+ e p k,j p l k,j l! l=2 f x + sup f x x Z+ l=2 e p = sup f x e p k,j + p k,j 1 + p k,j p k,j e p k,j k,j p l k,j + x Z+ l! l=2 = 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 = 2 f p k,j 1 e p k,j 2 f p 2 k,j. Using the properties of Trotter-Renyi distance we have for f K, l=2 d T R S nm, Z δn,m, f n m d T R X k,j, Z pk,j, f. k=1 Then The proof is complete. n d T R S nm, Z δn,m, f 2 f m k=1 p 2 k,j. Corollary 3.3. On account of the assumptions of Theorem 3.3, for all r {0, 1,..., n}, we have n m P S nm = r P Z δnm = r 2 p 2 k,j. Proof. Let us take the function fx such that f x + l = { 1 if m = r, 0 if m r. k=1 Set y = x + l, since x, l Z +, it follows that y Z +. Then, f = sup x According to Theorem 3.3 we conclude that f x = sup f y = 1. y d T R S nm, Z δn,m, f 2 n k=1 m p 2 k,j

10 On the other hand, by choosing the function fx, we deduce that d T R 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 ] l=0 = P Snm = r P Z δn,m = r. Combining 11 with the last equations, we can assert that The proof is streight-forward. P Snm = r P Z δn,m = r n 2 m k=1 Theorem 3.4. 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 d T R SNnM m, Z δnn,mm, f 2 f E Nn M n k=1 p 2 k,j. Proof. Using definition and properties of Trotter-Renyi operator, we have f ASNnMm 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 m=1 p 2 k,j. k=1 = 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 m=1 = 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. m=1 m=1 10

11 Thus, for all f K, and for x Z +, we obtain d T R S NnM m, Z δnn,mm, f P N n = n P M n = md T R S nm, Z δn,m, f m=1 2 f P N n = n n m P M n = m p 2 k,j m=1 k=1 = 2 f n M m P N n = n E p 2 k,j k=1 N n M m = 2 f E. p 2 k,j k=1 Then d T R S NnM m, Z δnn,mm, f 2 f E Nn M n k=1 p 2 k,j. Remark 3.3. Note that we have used the fact that min{n n, M m } P as min{n, m}. Corollary 3.4. According to the above Theorem 3.4, for all r {0, 1,..., n}, we conclude that P SNnMm = r P Z λnn,mm = r Nn M m 2E. Proof. Taking the function fx such that { 1 if m = r, f x + l = 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 3.4 we show that k=1 p 2 k,j N n M m d T R S NnMm, Z δn,m, f 2 p 2 k,j k=1 On the other hand, by choosing the function fx, we conclude that d T R 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 l=0 = P SNnM m = r P Z δnn,mm = r. = l ] 11

12 Combining 12 with the last equations, we have This finishes the proof. P S NnMm = r P Z δnn,mm = r N n M m 2 k=1 p 2 k,j. 4 Conclusions Trotter-Renyi distance method is showed the simple but efficient computation in use, particularly for establishing the bounds in Poisson approximations for independent Poisson-binomial distributed random variables. This tool will certainly be more effective than other tools when considering the Poisson approximation problems for various classes of independent discrete random variables.on vector spaces with finite and infinite dimensions. Acknowledgments This work was done partially while the first author was visiting the Institute for Mathematical Sciences, National University of Singapore in The visit was supported by the Institute. References [1] Arratia, R., Goldstein, L. and Gordon, L., 1990, Poisson approximation and the Chen-Stein method, Statistical Science, Vol. 5, [2] Barbour, A. D., Holst, L. and Janson, S., 1992, Poisson Approximation, Clarendon Press-Oxford. [3] Barbour, A. D. and Chen, L. H. Y., 2004, An introduction to Stein s method, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore. [4] Chen, L. H. Y., 1974, On the convergence of Poisson binomial to Poisson distribution, The Annals of Probability, Vol. 2, No. 1, [5] Chen, L. H. Y., 1975, Poisson approximation for dependent trials, The Annals of Probability, Vol. 3, [6] Chen, L. H. Y., and Leung, D., 2004, An introduction to Stein s method. Singapore University Press. [7] Cioczek, R., and Szynal, D., 1987, On the convergence rate in terms of the Trotter operator in the central limit theorem without moment conditions, Bulletin of the Polish Academy of Sciences Mathematics, Vol. 35, No. 9-10, [8] Deheuvels, P., Karr, A., Pfeifer, D., and Serfling R., 1988, Poisson approximations in selected metrics by coupling and semigroup methods with applications, Journal of Statistical Planning and Inference, 20,

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