Approximations Related to the Sums of m -dependent Random Variables
aa r X i v : . [ m a t h . P R ] M a y Approximations Related to the Sums of m -dependentRandom Variables Amit N. Kumar , Neelesh S. Upadhye , and P. Vellaisamy Department of Mathematics, Indian Institute of Technology Bombay,Powai, Mumbai-400076, India. Department of Mathematics, Indian Institute of Technology Madras,Chennai-600036, India. Email: [email protected] Email: [email protected] Email: [email protected]
Abstract
In this paper, we consider the sums of non-negative integer valued m -dependent random variables,and its approximation to the power series distribution. We first discuss some relevant results forpower series distribution such as Stein operator, uniform and non-uniform bounds on the solutionof Stein equation, and etc. Using Stein’s method, we obtain the error bounds for the approximationproblem considered. As special cases, we discuss two applications, namely, -runs and ( k , k ) -runs and compare the bound with the existing bounds. Keywords:
Power series distribution; m -dependent random variables; Stein’s method; Runs. MSC 2010 Subject Classifications :
Primary : 62E17, 62E20 ; Secondary : 60F05, 60E05.
The sums of m -dependent random variables (rvs) has special attention due to its applicability in manyreal-life applications such as runs and patterns, DNA sequences, and reliability theory, among manyothers. However, its distribution is difficult or sometimes intractable, especially if the setup is arisingfrom non-identical rvs concentrated on Z + = { , , , . . . } , the set of non-negative integers. Therefore,there is a need to approximate such a distribution with some known and easy-to-use distributions. In1his article, we consider power series distribution (PSD) approximation to the sums of m -dependentrvs. Approximations related to the sums of locally dependent rvs have been studied by several authorssuch as Barbour and Xia [4, 5], Fu and Johnson [9], Vellaisamy [26], Wang and Xia [27], and Soon[21], among many others.A sequence of rvs { X k } k ≥ is called m -dependent if σ ( X , X , . . . , X i ) and σ ( X j , X j +1 , . . . ) , for j − i > m , are independent, where σ ( X ) denotes the sigma-algebra generated by X . The sums of m -dependent rvs can be reduced to the sums of -dependent rvs, using rearrangement of rvs (see Section 3for details). We mainly focus on the sums of -dependent rvs concentrated on Z + , and obtain the errorbounds. Of course, the bound can directly apply for special distributions of PSD family. An advantageof approximation to PSD family is that we can obtain the error bounds for approximation to somespecific distributions such as Poisson and negative binomial distributions. For some related works, werefer the reader to Lin and Liu [15], ˇCekanaviˇcius and Vellaisamy [25], and references therein.For Z + -valued rvs X and X ∗ , the total variation distance is given by d T V ( X, X ∗ ) = 12 ∞ X k =0 | P ( X = k ) − P ( X ∗ = k ) | . (1.1)Hereafter, A denotes the indicator function of A ⊆ Z + . Let X be a rv concentrated on Z + , G = { f : Z + → R | f is bounded } and G X = { g ∈ G | g (0) = 0 and g ( x ) = 0 for x / ∈ supp ( X ) } , (1.2)associated with the rv X , where supp( X ) denotes the support of the rv X . We now briefly discussStein’s method (Stein [22]) which we use to derive our approximation results in Section 3. The Stein’smethod can be discussed in following three steps.(a) Identify a Stein operator, denoted by A X for a rv X , such that E [ A X g ( X )] = 0 , for g ∈ G X .(b) Solve the Stein equation A X g ( k ) = f ( k ) − E f ( X ) , for f ∈ G and g ∈ G X .(c) Replace k by a rv Y in Stein equation, and taking expectation and supremum to get d T V ( X, Y ) := sup f ∈H | E f ( X ) − E f ( Y ) | = sup f ∈H | E A X g f ( Y ) | , where g f is the solution of the Stein equation and H = { A | A ⊆ Z + } .For additional details on Stein’s method, see Barbour et al. [3], Barbour and Chen [2], Ley et al. [14],Reinert [19], Upadhye et al. [24], and the references therein.2his article is organized as follows. In Section 2, we discuss the PSD and its related results to Stein’smethod. In Section 3, we derive the error bound for PSD approximation to the sums of -dependent rvsand discuss some relevant remarks. In Section 4, we discuss two important applications of our resultsto the sums of -runs and ( k , k ) -runs. Let Z be a Z + -values rv. We say its distribution belongs to the PSD family, denoted by P , if P ( Z = k ) = p k is of the form p k = a k θ k γ ( θ ) , k ∈ Z + , (2.1)where θ > and a k , k ≥ , are called series parameter and coefficient function, respectively. Manydistributions such as Poisson, binomial, negative binomial, logarithmic series, and inverse sine distri-butions, among many others, belong to the PSD family. For more details, we refer the reader to Edwin[7], Noack [16], Patil [18], and the references therein.Next, we give a brief discussion about Stein’s method for PSD, in fact, many results follow fromEichelsbacher and Reinert [8]. The following proposition gives a Stein operator for PSD. Proposition 2.1.
Let the rv Z having distribution belonging to PSD family defined in (2.1) . Then aStein operator for Z is given by A Z g ( k ) = θ ( k + 1) a k +1 a k g ( k + 1) − kg ( k ) , g ∈ G Z , k ∈ Z + . (2.2) Proof.
From (2.1), it can be easily verified that θ ( k + 1) a k +1 a k p k − ( k + 1) p k +1 = 0 . (2.3)Let g ∈ G Z defined in (1.2), then ∞ X k =0 g ( k + 1) (cid:20) θ ( k + 1) a k +1 a k p k − ( k + 1) p k +1 (cid:21) = 0 . Rearranging the terms, we have ∞ X k =0 (cid:20) θ ( k + 1) a k +1 a k g ( k + 1) − kg ( k ) (cid:21) p k = 0 . This proves the result. 3ow, we discuss the solution of the Stein equation θ ( k + 1) a k +1 a k g ( k + 1) − kg ( k ) = f ( k ) − E f ( Z ) , f ∈ G , g ∈ G Z . (2.4)Next we describe discrete Gibbs measure (DGM), a large class of distributions, studied by Eichels-bacher and Reinert [8]. If a rv U has the distribution of the form P ( U = k ) = 1 C w e V ( k ) w k k ! , k ∈ Z + , (2.5)for some function V : Z + → R , w > , and C w = P ∞ k =0 e V ( k ) w k k ! , then we say the rv U belongs tothe DGM family. Observe here the support is Z + . Note that if we take a k = e V ( k ) /k ! ⇐⇒ V ( k ) =ln( a k k !) , θ = w , and γ ( θ ) = C w , which are valid choices, then the results derived by Eichelsbacherand Reinert [8] are valid for PSD family. Therefore, the solution of (2.4) can be directly obtained from (2 . and (2 . of Eichelsbacher and Reinert [8] and is given by g ( k ) = 1 ka k θ k k − X j =0 a j θ j [ f ( j ) − E f ( Z )]= − ka k θ k ∞ X j = k a j θ j [ f ( j ) − E f ( Z )] . Also, the Lemma . of Eichelsbacher and Reinert [8] can be written for PSD family in the followingmanner. Lemma 2.1.
Let G = { f : Z + → [0 , } , F ( k ) = P ki =0 p i and ¯ F ( k ) = P ∞ i = k p i . Assume that k F ( k ) F ( k − ≥ θ ( k + 1) a k +1 a k ≥ k ¯ F ( k + 1)¯ F ( k ) . Then, for f ∈ G and g f , the solution of (2.4) , we have sup f ∈B | ∆ g f ( k ) | = a k θ ( k + 1) a k +1 ¯ F ( k + 1) + 1 k F ( k − , where ∆ g f ( k ) = g f ( k + 1) − g f ( k ) .Moreover, sup f ∈B | ∆ g f ( k ) | ≤ k ∧ a k θ ( k + 1) a k +1 , (2.6)4 here x ∧ y denotes the minimum of x and y . Now, it is not easy to use direct form of the Stein operator (2.2) as a k is unknown and depends on k .So, we consider PSD family with Panjer’s recursive relation (see Panjer and Wang [17] for details),denoted by P , which is given by ( k + 1) p k +1 p k = a + bk = ⇒ θ ( k + 1) a k +1 a k = a + bk, for some a, b ∈ R . (2.7)Therefore, the stein operator (2.2) can be written as A Z g ( k ) = ( a + bk ) g ( k + 1) − kg ( k ) , k ∈ Z + . (2.8)Also, the bound (2.6) becomes sup f ∈B | ∆ g f ( k ) | ≤ k ∧ a + bk , k ≥ . (2.9)Note that if a, b ≥ (PSD family satisfies Panjer recursive relation with a, b ≥ , denoted by P ) thenthe bound (2.9) becomes uniform and is given by sup f ∈B | ∆ g f ( k ) | ≤ ∧ a , k ≥ . (2.10)Note that P ⊂ P ⊂ P . Also, observe that a = λ, b = 0 (cid:0) a k = 1 /k ! , θ = λ and γ ( θ ) = e θ (cid:1) and a = nq, b = q (cid:0) a k = (cid:0) n + k − k (cid:1) , θ = q and γ ( θ ) = (1 − θ ) − n (cid:1) for Poisson (with parameter λ ) andnegative binomial (with parameter n and p = 1 − q ) distributions, respectively, and hence the bounds(from (2.10)) are ∧ λ and ∧ nq , respectively, which are well-known bounds for Poisson and negativebinomial distributions. Many distributions satisfy the condition a, b ≥ . However, if the condition isnot satisfied, one can still use (2.9) to compute the uniform bound. For example, if a k = (cid:0) nk (cid:1) , θ = p/q ,and γ ( θ ) = (1 + θ ) n , then Z ∼ Bi ( n, p ) , and θ ( k +1) a k +1 a k = pq ( n − k ) , and hence a = np/q and b = − p/q ≤ . Therefore, the bound (2.9) is sup f ∈B | ∆ g f ( k ) | ≤ k ∧ qp ( n − k )= ( k if k ≥ np q ( n − k ) p if k ≤ np ≤ ( np if k ≥ np np if k ≤ np np , for all k, (2.11)which leads to a uniform bound for binomial distribution. Note here that the Stein operator (from (2.8))is A Z g ( k ) = pq ( n − k ) g ( k + 1) − kg ( k ) . (2.12)But, the well-known Stein operator for the binomial distribution is A Z g ( k ) = p ( n − k ) g ( k + 1) − qkg ( k ) , (2.13)which follows by multiplying q in (2.12). Also, the uniform bound will be changed and is given by /npq (that is, divided by q ), which is well-known bound with respect to the Stein operator (2.13) (seeUpadhye et al. [24]). Hence, throughout this article, we use k ∆ g k = sup k | ∆ g ( k ) | and the uniformbound for k ∆ g k can be obtained from (2.10) or may be computed explicitly for some applications.Next, let φ Z ( · ) be the probability generating function of Z . Then, using (2.7), it can be seen that φ ′ Z ( t ) = aφ Z ( t )1 − bt . Hence, mean and variance of the PSD are given by E ( Z ) = a − b and Var( Z ) = a (1 − b ) . (2.14)For more details, we refer the reader to Edwin [7], and Panjer and Wang [17]. Let Y , Y , . . . , Y n be a sequence of Z + -valued m dependent rvs and S n = P ni =1 Y i , the sums of m -dependent rvs. Then, grouping the consecutive summations in the following form Y ∗ i := min( im,n ) X j =( i − m +1 Y j , j = 1 , , . . . , ⌊ n/m ⌋ + 1 , where ⌊ x ⌋ denotes the greatest integer function of x , S n = P ⌊ n/m ⌋ +1 j =1 Y ∗ j become the sums of -dependent rvs. In this section, we derive an error bound for PSD approximation to the sums of -dependent rvs in total variation distance and discuss some relevant remarks. Throughout this section,6e assume X , X , . . . , X n , n ≥ , is a sequence of -dependent rvs and W n = n X i =1 X i . (3.1)For any Z + -valued rv Y , let D ( Y ) := 2 d T V ( Y, Y + 1) , where d T V ( X, X ∗ ) as defined in (1.1). Let N i,ℓ := { j : | j − i | ≤ ℓ } ∩ { , , . . . , n } and X N i,ℓ := X j ∈ N i,ℓ X j , for ℓ = 1 , . Note that X N i, − X N i, = X N i, − N i, . From (3.1), it can be verified that E ( W n ) = P ni =1 E ( X i ) and Var( W n ) = n X i =1 X | j − i |≤ [ E ( X i X j ) − E ( X i ) E ( X j )] = n X i =1 (cid:2) E ( X i X N i, ) − E ( X i ) E ( X N i, ) (cid:3) . (3.2)Now, the following theorem gives the error bound for Z -approximation to W n . Theorem 3.1.
Let Z ∈ P and W n be defined as in (3.1) . Assume that E ( Z ) = E ( W n ) , and τ =Var( W n ) − Var( Z ) . Then, for n ≥ , d T V ( W n , Z ) ≤ k ∆ g k ( | − b | " n X i =1 E ( X i ) E [ X N i, (2 X N i, − X N i, − D ( W n | X N i, , X N i, )]+ n X i =1 E [ X i X N i, (2 X N i, − X N i, − D ( W n | X N i, , X N i, )] + n X i =1 E [ X i ( X N i, − D ( W n | N i, )] + | τ (1 − b ) | ) . (3.3) Proof.
Consider the Stein operator given in (2.8) and taking expectation with respect to W n , we have E [ A Z g ( W n )] = a E [ g ( W n + 1)] + b E [ W n g ( W n + 1)] − E [ W n g ( W n )]= a E [ g ( W n + 1)] − (1 − b ) E [ W n g ( W n + 1)] + E [ W n ∆ g ( W n )]= (1 − b ) (cid:20) a (1 − b ) E [ g ( W n + 1)] − E [ W n g ( W n + 1)] (cid:21) + E [ W n ∆ g ( W n )] . Applying the first moment matching condition, E ( Z ) = a/ (1 − b ) = E ( W n ) , we get E [ A Z g ( W n )] = (1 − b ) h E ( W n ) E [ g ( W n + 1)] − E [ W n g ( W n + 1)] i + E [ W n ∆ g ( W n )] . (3.4)7et now W i,n := W n − X N i, so that X i and W i,n are independent. Consider the following expression from (3.4) E ( W n ) E [ g ( W n + 1)] − E [ W n g ( W n + 1) = n X i =1 E ( X i ) E [ g ( W n + 1)] − n X i =1 E [ X i g ( W n + 1)]= n X i =1 E ( X i ) E [ g ( W n + 1)] − n X i =1 E [ X i g ( W n + 1)] − n X i =1 E [ X i g ( W i,n + 1)] + n X i =1 E [ X i g ( W i,n + 1)]= n X i =1 E ( X i ) E [ g ( W n + 1) − g ( W i,n + 1)] − n X i =1 E [ X i ( g ( W n + 1) − g ( W i,n + 1))] . (3.5)It can be seen that g ( W n + 1) − g ( W i,n + 1) = g ( W i,n + X N i, + 1) − g ( W i,n + 1)= g ( W i,n + X N i, + 1) − g ( W i,n + X N i, )+ g ( W i,n + X N i, ) − g ( W i,n + X N i, − ... + g ( W i,n + 2) − g ( W i,n + 1)= X Ni, X j =1 ∆ g ( W i,n + j ) . (3.6)Using (3.6) in (3.5), we get E ( W n ) E [ g ( W n + 1)] − E [ W n g ( W n + 1)] = n X i =1 E ( X i ) E " X Ni, X j =1 ∆ g ( W i,n + j ) − n X i =1 E " X i X Ni, X j =1 ∆ g ( W i,n + j ) . (3.7)8ubstituting (3.7) in (3.4), we have E [ A Z g ( W n )] = (1 − b ) n X i =1 E ( X i ) E " X Ni, X j =1 ∆ g ( W i,n + j ) − n X i =1 E " X i X Ni, X j =1 ∆ g ( W i,n + j ) + n X i =1 E [ X i ∆ g ( W n )] . (3.8)Note that E ( Z ) = a/ (1 − b ) = E ( W n ) = P ni =1 E ( X i ) . Therefore, from (2.14), Var( Z ) = a (1 − b ) = 1(1 − b ) n X i =1 E ( X i ) . Hence, τ = Var( W n ) − Var( Z ) = n X i =1 E ( X i X N i, ) − n X i =1 E ( X i ) E ( X N i, ) − − b ) n X i =1 E ( X i ) . This implies (1 − b ) ( n X i =1 E ( X i X N i, ) − n X i =1 E ( X i ) E ( X N i, ) ) − n X i =1 E ( X i ) − τ (1 − b ) = 0 . (3.9)Next, define V i,n := W n − X N i, so that X N i, and V i,n are independent, and X i and V i,n are independent. From (3.9), we get (1 − b ) ( n X i =1 E ( X i X N i, ∆ g ( V i,n )) − n X i =1 E ( X i ) E ( X N i, ∆ g ( V i,n )) ) − n X i =1 E ( X i ∆ g ( V i,n )) − τ (1 − b ) E (∆ g ( V i,n )) = 0 . This is equivalent to (1 − b ) n X i =1 E " X i X Ni, X j =1 ∆ g ( V i,n ) − n X i =1 E ( X i ) E " X Ni, X j =1 ∆ g ( V i,n ) − n X i =1 E ( X i ∆ g ( V i,n )) − τ (1 − b ) E (∆ g ( V i,n )) = 0 . (3.10)9sing (3.10) in (3.8), we get E [ A Z g ( W n )] = (1 − b ) n X i =1 E ( X i ) E " X Ni, X j =1 ∆ g ( W i,n + j ) − n X i =1 E " X i X Ni, X j =1 ∆ g ( W i,n + j ) + (1 − b ) n X i =1 E " X i X Ni, X j =1 ∆ g ( V i,n ) − n X i =1 E ( X i ) E " X Ni, X j =1 ∆ g ( V i,n ) + n X i =1 E [ X i ∆ g ( W n )] − n X i =1 E ( X i ∆ g ( V i,n )) − τ (1 − b ) E (∆ g ( V i,n ))= (1 − b ) n X i =1 E ( X i ) E " X Ni, X j =1 (∆ g ( W i,n + j ) − ∆ g ( V i,n )) − n X i =1 E " X i X Ni, X j =1 (∆ g ( W i,n + j ) − ∆ g ( V i,n )) + n X i =1 E [ X i (∆ g ( W n ) − ∆ g ( V i,n ))] − τ (1 − b ) E (∆ g ( V i,n )) . (3.11)Note also that ∆ g ( W i,n + j ) − ∆ g ( V i,n ) = ∆ g ( V i,n + X N i, − N i, + j ) − ∆ g ( V i,n )= X Ni, − Ni, + j − X k =1 ∆ g ( V i,n + k ) . (3.12)and ∆ g ( W n ) − ∆ g ( V i,n ) = ∆ g ( V i,n + X N i, ) − ∆ g ( V i,n )= X Ni, − X k =1 ∆ g ( V i,n + k ) . (3.13)Substituting (3.12) and (3.13) in (3.11), we have E [ A Z g ( W n )] = (1 − b ) n X i =1 E ( X i ) E " X Ni, X j =1 X Ni, − Ni, + j − X k =1 ∆ g ( V i,n + k ) − n X i =1 E " X i X Ni, X j =1 X Ni, − Ni, + j − X k =1 ∆ g ( V i,n + k ) n X i =1 E " X i X Ni, − X j =1 ∆ g ( V i,n + j ) − τ (1 − b ) E (∆ g ( V i,n )) . (3.14)Consider first E " X i X Ni, − X j =1 ∆ g ( V i,n + j ) = E E " X i X Ni, − X j =1 ∆ g ( V i,n + j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X N i, = E E (cid:0) X i | X N i, (cid:1) E " X Ni, − X j =1 ∆ g ( W n − X N i, + j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X N i, , (3.15)since X i and V i,n are independent given X N i, . Observe that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E " X Ni, − X j =1 ∆ g ( W n − X N i, + j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X N i, = n i, ≤ k ∆ g k| n i, − | D ( W n | X N i, = n i, ) . (3.16)Using (3.16) in (3.15), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E " X i X Ni, − X j =1 ∆ g ( V i,n + j ) ≤ k ∆ g k E [ X i | X N i, − | D ( W n | X N i, )]= k ∆ g k E [ X i ( X N i, − D ( W n | X N i, )] , (3.17)since X i X N i, ≥ X i = ⇒ X i ( X N i, − ≥ .Consider next the following expression from (3.14) E " X Ni, X j =1 X Ni, − Ni, + j − X k =1 ∆ g ( V i,n + k ) = E E " X Ni, X j =1 X Ni, − X Ni, + j − X k =1 ∆ g ( W n − X N i, + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X N i, , X N i, . (3.18)Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E " X Ni, X j =1 X Ni, − Ni, + j − X k =1 ∆ g ( W n − X N i, + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X N i, = n i, , X N i, = n i, ≤ k ∆ g k n i, | n i, − n i, − | D ( W n | X N i, = n i, , X N i, = n i, ) . (3.19)11sing (3.19) in (3.18), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E " X Ni, X j =1 X Ni, − Ni, + j − X k =1 ∆ g ( V i,n + k ) ≤ k ∆ g k E [ X N i, | X N i, − X N i, − | D ( W n | X N i, , X N i, )]= k ∆ g k E [ X N i, (2 X N i, − X N i, − D ( W n | X N i, , X N i, )] , (3.20)since X N i, X N i, − X N i, ≥ and X N i, X N i, − X N i, ≥ which imply X N i, (2 X N i, − X N i, − ≥ .Similarly, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E " X i X Ni, X j =1 X Ni, − Ni, + j − X k =1 ∆ g ( V i,n + k ) ≤ k ∆ g k E [ X i X N i, (2 X N i, − X N i, − D ( W n | X N i, , X N i, )] . (3.21)Finally, using (3.17), (3.20) and (3.21) in (3.14), the proof follows. Remarks 3.1. (i) For n ≥ , we can use (3.4) to obtain the following crude upper bound for d T V ( W n , Z ) . d T V ( W n , Z ) ≤ (2 | − b |k g k + k ∆ g k ) n X i =1 E ( X i ) . (3.22) Note however that for n ≥ , the bound given in (3.3) would better than the one given in (3.22) .(ii) Assume D ( W n | X N i, ) ≤ c i ( n ) then D ( W n | X N i, , X N i, ) ≤ c i ( n ) . Therefore, the bound (3.3) becomes d T V ( W n , Z ) ≤ k ∆ g k ( n X i =1 c i ( n ) " | − b | h E ( X i ) E [ X N i, (2 X N i, − X N i, − E [ X i X N i, (2 X N i, − X N i, − i + E [ X i ( X N i, − + | τ (1 − b ) | ) =: d ( n ) . (3.23) Furthermore, let us denote L ( W ∗ i,n ) = L ( W n | X N i, ) and Z e = { X m | m ∈ { , . . . , ⌊ n/ ⌋}} =( X , X , . . . , X ⌊ n/ ⌋ ) . Then, L ( W ∗ i,n | Z e = z e ) can be written as sum of independent rvs, say ( z e ) j , for j = 1 , , . . . , n z e . Therefore, using (5 . of Röllin [20], we have D ( W ∗ i,n ) ≤ E [ E [ D ( W ∗ i,n ) | Z e ]] ≤ E " V / i,Z e , (3.24) where V i,z e = n ze X j =1 min (cid:26) , − D (cid:0) X ( z e ) j (cid:1)(cid:27) . (3.25) On the other hand, let m ∗ = ( ⌊ n/ ⌋ + 1 , if n is odd n/ , if n is even and Z o = { X m − | m ∈ { , . . . , m ∗ }} . (3.26) Then, applying the similar argument as above, we get D ( W ∗ i,n ) ≤ E [ E [ D ( W ∗ i,n ) | Z o ]] ≤ E " V / i,Z o , (3.27) where V i,z o is defined in a similar way as V i,z e . Hence, from (3.24) and (3.27) , we have D ( W ∗ i,n ) ≤ min ( E " V / i,Z o , E " V / i,Z e = c i ( n ) . Note that c i ( n ) = O ( n − / ) in general. For more details, we refer the reader to Section . andSection . of Röllin [20].(iii) The bound given in Theorem 3.1 can also be used for the case of matching the first two moments(i.e., τ = 0 ), whenever that is possible with the approximating distribution.(iv) From (3.8) , it can be easily verified that in the case of first moment matching, we have d T V ( W n , Z ) ≤ k ∆ g k ( | − b | n X i =1 [ E ( X i ) E ( X N i, ) + E ( X i X N i, )] + n X i =1 E ( X i ) ) =: d ( n ) . and then we have d T V ( W n , Z ) ≤ min { d ( n ) , d ( n ) } , where d ( n ) is defined in (3.23) .(v) Observe that if τ = 0 then the bound given in (3.3) is of optimal order O ( n − / ) and is com-parable with the existing bounds (Theorems . . , and . for Poisson, negative binomial, andbinomial, respectively) given by ˇCekanaviˇcius and Vellaisamy [25] with the relaxation of theconditions (3 . − (3 . . X ∈ P in the following corollary. Corollary 3.1.
Assume that the conditions of Theorem 3.1 hold. Then, for any X ∈ P and n ≥ , d T V ( W n , X ) ≤ min (cid:26) , a (cid:27) ( n X i =1 c i ( n ) " | − b | h E ( X i ) E [ X N i, (2 X N i, − X N i, − E [ X i X N i, (2 X N i, − X N i, − i + E [ X i ( X N i, − + | τ (1 − b ) | ) . (3.28) Example 3.1.
Assume that the conditions of Corollary 3.1 hold. Moreover, let Y ∼ Poi ( λ ) , the Poissonrv, so that a = λ and b = 0 . Then, for n ≥ , d T V ( W n , Y ) ≤ min (cid:26) , λ (cid:27) ( n X i =1 c i ( n ) " h E ( X i ) E [ X N i, (2 X N i, − X N i, − E [ X i X N i, (2 X N i, − X N i, − i + E [ X i ( X N i, − + | ¯ τ | ) , where ¯ τ = Var( W n ) − λ . Example 3.2.
Assume that the conditions of Corollary 3.1 hold. Moreover, let U ∼ N B ( α, p ) , thenegative binomial rv, so that a = α (1 − p ) and b = 1 − p . Then, for n ≥ , d T V ( W n , U ) ≤ min (cid:26) , α (1 − p ) (cid:27) ( n X i =1 c i ( n ) " p h E ( X i ) E [ X N i, (2 X N i, − X N i, − E [ X i X N i, (2 X N i, − X N i, − i + E [ X i ( X N i, − + | ¯ τ p | ) , where ¯ τ = Var( W n ) − α (1 − p ) /p . The distribution of runs and patterns has been applied successfully in many areas such as reliabilitytheory, machine maintenance, quality control, and statistical testing, among many others. Also, it is nottractable if the underlying setup is arising from non-identical trials. So, the approximation of the runshas been studied by several researchers which includes, among others, Fu and Johnson [9], Godboleand Schaffner [10], Kumar and Upadhye [13, 23], Vellaisamy [26], and Wang and Xia [27]. In thissection, we mainly focus on 2-runs and ( k , k ) -runs, however, the results can also be extended to othertypes of runs. 14 .1 -runs We consider here the setup similar to the one discussed in Chapter 5 of Balakrishnan and Koutras [1,p. 166] for -runs. Let η , η , . . . , η n +1 be a sequence of independent Bernoulli trials with successprobability P ( η i = 1) = p i = 1 − P ( η i = 0) , for i = 1 , , . . . , n + 1 . Assume that p i ≤ / , for all i ,and R n := n X i =1 X i , (4.1)where X i = η i η i +1 , ≤ i ≤ n , is a sequence of 1-dependent rvs. Observe that R n counts the number ofoverlapping success runs of length 2 in n + 1 trials. It is easy to see that E X i = P ( X i = 1) = p i p i +1 := a ( p i ) . Similarly, E ( X i X i +1 ) = p i p i +1 p i +2 := a ( p i ) and E ( X i X i +1 X i +2 ) = p i p i +1 p i +2 p i +3 := a ( p i ) .Now, consider the first term in (3.3). Then E ( X N i, (2 X N i, − X N i, − E ( X i − + X i + X i +1 ) + E [( X i − + X i + X i +1 )(2 X i − + 2 X i +2 − i +1 X j = i − a ( p j ) + 2[ a ( p i − ) a ( p i +1 ) + a ( p i − )( a ( p i ) + a ( p i +1 ))+ a ( p i +2 )( a ( p i − ) + a ( p i ))] := ¯ a ( p i ) . (4.2)Similarly, E ( X i X Ni, (2 X N i, − X N i, − E ( X i ( X i − + X i + X i +1 ) )+ E [ X i ( X i − + X i + X i +1 )(2 X i − + 2 X i +2 − a ( p i )( a ( p i − ) + a ( p i +2 )) + 2 a ( p i − )(1 + a ( p i +2 ))+ 2 a ( p i )(1 + a ( p i − )) + 2 i X j = i − a ( p j ) =: ¯ a ( p i ) . (4.3)and E ( X i ( X N i, − E X i i +2 X j = i − X j ! − E ( X i )= a ( p i ) X | j − i | =2 a ( p j ) + i X j = i − a ( p j ) =: ¯ a ( p i ) . (4.4)15ext, recall from Remarks 3.1 ( ii ) with W n = R n and W ∗ i,n = R ∗ i,n , L ( R ∗ i,n | Z e = z e ) can be written assum of independent rvs, say X ( z e ) j , for j ∈ { , , . . . , n z e } ∩ { ℓ : | ℓ − i | > } =: C i , for i = 1 , , . . . , n .Note that n z e = m ∗ defined in (3.26) and X ( z e ) j = X j − depends only on X j − (2 j
6∈ { , i + 4 } ) and X j (2 j = i − , j ≤ ⌊ n/ ⌋ ) , j ∈ C i , for all values of z e . So, for simplicity, let us write X ( z e ) j = X ( x j − ,x j )2 j − , j ∈ C i , where x j − and x j are corresponding values of the rvs X j − and X j , respectively. Note that we usethe same notation D (cid:0) X ( x j − ,x j )2 j − (cid:1) , for j − ∈ { , i − , i + 3 , m ∗ } , while X j − depends either X j − or X j , not both. Therefore, from (3.25), we have V i,z e = X j ∈C i min (cid:26) , − D (cid:16) X ( x j − ,x j )2 j − (cid:17)(cid:27) . Note that D (cid:0) X (1 , j − (cid:1) = D (cid:0) X (1 , j − (cid:1) = D (cid:0) X (0 , j − (cid:1) = D (cid:0) X (0 , j − (cid:1) = , for all j ∈ C i , except when j − ∈ { , i − , i + 3 , m ∗ } . For j − ∈ { , i − , i + 3 , m ∗ } , we have D (cid:0) X ( x j − ,x j )2 j − (cid:1) = p j − or p j ≤ / ⇒ − D (cid:0) X ( x j − ,x j )2 j − (cid:1) ≥ / . Hence, V i,z e = X j ∈C i min (cid:26) , − D (cid:16) X ( x j − ,x j )2 j − (cid:17)(cid:27) ≥
12 ( m ∗ − , for all values of z e .Next, from (3.24), we have D ( R ∗ i,n ) ≤ E " V / i,Z e ≤ m ∗ − − / , for all i. Similarly, D ( R ∗ i,n ) ≤ E " V / i,Z o ≤ ⌊ n/ ⌋ − − / , for all i. Therefore, ¯ c i ( n ) = ¯ c ( n ) = 4 min n ( m ∗ − − / , ( ⌊ n/ ⌋ − − / o ≤ m ∗ − − / , for all i. (4.5)Hence, using (4.2), (4.3), (4.4), (4.5), and Theorem 3.1 and Remarks 3.1 ( ii ) , we obtain the followingtheorem. Theorem 4.1.
Let Z ∈ P and R n be defined as in (4.1) . Assume that E ( Z ) = E ( R n ) , and τ = ar( R n ) − Var( Z ) . Then, for n ≥ and p i ≤ / , d T V ( R n , Z ) ≤ k ∆ g k ( ¯ c ( n ) n X i =1 " | − b | h a ( p i )¯ a ( p i ) + ¯ a ( p i ) i + ¯ a ( p i ) + | τ (1 − b ) | ) . (4.6) Remark 4.1.
Note that k ∆ g k is of O ( n − ) in general, and hence, if Var( Z ) = Var( R n ) then the abovebound become of order O ( n − / ) and is comparable with the bounds given by Barbour and Xia [4],Brown and Xia [5], Daly et al. [6], and Wang and Xia [27]. In fact, if p i = p , for all ≤ i ≤ n + 1 ,then a ( p ) = p , ¯ a ( p ) = 8 p + 10 p , ¯ a ( p ) = 4 p + 10 p + 4 p , and ¯ a ( p ) = 2 p + 4 p , for all ≤ i ≤ n + 1 , and hence, we have from (4.6) , d T V ( R n , Z ) ≤ n k ∆ g k ¯ c ( n ) (cid:20) | − b | (cid:2) p + 10 p + 12 p + 10 p (cid:3) + 2( p + p ) (cid:21) . (4.7) Now, let Z ∼ NB ( α, ¯ p ) , the negative binomial distribution with parameter α and ¯ p . Then, b = 1 − ¯ p with ¯ p = 1 / (1 + 2 p − p ) and k ∆ g k ≤ α (1 − ¯ p ) = p − p np , where α and ¯ p are obtained from first twomoments matching condition, and hence, for n ≥ and p ≤ / , we get d T V ( R n , NB ( α, ¯ p )) ≤ p ( m ∗ − / [4 + 11 p + 4 p − p ] . (4.8) Also, from Theorem . of Brown and Xia [5], for n ≥ and p < / , we have d T V ( R n , NB ( α, ¯ p )) ≤ . p p ( n − − p ) . (4.9) For n ≥ , we compare our bound with the one given in (4.9) , due to Brown and Xia [5]. Somenumerical computations are given in the following table. Table 1: Comparison of bounds. n p
From (4.8) From (4.9) n From (4.8) From (4.9) n From (4.8) From (4.9)20 0.05 0.344694 0.398900 25 0.303992 0.354924 30 0.263265 0.3228800.07 0.506847 0.576571 0.446997 0.513008 0.387111 0.4666920.09 0.683285 0.765878 0.602601 0.681445 0.521867 0.61992235 0.11 0.618205 0.723476 40 0.561012 0.675509 50 0.493157 0.6026500.13 0.763728 0.884669 0.693072 0.826015 0.609244 0.7369230.15 0.919907 1.057010 0.834802 0.986930 0.733832 0.88048217 t is clear from the above table that our bound given in (4.8) is better than the bound given in (4.9) ,which is due to Brown and Xia [5]. ( k , k ) -runs In this subsection, we consider the setup similar to Huang and Tsai [12] and Vellaisamy [26]. Let I , I , . . . be a sequence of independent Bernoulli trials. Here, we consider I , I , . . . , I ( n +1)( k + k − with success probability P ( I i = 1) = p i = 1 − P ( I i = 0) , for i = 1 , , . . . , ( n + 1)( k + k − . Define m := k + k − and Y j := (1 − I j ) . . . (1 − I j + k − ) I j + k . . . I j + k + k − , j = 1 , , . . . , nm. (4.10)Note that Y , Y , . . . , Y nm is a sequence of m -dependent rvs. Now, let us also define X i := im X j =( i − m +1 Y j , for i = 1 , , . . . , n. Then X , X , . . . , X n become a sequence of -dependent rvs, that is, we reduced m -dependent to -dependent sequence of rvs. From (4.10), it is clear that Y i , for i = 1 , , . . . , nm , are Bernoulli rvs andif Y i = 1 then Y j = 0 , for all j such that | j − i | ≤ m and j = i . Therefore, X i , i = 1 , , . . . , n , are alsoBernoulli rvs. Next, let R ′ n = nm X i =1 Y i = n X i =1 X i , (4.11)the sum of the corresponding -dependent rvs ( X i ’s). The distribution of R ′ n is called the distributionof ( k , k ) -runs or modified distribution of order k or distribution of order ( k , k ) . For more details.we refer the reader to Balakrishnan and Koutras [1], Huang and Tsai [12], Upadhye and Kumar [23],Vellaisamy [26] and reference therein.Next, note that E ( Y j ) = P ( Y j = 1) = (1 − p j ) . . . (1 − p j + k − ) p j + k . . . p j + k + k − =: a ( p j ) , for j = 1 , , . . . , nm, and hence E X i = im X j =( i − m +1 E Y j = im X j =( i − m +1 a ( p j ) =: a ∗ ( p i ) , for i = 1 , , . . . , n. (4.12)18lso, E ( X i X i +1 ) = im − X ℓ =( i − m +1 a ( p ℓ ) ( i +1) m X ℓ = ℓ + m +1 a ( p ℓ ) + ( i +1) m X ℓ = im +2 a ( p ℓ ) ℓ − m − X ℓ =( i − m +1 a ( p ℓ ) =: a ∗ ( p i p i +1 ) . (4.13)and E ( X i X i +1 X i +2 ) = im − X ℓ =( i − m +1 a ( p ℓ ) ( i +1) m − X ℓ = ℓ + m +1 a ( p ℓ ) ( i +2) m X ℓ = ℓ + m +1 a ( p ℓ )+ ( i +1) m − X ℓ = im +2 a ( p ℓ ) ℓ − m − X ℓ =( i − m +1 a ( p ℓ ) ( i +2) m X ℓ = ℓ + m +1 a ( p ℓ )+ ( i +2) m X ℓ =( i +1) m +3 a ( p ℓ ) ℓ − m − X ℓ = im +2 a ( p ℓ ) ℓ − m − X ℓ =( i − m +1 a ( p ℓ )=: a ∗ ( p i p i +1 p i +2 ) . (4.14)Using the steps similar to (4.2)-(4.4) with (4.12), (4.13) and (4.14), we have E ( X N i, (2 X N i, − X N i, − ≤ i +1 X j = i − a ∗ ( p j p j +1 ) + 2[ a ∗ ( p i − ) a ∗ ( p i +1 )+ a ∗ ( p i − )( a ∗ ( p i ) + a ∗ ( p i +1 )) + a ∗ ( p i +2 )( a ∗ ( p i − ) + a ∗ ( p i ))]:= a ∗ ( p i ) , (4.15) E ( X i X N i, (2 X N i, − X N i, − ≤ a ∗ ( p i )( a ∗ ( p i − ) + a ∗ ( p i +2 )) + 2 a ∗ ( p i − p i )(1 + a ∗ ( p i +2 ))+ 2 a ∗ ( p i p i +1 )(1 + a ∗ ( p i − )) + 2 i X j = i − a ∗ ( p j p j +1 p j +2 )=: a ∗ ( p i ) . (4.16)and E ( X i ( X N i, − ≤ a ∗ ( p i ) X | j − i | =2 a ∗ ( p j ) + i X j = i − a ∗ ( p j p j +1 ) =: a ∗ ( p i ) . (4.17)19ext, from Subsection 4.1, following the discussion about Remarks 3.1 ( ii ) , we have V i,z e = X j ∈C i min (cid:26) , − D (cid:16) X ( x j − ,x j )2 j − (cid:17)(cid:27) . Note that D (cid:16) X ( x j − ,x j )2 j − (cid:17) = 12 h P (cid:16) X ( x j − ,x j )2 j − = 0 (cid:17) + (cid:12)(cid:12)(cid:12) P (cid:16) X ( x j − ,x j )2 j − = 0 (cid:17) − P (cid:16) X ( x j − ,x j )2 j − = 1 (cid:17)(cid:12)(cid:12)(cid:12)i ≤ h P (cid:16) X ( x j − ,x j )2 j − = 0 (cid:17)i ≤
12 [1 + ¯ a ( p j − )] , (4.18)where ¯ a ( p j − ) = max ≤ x j − , x j ≤ P (cid:16) X ( x j − ,x j )2 j − = 0 (cid:17) . (4.19)Next, using (4.18), we have V / i,z e ≤
12 min ( , X j ∈C i (1 − ¯ a ( p j − )) )! − / , for all z e Therefore, from (3.24), we have D ( R ′ ∗ i,n ) ≤ E " V / i,Z e ≤
12 min ( , X j ∈C i (1 − ¯ a ( p j − )) )! − / =: V ∗ i,e . Similarly, D ( R ′ ∗ i,n ) ≤ E " V / i,Z o ≤
12 min ( , X j ∈D i (1 − ¯ a ( p j − )) )! − / =: V ∗ i,o , where D i = { , , . . . , ⌊ n/ ⌋} ∩ { ℓ : | ℓ − i | > } . Therefore, c ∗ i ( n ) = min { V ∗ i,e , V ∗ i,o } . (4.20)Using (4.12), (4.15)-(4.17), (4.20), Theorem 3.1 and Remarks 3.1 ( ii ) , the following result is estab-lished. Theorem 4.2.
Let Z ∈ P and R ′ n be defined as in (4.11) . Assume that E ( Z ) = E ( R ′ n ) , and τ = ar( R ′ n ) − Var( Z ) . Then, for n ≥ m , a ( p j − ) ≤ / defined in (4.19) , d T V ( R ′ n , Z ) ≤ k ∆ g k ( n X i =1 c ∗ i ( n ) " | − b | h a ∗ ( p i ) a ∗ ( p i ) + a ∗ ( p i ) i + a ∗ ( p i ) + | τ (1 − b ) | ) . Remark 4.2.
Note that the above bound is comparable with the existing bounds given by Upadhyeand Kumar [23] and order improvement over the bounds given by Barbour et al. [3], Godbole [11],Godbole and Schaffner [10] (with k = 1 ), and Vellaisamy [26]. Also, note that we have used a slightlydifferent form of ( k , k ) -runs, that is, we use I , I , . . . , I ( n +1)( k + k − instead of I , I , . . . , I n , so that X , X , . . . , X n become a sequence of 1-dependent rvs and we can directly apply our result. However,we can also use some other forms and derive the corresponding results. References [1] Balakrishnan, N. and Koutras, M. V. (2002).
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