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Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 RESEARCH Open Access On bounds in Poisson approximation for integer-valued independent random variables Tran Loc Hung* and Le Truong Giang * Correspondence: [email protected] University of Finance and Marketing, Ho Chi Minh City, Vietnam Abstract The main aim of this note is to establish some bounds in Poisson approximation for row-wise arrays of independent integer-valued random variables via the Trotter-Renyi distance. Some results related to random sums of independent integer-valued random variables are also investigated. MSC: 60F05; 60G50; 41A36 Keywords: Poisson approximation; random sums; Le Cam’s inequality; Trotter’s operator; Renyi’s operator; probability distance; integer-valued random variable 1 Introduction Let {Xn,j , j = , , . . . , n; n = , , . . .} be a row-wise triangular array of independent integervalued random variables with success probabilities P(Xn,j = ) = pn,j ; P(Xn,j = ) = – pn,j – qn,j ; pn,j , qn,j ∈ (, ); pn,j + qn,j ∈ (, ); j = , , . . . , n; n = , , . . . . Set Sn = nj= Xn,j and λn = E(Sn ) = nj= pn,j . Suppose that limn→∞ λn = λ ( < λ < +∞). We will denote by Zλ the Poisson random variable with mean λ. It has long been known that in the case of all qn,j = (j = , , . . . , n; n = , , . . .), the partial sum Sn is said to be a Poisson-binomial random variable, and the probability distributions of Sn , n = , , . . . , are usually approximated by the distribution of Zλ . Specially, under the assumptions that limn→∞ max≤j≤n pn,j = , the well-known Poisson approximation theorem states that d → Zλ Sn − as n → ∞. () d → means the convergence in distribution. It is to Here, and from now on, the notation − be noticed that, for the information on the quality of the Poisson approximation, Le Cam () [] established the remarkable inequality ∞ n P(Sn = k) – P(Zλ = k) ≤ pn,j . () j= k= It is to be noticed that another inequality in Poisson approximation is usually expressed in terms of the total variation distance dTV (Sn , Zλ ) dTV (Sn , Zλ ) ≤ n pn,j , () j= © 2014 Hung and Giang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 Page 2 of 11 where for the distributions P and Q on Z+ = {, , , . . .}, the total variation distance between P and Q will be defined as follows: dTV (P, Q) := P(x) – Q(x). x∈Z () + (For other surveys, see [–], and [].) In recent years many powerful tools for establishing the Le Cam’s inequality for a wide class of discrete independent random variables have been demonstrated, like the coupling technique, the Stein-Chen method, the semi-group method, the operator method, etc. Results of this nature may be found in [–], and []. The main aim of this paper is to establish the bounds of the Le Cam-style inequalities for independent discrete integer-valued random variables using the Trotter-Renyi distance based on Trotter-Renyi operator (see [, ], for more details). It is to be noticed that during the last several decades the Trotter-operator method has been used in many areas of probability theory and related fields. For a deeper discussion of Trotter’s operator we refer the reader to [–], and []. The results obtained in this paper are extensions of known results in [, , –], and []. The present paper is also a continuation of []. This paper is organized as follows. The second section deals with the definition and properties of Trotter-Renyi distance, based on Trotter’s operator and Renyi’s operator. Section gives some results on Le Cam’s inequalities, based on the Trotter-Renyi distance, for independent integer-valued distributed random variables. The random versions of these results are also given in this section. 2 Preliminaries In the sequel we shall recall some properties of Trotter-Renyi operator, 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). Based on Renyi’s definition ([], Chapter , Section , p.), we redefine the Trotter-Renyi operator as follows. Definition . The operator AX associated with a discrete random variable X is called the Trotter-Renyi operator, defined by ∞ f (x + k)P(X = k), (AX f )(x) = E f (X + x) = ∀f ∈ K, ∀x ∈ Z+ , () k= where by K is denoted the class of all real-valued bounded functions f on the set of all non-negative integers Z+ := {, , , . . .}. The norm of the function f ∈ K is defined by f = supx∈Z+ |f (x)|. It is to be noticed that Renyi’s operator defined in [] actually is a discrete form of Trotter’s operator (we refer the readers to [, , –], and [], for a more general and detailed discussion of Trotter’s operator). We shall need in the sequence the following main properties of Trotter-Renyi operator, for all functions f , g ∈ K and for α ∈ R: Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 . . . . . Page 3 of 11 AX (f + g) = AX (f ) + AX (g). AX (αf ) = αAX (f ). AX (f ) ≤ f . AX (f ) + AY (f ) ≤ AX (f ) + AY (f ). Suppose that AX , AY are operators associated with two independent random variables X and Y . Then, for all f ∈ K, AX+Y (f ) = AX AY (f ) = AY AX (f ). In fact, for all x ∈ Z+ AX+Y f (x) = ∞ f (x + l)P(X + Y = l) = l= ∞ f (x + k + r)P(Y = k)P(X = r) r,k= = AX AY f (x) = AX AY f (x). . . Suppose that AX , AX , . . . , AXn are the operators associated with the independent random variables X , X , . . . , Xn . Then, for all f ∈ K, ASn (f ) = AX AX · · · AXn (f ) is the operator associated with the partial sum Sn = X + X + · · · + Xn . Suppose that AX , AX , . . . , AXn and AY , AY , . . . , AYn are operators associated with independent random variables X , X , . . . , Xn and Y , Y , . . . , Yn . Moreover, assume that all Xi and Yj are independent for i, j = , , . . . , n. Then, for every f ∈ K, n AX (f ) – AY (f ). An X (f ) – An Y (f ) ≤ k k k= k k= k () k= Clearly, AX AX · · · AXn – AY AY · · · AYn = n AX AX · · · AXk– (AXk – AYk )AYk+ · · · AYn . k= Accordingly, n An X (f ) – An Y (f ) ≤ AX . . . AX (AX – AY )AY · · · AY (f ) n k– k k k+ k k k= k= k= n AY · · · AY (AX – AY )(f ) ≤ n k+ k k k= ≤ n AX (f ) – AY (f ). k k k= . AnX (f ) – AnY (f ) ≤ nAX (f ) – AY (f ). Lemma . The equation AX f (x) = AY f (x) for f ∈ K, x ∈ Z+ shows that X and Y are identically distributed random variables. Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 Page 4 of 11 Let AX , AX , . . . , AXn , . . . be a sequence of Trotter-Renyi’s operators associated with the independent discrete random variables X , X , . . . , Xn , . . . , and assume that AX is a TrotterRenyi 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 , X , . . . , Xn , . . . to converge in distribution to a random variable X is that lim AXn (f ) – AX (f ) = , n→∞ for all f ∈ K. Proof Since limn→∞ AXn (f ) – AX (f ) = , for all f ∈ K, we conclude that ∞ lim f (x + k) P(Xn = k) – P(X = k) = , n→∞ for all f ∈ K and for all x ∈ Z+ . k= Taking f (x) = ⎧ ⎨, if ≤ x ≤ t, ⎩, if x > t, then we recover t P(Xn = k) – P(X = k) = . lim n→∞ k= d It follows that P(Xn ≤ t) – P(X ≤ t) → as n → +∞. We infer that Xn − → X as n → +∞. Before stating the definition of the Trotter-Renyi distance we firstly need the definition of a probability metric. Let (, A, P) be a probability space and let Z(, A) be a space of real-valued A-measurable random variables X : → R. Definition . A functional d(X, Y ) : Z(, A) × Z(, A) → [, ∞) is said to be a probability metric in Z(, A) if it possesses for the random variables X, Y , Z ∈ Z(, A) the following properties (see [, ] and [] for more details): . P(X = Y ) = ⇒ d(X, Y ) = ; . d(X, Y ) = d(Y , X); . d(X, Y ) ≤ d(X, Z) + d(Z, Y ). We now return to the definition of a probability distance based on the Trotter-Renyi operator (see [, ], and []). Definition . The Trotter-Renyi distance dTR (X, Y ; f ) of two random variables X and Y with respect to the function f ∈ K is defined by dTR (X, Y ; f ) := AX f – AY f = sup Ef (X + x) – Ef (Y + x). x∈Z+ () Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 Page 5 of 11 Based on the properties of the Trotter-Renyi operator, some properties of the TrotterRenyi distance are summarized in the following (see [, , , ], and [] for more details) and we shall omit the proofs. . It is easy to see that dTR (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 dTR (X, Y ; f ) = if P(X = Y ) = . (b) dTR (X, Y ; f ) = dTR (Y , X; f ) for every f ∈ K. (c) dTR (X, Y ; f ) ≤ dTR (X, Z; f ) + dTR (Z, Y ; f ) for every f ∈ K. . If dTR (X, Y ; f ) = for every f ∈ K, then FX ≡ FY . . Let {Xn , n ≥ } be a sequence of random variables and let X be a random variable. The condition lim dTR (Xn , X; f ) = , n→+∞ for all f ∈ K, d . → X as n → ∞. implies that Xn − Suppose that X , X , . . . , Xn ; Y , Y , . . . , Yn are independent random variables (in each group). Then, for every f ∈ K, dTR n Xj , j= n ≤ Yj ; f j= n dTR (Xj , Yj ; f ). () j= Moreover, if the random variables are identically (in each group), then we have dTR n Xj , j= . ≤ ndTR (X , Y ; f ). Yj ; f j= Suppose that X , X , . . . , Xn ; Y , Y , . . . , Yn are independent random variables (in each group). Let {Nn , n ≥ } be a sequence of positive integer-valued random variables that are independent of X , X , . . . , Xn and Y , Y , . . . , Yn . Then, for every f ∈ K, dTR N n Xj , j= . n Nn ≤ Yj ; f j= ∞ k= P(Nn = k) k dTR (Xj , Yj ; f ). () j= Suppose that X , X , . . . , Xn ; Y , Y , . . . , Yn are independent identically distributed random variables (in each group). Let {Nn , n ≥ } be a sequence of positive integer-valued random variables that are independent of X , X , . . . , Xn and Y , Y , . . . , Yn . Moreover, suppose that E(Nn ) < +∞, n ≥ . Then, for every f ∈ K, we have dTR N n j= Xj , Nn Yj ; f ≤ E(Nn ) · dTR (X , Y ; f ). j= Finally, we emphasize that the Trotter-Renyi distance in () and the total variation distance in () have a close relationship if the function f is chosen as an indicator function of Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 Page 6 of 11 a set A ∈ Z+ , namely ⎧ ⎨, if x ∈ A, f (x) = χA (x) = ⎩, if x ∈/ A. Then dTR (X, Y , χA ) = dTV (X, Y ), where we denote by dTV (X, Y ) the total variation distance between two integer-valued random variables X and Y , defined as follows: P(X = k) – P(Y = k). dTV (X, Y ) = sup P(X ∈ A) – P(Y ∈ A) = A⊆Z+ k∈Z+ For a deeper discussion of the total variation distance, we refer the reader to [–], and []. 3 Main results Let {AXn,j , j = , , . . . , n; n = , , . . .} be a sequence of operators associated with the integervalued random variables Xn,j , j = , , . . . , n; n = , , . . . , and let {AZpn,j , j = , , . . . , n; n = , , . . .} be a sequence of operators associated with the Poisson random variables with parameters pn,j , j = , , . . . , n; n = , , . . . . Since Zλn is a Poisson random variable with positive d parameter λn = nj= pn,j , we can write Zλn = nj= Zpn,j , where Zpn, , Zpn, , . . . , Zpn,n are independent Poisson random variables with positive parameters pn, , pn, , . . . , pn,n , and the d notation = denotes coincidence of distributions. Theorem . Let {Xn,j , j = , , . . . , n; n = , , . . .} be a row-wise triangular array of independent, integer-valued random variables with probabilities P(Xn,j = ) = pn,j , P(Xn,j = ) = – pn,j – qn,j ; pn,j , qn,j ∈ (, ); pn,j + qn,j ∈ (, ); j = , , . . . , n; n = , , . . . . Let us write Sn = nj= Xn,j and λn = nj= pn,j . We will denote by Zλn the Poisson random variable with parameter λn . Then, for all functions f ∈ K, dTR (Sn , Zλn ; f ) ≤ f n pn,j + qn,j . j= Proof Applying (), we have dTR (Sn , Zλn , f ) ≤ n dTR (Xn,j , Zpn,j ; f ) = j= n AX (f ) – AZ (f ). pn,j n,j k= Moreover, for all f ∈ K, for all x ∈ Z+ and r ∈ {, , . . . , n} we conclude that AXnj f (x) – AZpn,j f (x) = ∞ f (x + r) P(Xnj = r) – P(Zλpn,j = r) r= = ∞ r= e–pn,j prn,j f (x + r) P(Xnj = r) – r! Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 Page 7 of 11 = f (x) – pn,j – qn,j – e–pn,j + f (x + ) pn,j – pn,j e–pn,j ∞ e–pn,j prn,j . + f (x + r) P(Xn,j = r) – r! r= Therefore, for all functions f ∈ K , and for all x ∈ Z+ , we have AX f (x) – AZ f (x) pn,j n,j = f (x) – pn,j – qn,j – e–pn,j + f (x + ) pn,j – pn,j e–pn,j ∞ e–pn,j prn,j + f (x + r) P(Xn,j = r) – r! r= = f (x) – pn,j – qn,j – e–pn,j + f (x + ) pn,j – pn,j e–pn,j ∞ e–pn,j prn,j f (x + r) P(Xn,j = r) – + r! r= ≤ f (x) – pn,j – qn,j – e–pn,j + f (x + ) pn,j – pn,j e–pn,j ∞ ∞ e–pn,j prn,j f (x + r)P(Xn,j = r) + f (x + r) + r! r= ≤ e–pn,j r= + pn,j + qn,j – sup f (x) + pn,j – pn,j e–pn,j sup f (x) x∈Z+ x∈Z+ ∞ ∞ –p e n,j prn,j + sup f (x) P(Xn,k = r) + sup f (x) x∈Z+ r! x∈Z+ r= r= = sup f (x) e–pn,j + pn,j + qn,j – + pn,j – pn,j e–pn,j + qn,j + – e–pn,j – pn,j e–pn,j x∈Z+ = f pn,j – pn,j e–pn,j + qn,j ≤ f pn,j + qn,j . One infers that ∀f ∈ K, AX (f ) – AZ (f ) ≤ f p + qn,j . pn,j n,j n,j Therefore, applying (), we can assert that dTR (Sn , Zλn ; f ) ≤ f n pn,j + qn,j . j= This completes the proof. Corollary . Under the assumptions of Theorem ., let r ∈ {, , . . . , n}, we have n P(Sn = r) – P(Zλ = r) ≤ pn,j + qn,k . n j= Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 Page 8 of 11 Remark . We consider Corollary . and assume that the following conditions are satisfied: (i) lim n→∞ n qn,j = , j= lim max pn,j = , (ii) n→∞ ≤k≤n (iii) lim λn = lim n→∞ n n→∞ pn,j = λ ( < λ < +∞). j= d Then Sn → Zλ as n → ∞. Theorem . Let {Xn,j , j = , , . . . , n; n = , , . . .} be a row-wise triangular array of independent, integer-valued random variables with probabilities P(Xn,j = ) = pn,j , P(Xn,j = ) = – pn,j – qn,j ; pn,j , qn,j ∈ (, ); pn,j + qn,j ∈ (, ); j = , , . . . , n; n = , , . . . . Moreover, we suppose that Nn , n = , , . . . are positive integer-valued random variables, independent of all n N n Xn,j , j = , , . . . , n; n = , , . . . . Let us write SNn = N j= Xn,j and λNn = j= pn,j . We will denote by ZλNn the Poisson random variable with parameter λNn . Then, for all functions f ∈ K, N n pNn ,j + qNn ,j . dTR (SNn , ZλNn ; f ) ≤ f E j= Proof According to Theorem . and (), for all functions f ∈ K, and for all x ∈ Z+ , we have dTR (SNn , ZλNn ; f ) ≤ ∞ P(Nn = m)dTR (Sm , Zλm ; f ) m= ≤ ∞ P(Nn = m)f m= = f ∞ pNn ,j + qNn ,j j= P(Nn = m) m= = f E m m pNn ,j + qNn ,j j= N n pNn ,j + qNn ,j . j= Therefore, N n pNn ,j + qNn ,j . dTR (SNn , ZλNn ; f ) ≤ f E j= The proof is complete. Corollary . According to Theorem ., let r ∈ {, , . . . , n}, we have N n P(SN = r) – P(Zλ = r) ≤ E pNn ,j + qNn ,j . n Nn j= Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 Page 9 of 11 Theorem . Let {Xk,j } (k = , , . . . ; j = , , . . .) be a double array of independent integervalued random variables with probabilities P(Xk,j = ) = pk,j , P(Xk,j = ) = – pk,j – qk,j , pn,k ∈ (, ); k = , , . . . ; j = , , . . . . Assume that for every k = , , . . . the random variables Xk, , Xk, , . . . , are independent, and for every j = , , . . . the random variables X,j , X,j , . . . are independent. Set Snm = nk= m j= Xk,j . Let us denote by Zδn,m the Poisson random variable p with mean δn,m = nk= m j= k,j . Then, for all f ∈ K, dTR (Snm , Zδn,m , f ) ≤ f n m pk,j + qk,j . k= j= Proof Applying the inequality in (), we have dTR (Snm , Zδnm , f ) ≤ n dTR (Skm , Zμk,m , f ) k= ≤ n m dTR (Sk,j , Zλk,j , f ). k= j= According to Theorem ., for all functions f ∈ K, and for all x ∈ Z+ , we conclude that dTR (Sk,j , Zλk,j , f ) ≤ f pk,j + qk,j . Therefore, dTR (Snm , Zδnm , f ) ≤ f n m pk,j + qk,j . k= j= This completes the proof. Theorem . Let {Xk,j , k = , , . . . ; j = , , . . .} be a double array of independent integervalued random variables with P(Xk,j = ) = pk,j ; P(Xk,j = ) = – pk,j – qk,j ; pk,j , qk,j ∈ (, ); pk,j + qk,j ∈ (, ); k = , , . . . ; n = , , . . . . Assume that for every k = , , . . . the random variables Xk, , Xk, , . . . , are independent, and for every j = , , . . . the random vari ables X,j , X,j , . . . are independent. Set Snm = nk= m j= Xk,j . Suppose that Nn , Mm are non-negative integer-valued random variables independent of all Xn,m , n ≥ ; m ≥ . Let us denote by ZδNn Mm the Poisson random variable with mean δNn Mm = E(SNn Mm ) = N n Mm j= pk,j . Then, for all functions f ∈ K, k= dTR (SNn Mm , ZδNn Mm , f ) ≤ f E N M n n pk,j + qk,j . k= j= Proof According to Definition ., we have (ASNn Mm f )(x) := E f (SNn Mm + x) = ∞ n= P(Nn = n) ∞ m= P(Mn = m)(ASnm f )(x) Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 Page 10 of 11 and (AZδN n Mm f )(x) := E f (ZδNn Mm + x) = ∞ P(Nn = n) n= ∞ P(Mn = m)(AZδnm f )(x). m= Therefore, for all functions f ∈ K, and for all x ∈ Z+ , we have AS Nn Mm (f ) – AZδN ≤ ∞ P(Nn = n) n= n Mm ∞ P(Mn = m)ASnm (f ) – AZδn,m (f ) m= ≤ f ∞ P(Nn = n) n= = f (f ) ∞ ∞ n m pk,j + qk,j P(Mn = m) m= k= j= P(Nn = n)E n= n M m pk,j + qk,j k= j= N M n m pk,j + qk,j . = f E k= j= Thus, dTR (SNn Mm , ZδNn ,Mm , f ) ≤ f E N M n n pk,j + qk,j . k= j= The proof is straightforward. Remark . In the case of all probabilities qn,j = , j = , , . . . , n; n = , , . . . the partial sum Sn = nj= Xn,j will become a Poisson-binomial random variable, and one concludes that the results of Theorems ., ., ., and . are extensions of results in [] (see [] for more details). We conclude this paper with the following comments. The Trotter-Renyi distance method is based on the Trotter-Renyi operator and it has a big application in the Poisson approximation. Using this method it is possible to establish some bounds in the Poisson approximation for sums (or random sums) of independent integer-valued random vectors. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly to this work. All authors drafted the manuscript, read and approved the final version of the manuscript. Acknowledgements The authors wish to express their gratitude to the referees for valuable remarks and comments, improving the previous version of this paper. The research was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED, Vietnam) under grant 101.01-2010.02. Received: 23 January 2014 Accepted: 10 July 2014 Published: 18 August 2014 Hung and Giang Journal of Inequalities and Applications 2014, 2014:291 http://www.journalofinequalitiesandapplications.com/content/2014/1/291 References 1. Le Cam, L: An approximation theorem for the Poisson binomial distribution. Pac. J. Math. 10(4), 1181-1197 (1960) 2. Barbour, AD, Holst, L, Janson, S: Poisson Approximation. Clarendon, Oxford (1992) 3. Chen, LHY, Leung, D: An Introduction to Stein’s Method. Singapore University Press, Singapore (2004) 4. Neammanee, KA: Nonuniform bound for the approximation of Poisson binomial by Poisson distribution. Int. J. Math. Math. Sci. 48, 3041-3046 (2003) 5. 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