1 Introduction and Preliminaries

Sums of random variables (rvs) have always a special attraction as it raises in relevant theoretical challenges. Moreover, several linear statistics can be represented as weighted sums of rvs. Also, it is difficult to find the exact distribution of weighted sums of rvs, especially, if the underlying rvs are non-identical. So, it is of interest to study the behavior of such distributions. Many researchers studied the limiting behavior of weighted sums of rvs such as Chow and Lai [7], Olvera-Cravioto [18], and Zhengyan [28], among many others. But if weights are natural numbers, then it is also difficult to get asymptotic limits. Therefore, the study of the proximity of such distributions with a suitable distribution is of interest when the summation is taken over a finite set.

In this paper, we consider weighted sums of \({\mathbb {Z}}_+\)-valued rvs, where \({\mathbb {Z}}_+=\{0,1,2,\ldots \}\), the set of nonnegative integers, and propose its approximation with pseudo-binomial and negative binomial distributions by matching the first two moments. Also, we assume weights are natural numbers. Let \(X_1\sim \) PB(N, p), the pseudo-binomial distribution (see Čekanavičius and Roos [24], p. 370), and \(X_2\sim \) NB\((r,{\bar{p}})\), the negative binomial distribution, then their probability mass functions are given by

$$\begin{aligned} {\mathbb {P}}(X_1=k)=\frac{1}{\delta }\left( {\begin{array}{c}N\\ k\end{array}}\right) {p}^{k}q^{N-k}, \quad k=0,1,\ldots ,\left\lfloor N\right\rfloor \end{aligned}$$

and

$$\begin{aligned} {\mathbb {P}}(X_2=k)=\left( {\begin{array}{c}r+k-1\\ k\end{array}}\right) {\bar{p}}^r {\bar{q}}^{k}, \quad k=0,1,\ldots , \end{aligned}$$

respectively, where \(N>1\), \(r>0\), \(0<q=1-p<1\), \(0<{\bar{q}}=1-{\bar{p}}<1\), \(\delta =\sum _{k=0}^{\left\lfloor N\right\rfloor }\left( {\begin{array}{c}N\\ k\end{array}}\right) {p}^{k}q^{N-k}\), \(\left( {\begin{array}{c}N\\ k\end{array}}\right) =N(N-1)\cdots (N-k+1)/k!\), and \(\left\lfloor N\right\rfloor \) is the greatest integer function of N. The study of asymptotic behavior for weighted sums of rvs is discussed widely in the literature under certain conditions on weights; for example, the sum of squares of weights is finite (see Chow and Lai [7]), weights are normalized (see Etemadi [9]) and geometrically weighted (see Bhati and Rattihalli [6]), among many others. However, we consider weights are natural numbers which do not satisfy these types of conditions and obtain error bounds for pseudo-binomial and negative binomial approximations. This study of proximity is useful to identify the behavior of such distributions over a finite set. We use the total variation distance metric and Stein’s method to derive our approximation results.

Next, let \({\mathcal {G}}=\{g:{{\mathbb {Z}}}_+\rightarrow {{\mathbb {R}}}|~g~\text {is bounded}\}\) and \({\mathcal {G}}_Y=\{g\in {\mathcal {G}}|~g(0)=0~\text {and}~g(y)=0,~\text {for}~y\notin \mathbf{\text {S}}(Y)\}\), for a \({\mathbb {Z}}_+\)-valued rv Y, where S(Y) is the support of the rv Y. We discuss briefly Stein’s method (Stein [20]) which can be carried out mainly in the three steps. First, compute a Stein operator \({\mathcal {A}}_Y\) which satisfies \({{\mathbb {E}}}[{\mathcal {A}}_Y g(Y)]=0\), for \(g \in {\mathcal {G}}_Y\). Second, find the solution of Stein equation

$$\begin{aligned} {\mathcal {A}}_Yg(k)=f(k)-{{\mathbb {E}}}f(Y),~f \in {\mathcal {G}}~\text {and}~g \in {\mathcal {G}}_Y. \end{aligned}$$
(1.1)

Finally, use a rv Z in place of k in (1.1) and take expectation and supremum which leads to the total variation distance between Y and Z as follows:

$$\begin{aligned} d_{TV}(Y,Z):=\sup _{f \in {\mathcal {H}}}|{{\mathbb {E}}}f(Y)-{\mathbb E}f(Z)|=\sup _{f \in {\mathcal {H}}}|{{\mathbb {E}}}{\mathcal {A}}_Yg(Z)|, \end{aligned}$$
(1.2)

where \({\mathcal {H}}=\{I(A)| A \subseteq {{\mathbb {Z}}}_+\}\) and I(A) is the indicator function of the set A.

Next, consider a rv X and its Stein operator of the form

$$\begin{aligned} {\mathcal {A}}_Xg(k)=(\alpha +\beta k)g(k+1)-kg(k), \quad k\in {\mathbb {Z}}_+,~g\in {\mathcal {G}}_X, \end{aligned}$$
(1.3)

which represent pseudo-binomial Stein operator if \(\alpha =Np/q\) and \(\beta =-p/q\) and negative binomial Stein operator if \(\alpha =r {\bar{q}}\) and \(\beta ={\bar{q}}\), respectively. For details, see (5) and (6) of Upadhye et al. [23]. Also, the upper bound for the solution of (1.1) (say \(g_f\)) is given by

$$\begin{aligned} \Vert \Delta g_f\Vert \le \left\{ \begin{array}{ll} 1/\left\lfloor N\right\rfloor p, &{} \text {if }X\sim \text { PB}(N,p);\\ 1/r {\bar{q}}, &{} \text {if }X\sim \text { NB}(r,{\bar{p}}), \end{array}\right. \end{aligned}$$
(1.4)

where \(\Delta g_f(k)=g_f(k+1)-g_f(k)\) and \(\Vert \Delta g_f\Vert =\sup _k |\Delta g_f(k)|\). See (2.8), (2.10), and (2.11) of Kumar et al. [14] and (57) of Čekanavičius and Roos [24] for more details. Observe that

$$\begin{aligned} \frac{\alpha }{1-\beta }=\left\{ \begin{array}{ll} N p, &{} \text {if } X\sim \text { PB }(N,p);\\ r {\bar{q}}/{\bar{p}}, &{} \text {if } X\sim \text { NB}(r,{\bar{p}}) \end{array}\right. \quad \text {and}\quad \frac{\alpha }{(1-\beta )^2}=\left\{ \begin{array}{ll} N pq, &{} \text {if }X\sim \text {PB}(N,p);\\ r {\bar{q}}/{\bar{p}}^2, &{} \text {if }X\sim \text {NB}(r,{\bar{p}}) \end{array}\right. \end{aligned}$$

are mean and variance of pseudo-binomial and negative binomial distributions, respectively. For more details, we refer the reader to Brown and Xia [5], Eichelsbacher and Reinert [8], Kumar et al. [14], Ley et al. [16], Upadhye and Barman [22], Upadhye et al. [23], and references therein.

This paper is organized as follows: In Sect. 2, we present our main results and discuss some relevant remarks and applications. In Sect. 3, we give the proofs of our main results.

2 Main Results

Let \(J\subset {\mathbb {N}}=\{1,2,\ldots \}\) be finite and \(\{\eta _i,i\in J\}\) be a collection of \({\mathbb {Z}}_+\)-valued random variables. Also, for each i, let \(w_i\in {\mathbb {N}}\), \({\mathbb {E}}(\omega _i \eta _i)^3<\infty \), and \(i\in A_i\subseteq B_i\subset J\) be such that \(\eta _i\) is independent of \(\eta _{A_i^c}\) and \(\eta _{A_i}\) is independent of \(\eta _{B_i^c}\), where \(\eta _A\) is the collection of random variables \(\{\eta _i,i\in A\}\) and \(A^c\) denotes the complement of the set A. See Section 3 of Röllin [19] for a similar type of locally dependent structure. In addition, if \(A_i=B_i=\{i\}\), then our locally dependent structure reduced to the independent collection of random variables. Now, let \(w_i=1\) for at least one \(i\in J\) and define

$$\begin{aligned} W:=\sum _{i\in J}\omega _i \eta _i, \end{aligned}$$
(2.1)

the weighted sum of locally dependent random variables. For a set \(A\subset J\), we let \(\eta _{A}^{*}=\sum _{i\in A}\omega _i \eta _i\). For any random variables Z, we define \({\mathcal {D}}(Z):=2d_{TV}({\mathcal {L}}(Z),{\mathcal {L}}(Z+1))\). Throughout this section, let X be a random variable having Stein operator (1.3) and

$$\begin{aligned} \alpha =\frac{({\mathbb {E}}W)^2}{\mathrm {Var}(W)}\quad \text {and}\quad \beta =\frac{\mathrm {Var}(W)-{\mathbb {E}}W}{\mathrm {Var}(W)} \end{aligned}$$
(2.2)

so that \({\mathbb {E}}X={\mathbb {E}}W\) and \(\mathrm {Var}(X)=\mathrm {Var}(W)\).

2.1 Locally Dependent Random Variables

In this subsection, we consider \(\{i\}\subset A_i\subset B_i\) and discuss the approximation result for the weighted sum of locally dependent random variables.

Theorem 2.1

Let W be the weighted sum of locally random variables as defined in (2.1) and X be a random variable having Stein operator (1.3) satisfying (2.2). Then,

$$\begin{aligned} d_{TV}({\mathcal {L}}(W),{\mathcal {L}}(X)) \le&\Vert \Delta g\Vert \left\{ \frac{(1-\beta )}{2}\left[ \sum _{i\in J} \omega _i{\mathbb {E}}\eta _i {\mathbb {E}}[\eta _{A_i}^*(2\eta _{B_i}^{*}-\eta _{A_i}^{*}-1){\mathcal {D}}(W|\eta _{A_i},\eta _{B_i})]\right. \right. \nonumber \\&\left. +\sum _{i\in J}\omega _i{\mathbb {E}}[\eta _i(\eta _{A_i}^{*}-1)(2\eta _{B_i}^{*}-\eta _{A_i}^{*}-2){\mathcal {D}}(W|\eta _i,\eta _{A_i},\eta _{B_i})]\right] \nonumber \\&+\sum _{i\in J}\omega _i\bigr |(1-\beta )[{\mathbb {E}}(\eta _i) {\mathbb {E}}(\eta _{A_i}^*)\nonumber \\&-{\mathbb {E}}(\eta _i(\eta _{A_i}^{*}-1))]+\beta {\mathbb {E}}(\eta _i)\bigr |{\mathbb {E}}[\eta _{B_i}^{*}{\mathcal {D}}(W|\eta _{B_i})]\nonumber \\&\left. +|\beta | \sum _{i\in J} \omega _i{\mathbb {E}}[\eta _i(\eta _{B_i}^{*}-1){\mathcal {D}}(W|\eta _{B_i})]\right\} , \end{aligned}$$
(2.3)

where the upper bound of \(\Vert \Delta g\Vert \) is given in (1.4).

Remark 2.1

  1. (i)

    Observe that W can be represented as a conditional sum of independent random variables. Therefore, Subsections 5.3 and 5.4 of Röllin [19] are useful to find the upper bound of \({\mathcal {D}}(W|\cdot )\).

  2. (ii)

    The choice of parameters in (2.2) is valid if \({\mathbb {E}}W>\mathrm {Var}(W)\) and \({\mathbb {E}}W<\mathrm {Var}(W)\) for pseudo-binomial and negative binomial approximations, respectively.

Next, we discuss some applications of Theorem 2.1.

Example 2.1

((1,1)-runs) Let \(J=\{1,2,\ldots ,n\}\) and \(\{\zeta _i,i \in J\}\) be a sequence of independent Bernoulli trials with success probability \(p_i={\mathbb {P}}(\zeta _i=1)=1-{\mathbb {P}}(\zeta _i=0)\). Also, let \(A_i=\{j:|j-i|\le 1\}\cap J\), \(B_i=\{j:|j-i|\le 2\}\cap J\), \({\bar{\zeta }}_i=(1-\zeta _{i-1})\zeta _i\), and \(W_n=\sum _{i=2}^{n}{\bar{\zeta }}_i\). Then, the distribution of \(W_n\) is known as the distribution of (1, 1)-runs and it adopted our locally dependent structure with \(\omega _i=1\). For more details, see Huang and Tsai [12], Upadhye et al. [23], Vellaisamy [25], and reference therein.

Next, it can be easily verified that

$$\begin{aligned}&\sum _{i=2}^{n}(1-p_{i-1})p_i={\mathbb {E}}(W_n)>\mathrm {Var}(W_n)\\&=\sum _{i=2}^{n}(1-p_{i-1})p_i-\sum _{i=2}^{n}\sum _{j\in A_i}(1-p_{i-1})(1-p_{j-1})p_ip_j. \end{aligned}$$

Therefore, the pseudo-binomial approximation to \(W_n\) is suitable in the view of the valid choice of parameters. Now, let \({\mathcal {D}}(W_i^\star )={\mathcal {D}}(W_n|{\bar{\zeta }}_{B_i})\) and \({\bar{\zeta }}_{e}=\{{\bar{\zeta }}_{2k},1\le k\le \left\lfloor n/2\right\rfloor \}\) then \({\mathcal {L}}(W_i^\star |{\bar{\zeta }}_e=k)\) can be represented as the sum of independent random variables (say \(\zeta _i^{k}\), \(i\in \{1,2,\ldots ,n_k\}=:{\mathcal {F}}_k\) and \(k\in \{0,1\}^{\left\lfloor n/2\right\rfloor }\)), and therefore, from (5.11) if Röllin [19], we have

$$\begin{aligned} {\mathcal {D}}(W_i^\star )\le {\mathbb {E}}\{{\mathbb {E}}[W_i^\star |{\bar{\zeta }}_e]\}\le {\mathbb {E}}\left[ \frac{2}{V_{{\bar{\zeta }}_e}^{1/2}}\right] , \end{aligned}$$

where \(V_{{\bar{\zeta }}_e=k}=\sum _{j \in {\mathcal {F}}_k}\min \left\{ 1/2,1-{\mathcal {D}}(\zeta _j^k)\right\} \). Let \(1/2 \ge {\mathbb {P}}(\zeta _i^k=1)=1-{\mathbb {P}}(\zeta _i^k=0)=\sum _{k_1,k_2\in \{0,1\}}\) \({\mathbb {P}}({\bar{\zeta }}_i=1|{\bar{\zeta }}_{i-1}=k_1,{\bar{\zeta }}_{i+1}=k_2)={\mathbb {P}}({\bar{\zeta }}_i=1|{\bar{\zeta }}_{i-1}=0,{\bar{\zeta }}_{i+1}=0)=(1-p_{i-1})p_i\). Therefore,

$$\begin{aligned} 1-{\mathcal {D}}(\zeta _i^k)&=1-\frac{1}{2}\sum _{m=0}^{1}|{\mathbb {P}}(\zeta _i^k=m-1)-{\mathbb {P}}(\zeta _i^k=m)|\\&=1-\frac{1}{2}\{2{\mathbb {P}}(\zeta _i^k=0)-{\mathbb {P}}(\zeta _i^k=1)\}=\frac{3}{2}(1-p_{i-1})p_i. \end{aligned}$$

Hence, \(V_{{\bar{\zeta }}_e=k}=\frac{1}{2}\sum _{j \in {\mathcal {F}}_k}\min \left\{ 1,3(1-p_{i-1})p_i\right\} \) and, from Theorem 2.1 with the pseudo-binomial setting, we have

$$\begin{aligned} d_{TV}({\mathcal {L}}(W_n),\text {PB}({\hat{n}},{\hat{p}})) \le&\frac{1}{\left\lfloor {\hat{n}}\right\rfloor {\hat{p}}{\hat{q}}}\left\{ \sum _{i=2}^{n}\big [ {\mathbb {E}}{\bar{\zeta }}_i {\mathbb {E}}[{\bar{\zeta }}_{A_i}^*(2{\bar{\zeta }}_{B_i}^{*}-{\bar{\zeta }}_{A_i}^{*}-1)]+{\mathbb {E}}[{\bar{\zeta }}_i({\bar{\zeta }}_{A_i}^{*}-1)(2{\bar{\zeta }}_{B_i}^{*}-{\bar{\zeta }}_{A_i}^{*}-2)]\big ]\right. \\&+2\sum _{i=2}^n\bigr |{\mathbb {E}}({\bar{\zeta }}_i) {\mathbb {E}}({\bar{\zeta }}_{A_i^*})-{\mathbb {E}}({\bar{\zeta }}_i({\bar{\zeta }}_{A_i}^{*}-1))-{\hat{p}}{\mathbb {E}}({\bar{\zeta }}_i)\bigr |{\mathbb {E}}[{\bar{\zeta }}_{B_i}^{*}]\\&\left. +2{\hat{p}} \sum _{i=2}^{n}{\mathbb {E}}[{\bar{\zeta }}_i({\bar{\zeta }}_{B_i}^{*}-1)]\right\} \left( \frac{1}{2}\sum _{j \in {\mathcal {F}}}\min \left\{ 1,3(1-p_{i-1})p_i\right\} \right) ^{-1/2}, \end{aligned}$$

where \({\hat{n}}=\left( \sum _{i=2}^{n}(1-p_{i-1})p_i\right) ^2/\sum _{i=2}^{n}\sum _{j\in A_i}(1-p_{i-1})(1-p_{j-1})p_ip_j\), \({\hat{p}}=\sum _{i=2}^{n}\sum _{j\in A_i}(1-p_{i-1})(1-p_{j-1})p_ip_j/\sum _{i=2}^{n}(1-p_{i-1})p_i\), and \({\mathcal {F}}=\min _{k}{\mathcal {F}}_k\). Note that the above bound is of \(O(n^{-1/2})\) which is an improvement over (77) of Upadhye et al. [23], Theorem 2.1 of Vellaisamy [25], which are of O(1), and Theorem 2.1 of Godbole [11], which is of order O(n).

Example 2.2

(Collateralized Debt Obligation (CDO)) A CDO is a type of asset-backed security that transferred pool of assets into a product and sold to investors. These assets divided into a set of repayment which is called tranches. The tranches have different payment priorities and interest rates. The basic tranches used in CDO are senior, mezzanine, and equity. Investors can invest in their interested tranches. For more details, see Neammanee and Yonghint [17], Yonghint et al. [27], and reference therein.

In Yonghint et al. [27], it is demonstrated that the locally dependent CDO occurs in the borrower’s related assets that arise from several groups. If the element of groups have weights in terms of economy. Then, the weighted locally dependent CDO is also useful in applications.

We consider the CDO similar to discussed by Yonghint et at. [27]. Let the CDO tranche pricing is based on \({\bar{n}}\) assets and the recovery rate of ith assets is \(R_i>0\). The percentage cumulative loss in CDO up to time T is

$$\begin{aligned} L(T)=\frac{1}{{\bar{n}}}\sum _{i=1}^{{\bar{n}}}(1-R_i)\omega _i I_i, \end{aligned}$$

where \(I_i=I(\tau _i \le T)\), and \(\tau _i\) is the default time of the ith asset. Assume the recovery rate is constant, say R; then, the CDO pricing problem is reduced to calculate

$$\begin{aligned} {\mathbb {E}}[(L(T ) - z^*)^+]=\frac{1-R}{{\bar{n}}}{\mathbb {E}}[({\overline{W}}_{{\bar{n}}}-z)^+], \end{aligned}$$
(2.4)

where \(z^*=(1-R)z/{\bar{n}}>0\) is the attachment or the detachment point of the tranche, \({\overline{W}}_{{\bar{n}}}=\sum _{i=1}^{{\bar{n}}} \omega _i I_i\), and \((a)^+=\max (a,0)\). Note that, from (2.4), it is sufficient to deal with \({\mathbb {E}}[({\overline{W}}_{{\bar{n}}}-z)^+]\). For additional details, see Yonghint et al. [27] and reference therein.

We are interested to approximate \({\mathbb {E}}[({\overline{W}}_{{\bar{n}}}-z)^+]\) by \({\mathbb {E}}[(\text {PB}(N,p)-z)^+]\). First, let us modify the Stein equation (1.1) as

$$\begin{aligned} (k-z)^+-{\mathbb {E}}[(\text {PB}(N,p)-z)^+]={\mathcal {A}}g(k). \end{aligned}$$
(2.5)

Here, \(f:{\mathbb {Z}}_+\rightarrow {\mathbb {R}}\) such that \(f(k)=(k-z)^+\). Using the rv \({\overline{W}}_{{\bar{n}}}\) in place of k and taking expectation, we get

$$\begin{aligned} {\mathbb {E}}[({\overline{W}}_{{\bar{n}}}-z)^+]-{\mathbb {E}}[(\text {PB}(N,p)-z)^+]={\mathbb {E}}[{\mathcal {A}}g({\overline{W}}_{{\bar{n}}})]. \end{aligned}$$

Therefore, it is enough to deal with the right-hand side, that is, \({\mathbb {E}}[{\mathcal {A}}g({\overline{W}}_{{\bar{n}}})]\).

Next, we move to find the upper bound for \(\Vert \Delta g\Vert \). Following the steps similar to Lemma 1 of Neammanee and Yonghint [17], for \(z\ge 0\), we have

$$\begin{aligned}&{\mathbb {E}}[(\text {PB}(N,p)-z)^+]=\frac{1}{\delta }\sum _{m=1}^{\left\lfloor N\right\rfloor }(m-z)^+\left( {\begin{array}{c}N\\ m\end{array}}\right) p^m q^{N-m}\nonumber \\&\le \frac{1}{\delta }\sum _{m=1}^{\left\lfloor N\right\rfloor }m\left( {\begin{array}{c}N\\ m\end{array}}\right) p^m q^{N-m}= Np. \end{aligned}$$
(2.6)

It can be easily verified that (2.5) has a solution

$$\begin{aligned}&g(k)=-\sum _{j=k}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)](k-1)!}{[N(N-1)\cdots (N-k+1)]j!}\left( \frac{p}{q}\right) ^{j-k}\\&\times \big ((j-z)^+-{\mathbb {E}}[(\text {PB}(N,p)-z)^+]\big ), \end{aligned}$$

for \(k\ge 1\). For details, see (2.6) of Eichelsbacher and Reinert [8]. Now, following the steps similar to Lemma 2 Neammanee and Yonghint [17], for \(k\ge 1\), we get

$$\begin{aligned} 0<\sum _{j=k}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)](k-1)!}{[N(N-1)\cdots (N-k+1)]j!}\left( \frac{p}{q}\right) ^{j-k}(j-z)^+&\le 1+\sum _{j=1}^{\left\lfloor N-k\right\rfloor }\left( {\begin{array}{c}N-k\\ j\end{array}}\right) \left( \frac{p}{q}\right) ^j\nonumber \\&\le q^{\left\lceil k-N\right\rceil }\le q^{\left\lceil 1-N\right\rceil }, \end{aligned}$$
(2.7)

where \(\left\lceil x\right\rceil \) denote the least integer more than or equal to x. Also, for \(k\le Np\), we have

$$\begin{aligned} \sum _{j=k}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)](k-1)!}{[N(N-1)\cdots (N-k+1)]j!}\left( \frac{p}{q}\right) ^{j-k}&\le \sum _{j=k}^{\left\lfloor N\right\rfloor }\frac{(N-k)\cdots (N-j+1)}{(j-k+1)!}\left( \frac{p}{q}\right) ^{j-k}\nonumber \\&\le \frac{1}{N-k+1}\sum _{j=0}^{\left\lfloor N-k\right\rfloor }\left( {\begin{array}{c}N-k+1\\ j+1\end{array}}\right) \left( \frac{p}{q}\right) ^j\nonumber \\&\le \frac{q^{\left\lceil k-N\right\rceil }-q}{(N-k)p}\le \frac{q^{-\left\lceil N\right\rceil }}{Np} \end{aligned}$$
(2.8)

and, for \(k\ge np\),

$$\begin{aligned} \sum _{j=k}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)](k-1)!}{[N(N-1)\cdots (N-k+1)]j!}\left( \frac{p}{q}\right) ^{j-k}&=\frac{1}{k}\left( 1+ \sum _{j=k+1}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)]k!}{[N(N-1)\cdots (N-k+1)]j!}\left( \frac{p}{q}\right) ^{j-k}\right) \nonumber \\&\le \frac{1}{k}\left( 1+ \sum _{j=1}^{\left\lfloor N-k\right\rfloor }\left( {\begin{array}{c}N-k\\ j\end{array}}\right) \left( \frac{p}{q}\right) ^{j}\right) \le \frac{q^{-\left\lceil N\right\rceil }}{Np}. \end{aligned}$$
(2.9)

Combining (2.8) and (2.9), for \(k\ge 1\), we have

$$\begin{aligned} 0< \sum _{j=k}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)](k-1)!}{[N(N-1)\cdots (N-k+1)]j!}\left( \frac{p}{q}\right) ^{j-k} \le \frac{q^{-\left\lceil N\right\rceil }}{Np}. \end{aligned}$$
(2.10)

Therefore, from (2.6) and (2.10), we have

$$\begin{aligned} \sum _{j=k}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)](k-1)!}{[N(N-1)\cdots (N-k+1)]j!}\left( \frac{p}{q}\right) ^{j-k} {\mathbb {E}}[(\text {PB}(N,p)-z)^+]\le q^{-\left\lceil N\right\rceil }. \end{aligned}$$
(2.11)

Next, observe that

$$\begin{aligned} \Delta g(k)=g(k+1)-g(k)=C_k+D_k, \end{aligned}$$

where

$$\begin{aligned} C_k=&\sum _{j=k}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)](k-1)!}{[N(N-1)\cdots (N-k+1)]j!}\left( \frac{p}{q}\right) ^{j-k}(j-z)^+\\&-\sum _{j=k+1}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)]k!}{[N(N-1)\cdots (N-k)]j!}\left( \frac{p}{q}\right) ^{j-k-1}(j-z)^+ \end{aligned}$$

and

$$\begin{aligned} D_k=&\sum _{j=k+1}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)]k!}{[N(N-1)\cdots (N-k)]j!}\left( \frac{p}{q}\right) ^{j-k-1} {\mathbb {E}}[(\text {PB}(N,p)-z)^+]\\&-\sum _{j=k}^{\left\lfloor N\right\rfloor }\frac{[N(N-1)\cdots (N-j+1)](k-1)!}{[N(N-1)\cdots (N-k+1)]j!}\left( \frac{p}{q}\right) ^{j-k}{\mathbb {E}}[(\text {PB}(N,p)-z)^+]. \end{aligned}$$

Using (2.7) and (2.11), we have

$$\begin{aligned} |\Delta g|\le |C_k|+|D_k|\le q^{-\left\lceil N\right\rceil }(1+q). \end{aligned}$$

Hence, from (3.5), we have

$$\begin{aligned} \left| {\mathbb {E}}[{\mathscr {A}}g({\overline{W}}_{{\bar{n}}})]\right| \le&\frac{(1+q)}{q^{\left\lceil N\right\rceil +1}}\left\{ \frac{1}{2}\left[ \sum _{i=1}^{{\bar{n}}} \omega _i{\mathbb {E}}I_i {\mathbb {E}}[I_{A_i}^*(2I_{B_i}^{*}-I_{A_i}^{*}-1){\mathcal {D}}({\overline{W}}_{{\bar{n}}}|I_{A_i},I_{B_i})]\right. \right. \nonumber \\&\left. +\sum _{i=1}^{{\bar{n}}}\omega _i{\mathbb {E}}[I_i(I_{A_i}^{*}-1)(2I_{B_i}^{*}-I_{A_i}^{*}-2){\mathcal {D}}({\overline{W}}_{{\bar{n}}}|I_i,I_{A_i},I_{B_i})]\right] \nonumber \\&+\sum _{i=1}^{{\bar{n}}}\omega _i\bigr |{\mathbb {E}}(I_i) {\mathbb {E}}(I_{A_i^*})-{\mathbb {E}}(I_i(I_{A_i}^{*}-1))-p{\mathbb {E}}(I_i)\bigr |{\mathbb {E}}[I_{B_i}^{*}{\mathcal {D}}(W|I_{B_i})]\nonumber \\&\left. +p\sum _{i=1}^{{\bar{n}}} \omega _i{\mathbb {E}}[I_i(I_{B_i}^{*}-1){\mathcal {D}}({\overline{W}}_{{\bar{n}}}|I_{B_i})]\right\} , \end{aligned}$$
(2.12)

where \(q=1-p=\mathrm {Var}({\overline{W}}_{{\bar{n}}})/{\mathbb {E}}{\overline{W}}_{{\bar{n}}}\), \(N=({\mathbb {E}}{\overline{W}}_{{\bar{n}}})^2/({\mathbb {E}}{\overline{W}}_{{\bar{n}}}-\mathrm {Var}({\overline{W}}_{{\bar{n}}}))\), \(I_{A}^{*}=\sum _{i\in A}\omega _i I_i\), \(I_{A}\) is the collection of random variables \(\{I_i:i\in A\}\), for \(A\subset \{1,2,\ldots ,{\bar{n}}\}\), and \({\mathcal {D}}({\overline{W}}_{{\bar{n}}}|\cdot )\) can be computed subject to the exact structure of dependency. For example, if the dependency structure is the same as discussed in Example 2.1 and \(\omega _i=1\), \(1\le i \le n\), then

$$\begin{aligned} \left| {\mathbb {E}}[{\mathscr {A}}g({\overline{W}}_{{\bar{n}}})]\right| \le&\frac{(1+q)}{q^{\left\lceil N\right\rceil +1}}\left\{ \sum _{i=1}^{{\bar{n}}}\big [ {\mathbb {E}}I_i {\mathbb {E}}[I_{A_i}^*(2I_{B_i}^{*}-I_{A_i}^{*}-1)]+{\mathbb {E}}[I_i(I_{A_i}^{*}-1)(2I_{B_i}^{*}-I_{A_i}^{*}-2)]\big ]\right. \\&+2\sum _{i=1}^{{\bar{n}}}\bigr |{\mathbb {E}}(I_i) {\mathbb {E}}(I_{A_i^*})-{\mathbb {E}}(I_i(I_{A_i}^{*}-1))-p{\mathbb {E}}(I_i)\bigr |{\mathbb {E}}[I_{B_i}^{*}]\\&\left. +2p \sum _{i=1}^{{\bar{n}}}{\mathbb {E}}[I_i(I_{B_i}^{*}-1)]\right\} \left( \frac{1}{2}\sum _{j \in {\mathcal {F}}}\min \left\{ 1,3(1-{\mathbb {E}}(I_{i-1})){\mathbb {E}}I_i\right\} \right) ^{-1/2}, \end{aligned}$$

which is an improvement over the bound given in Theorem 2(1) of Yonghint et al. [27].

2.2 Independent Random Variables

In this subsection, we consider \(B_i=A_i=\{i\}\) in the earlier discussed setup and obtain approximation results for \(W^*=\sum _{i\in J}\omega _i \eta _i\), the weighted sum of independent random variables. To simplify the presentation, let us define \(p_i(k):={\mathbb {P}}(\eta _i=k)\) and \(\gamma :=2 \max _{i \in J}d_{TV}(W_i,W_i+1)\) where \(W_i=W^*-\omega _i \eta _i\).

Theorem 2.2

Let \(W^*\) be the weighted sum of independent random variables and X be a random variable having Stein operator (1.3) satisfying (2.2). Then,

$$\begin{aligned} d_{TV}({\mathcal {L}}(W^*),{\mathcal {L}}(X))\le \gamma \Vert \Delta g\Vert \sum _{i\in J}\omega _i\left( \sum _{k=1}^{\infty }h_i(k)+d_i\right) , \end{aligned}$$
(2.13)

where \(d_i={\mathbb {E}}(\omega _i\eta _i)|(1-\beta )[{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _i-1))]+\beta {\mathbb {E}}(\eta _i)|\) and

$$\begin{aligned} h_i(k)=\left\{ \begin{array}{ll} \frac{k(k-1)}{2}|(1-\beta ){\mathbb {E}}\eta _i p_i(k)+\beta k p_i(k)-(k+1)p_i(k+1)|, &{} \text {if }\omega _i=1;\\ \sum _{\ell =1}^{\omega _i k-1}|(1-\beta )\ell {\mathbb {E}}\eta _i+\beta \ell k-(\ell -1)k|p_i(k), &{} \text {if }\omega _i\ge 2. \end{array}\right. \end{aligned}$$

Remark 2.1

Note that Remark 4.1 of Vellaisamy et al. [26] can be applied to our results, and hence for \(\gamma _j=\min \{1/2,1-d_{TV}(\omega _j\eta _j,\omega _j\eta _j+1)\}\), \(\gamma ^*=\max _{j \in J}\gamma _j\), we have

$$\begin{aligned} \gamma \le \sqrt{\frac{2}{\pi }}\left( \frac{1}{4}+\sum _{j\in J}\gamma _j-\gamma ^*\right) ^{-1/2}. \end{aligned}$$

Therefore, if \(\eta _J\) is of O(n), then the bound (2.13) is of \(O(n^{-1/2})\). Observe that the above bound for \(\gamma \) is useful when \(w_j=1\) for many values of j.

Corollary 2.1

Assume the conditions of Theorem 2.2 hold with \(X\sim \) PB(N, p) and \({\mathbb {E}}W^*>\mathrm {Var}(W^*)\). Then,

$$\begin{aligned} d_{TV}({\mathcal {L}}(W^*),\text {PB}(N,p))\le \frac{\gamma }{\left\lfloor N\right\rfloor pq} \sum _{i\in J}\omega _i\left( \sum _{k=1}^{\infty }h_i(k)+d_i\right) , \end{aligned}$$

where \(d_i={\mathbb {E}}(\omega _i\eta _i)|{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _i-1))-p{\mathbb {E}}(\eta _i)|\) and

$$\begin{aligned} h_i(k)=\left\{ \begin{array}{ll} \frac{k(k-1)}{2}|{\mathbb {E}}\eta _i p_i(k)-pkp_i(k)-q(k+1)p_i(k+1)|, &{} \text {if }\omega _i=1;\\ \sum _{\ell =1}^{\omega _i k-1}|\ell {\mathbb {E}}\eta _i-p\ell k-q(\ell -1)k|p_i(k), &{} \text {if } \omega _i\ge 2. \end{array}\right. \end{aligned}$$

Remark 2.2

  1. (i)

    If \(J=\{1,2,\ldots ,n\}\), \(\omega _i=1\), and \(\eta _i\sim \) Ber(p), for all \(1\le i \le n\). Then, \(h_i(k)=d_i=0\), and hence, \(d_{TV}({\mathcal {L}}(W^*), \text {PB}(N,p))=0\), as expected.

  2. (ii)

    Let \(\omega _i=1\) and \(\eta _i\sim \) Ber\((p_i)\), for \(i\in J=\{1,2,\ldots ,n\}\). Then, from Corollary 2.1, we have

    $$\begin{aligned} d_{TV}({\mathcal {L}}(W^*),PB(N,p))\le \sqrt{\frac{2}{\pi }}\left( \frac{1}{4}+\sum _{i=1}^{n}{\bar{\gamma }}_i-{\bar{\gamma }}^*\right) ^{-1/2}\frac{1}{\left\lfloor N\right\rfloor pq}\sum _{i=1}^{n}p_i^2|p-p_i|, \end{aligned}$$
    (2.14)

    where \({\bar{\gamma }}_j=\frac{1}{2}\min \{1,1+p_i-|1-2p_i|\}\) and \({\bar{\gamma }}^*=\max _{1\le i\le n}{\bar{\gamma }}_i\). The bound given in (2.14) is of \(O(n^{-1/2})\) and is an order improvement over Theorem 1 of Barbour and Hall [2], Theorem 9.E of Barbour et al. [3], and the bounds discussed by Kerstan [13] and Le Cam [15].

  3. (iii)

    Consider the setup of CDO discussed in Example 2.2 under independent Bernoulli trials and unit weights, that is, \({\overline{W}}_n^*=\sum _{i=1}^{n}I_i\) where \(I_i\), for \(1\le i\le n\), are independent Bernoulli trials. Using \(A_i=B_i=\{i\}\) in (3.1), routine calculations lead to

    $$\begin{aligned} \big |{\mathbb {E}}[{\mathscr {A}}g({\overline{W}}_n^*)]\big |\le \frac{(1+q)}{q^{\left\lceil N\right\rceil +1}}\sum _{i=1}^{n}p_i|p-p_i|, \end{aligned}$$
    (2.15)

    where \(p=\frac{1}{N}\sum _{i=1}^{n}p_i\). Note that if \(p_i=p\) in (2.15), for \(1\le i\le n\), then \(\big |{\mathbb {E}}[{\mathscr {A}}g({\overline{W}}_n^*)]\big |=0\), as expected. Also, from (3.8) with \(\omega _i=1\), for \(1\le i \le n\), we have

    $$\begin{aligned} \big |{\mathbb {E}}[{\mathscr {A}}g({\overline{W}}_n^*)]\big |\le \sqrt{\frac{2}{\pi }}\left( \frac{1}{4}+\sum _{i=1}^{n}{\tilde{\gamma }}_i-{\tilde{\gamma }}^*\right) ^{-1/2}\frac{1+q}{q^{\left\lceil N\right\rceil +1}}\sum _{i=1}^{n}p_i^2|p-p_i|, \end{aligned}$$
    (2.16)

    where \({\tilde{\gamma }}_i=\frac{1}{2}\min \{1,1+p_i-|1-2p_i|\}\) and \({\tilde{\gamma }}^*=\max _{1\le i\le n}{\bar{\gamma }}_i\), \(q=1-p=\sum _{i=1}^{n}p_iq_i/\sum _{i=1}^{n}p_i\), and \(N=(\sum _{i=1}^{n}p_i)^2/\sum _{i=1}^{n}p_i^2\). For Poisson approximation, the existing bound given in (4) of Neammanee and Yonghint [17] is

    $$\begin{aligned} \big |{\mathbb {E}}[{\mathscr {A}}g({\overline{W}}_n^*)]\big |\le \left( 2\exp \left( \sum _{i=1}^{n}p_i\right) -1\right) \sum _{i=1}^{n}p_i^2. \end{aligned}$$
    (2.17)

    Note that, for small values of \(p_i\), the bound given in (2.15) is better than the bound given in (2.17). For instance, let \(n=50\) and \(p_i\), \(1\le i\le 50\), be defined as follows:

    i

    \(p_i\)

    i

    \(p_i\)

    i

    \(p_i\)

    0-10

    0.05

    21-30

    0.15

    41-50

    0.25

    11-20

    0.10

    31-40

    0.20

    		  

    Next, the following table gives a comparison between (2.15), (2.16), and (2.17).

    n

    From (2.15)

    From (2.16)

    From (2.17)

    10

    \(7.14 \times 10^{-17}\)

    \(6.00\times 10^{-18}\)

    0.0574

    20

    0.3711

    0.01630

    0.9954

    30

    4.9800

    0.36415

    13.7099

    40

    111.8440

    11.7054

    221.8700

    50

    3311.4600

    897.600

    4970.7400

    Observe that our bounds are better than existing bounds for various values of N and \(p_i\).

Corollary 2.2

Assume the conditions of Theorem 2.2 hold with \(X\sim \) NB\((r,{\bar{p}})\) and \({\mathbb {E}}W^*<\mathrm {Var}(W^*)\). Then,

$$\begin{aligned} d_{TV}({\mathcal {L}}(W^*),\text {NB}(r,{\bar{p}}))\le \frac{\gamma }{r {\bar{q}}} \sum _{i\in J}\omega _i\left( \sum _{k=1}^{\infty }h_i(k)+d_i\right) , \end{aligned}$$
(2.18)

where \(d_i={\mathbb {E}}(\omega _i\eta _i)|{\bar{p}}[{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _i-1))]+{\bar{q}}{\mathbb {E}}(\eta _i)|\) and

$$\begin{aligned} h_i(k)=\left\{ \begin{array}{ll} \frac{k(k-1)}{2}|{\bar{p}}{\mathbb {E}}\eta _i p_i(k)+{\bar{q}} kp_i(k)-(k+1)p_i(k+1)|, &{} \text {if } \omega _i=1;\\ \sum _{\ell =1}^{\omega _i k-1}|{\bar{p}}\ell {\mathbb {E}}\eta _i+{\bar{q}}\ell k-(\ell -1)k|p_i(k), &{} \text {if } \omega _i\ge 2. \end{array}\right. \end{aligned}$$

Remark 2.3

  1. (i)

    If \(J=\{1,2,\ldots ,n\}\) and \(\omega _i=1\), for all i, then, from Corollary 2.2, we have

    $$\begin{aligned}&d_{TV}({\mathcal {L}}(W^*),\text {NB}(r,{\bar{p}}))\nonumber \\&\le \frac{\gamma }{r {\bar{q}}} \sum _{i=1}^{n}\left( \sum _{k=2}^{\infty }\frac{k(k-1)}{2}|{\bar{p}}{\mathbb {E}}\eta _i p_i(k)+{\bar{q}}kp_i(k)-(k+1)p_i(k+1)|+d_i\right) , \end{aligned}$$
    (2.19)

    which is an improvement over the bound given in (17) of Vellaisamy et al. [26]. Also, if \(\eta _i\sim \) Geo\(({\bar{p}})\), the geometric distribution, for \(1\le i\le n\), then \(d_{TV}({\mathcal {L}}(W^*),\text {NB}(n,{\bar{p}}))=0\), as expected.

  2. (ii)

    If \(\eta _i\sim \text {NB}(n_i,p_i)\), \(1\le i \le n\), then the bound given in (2.19) leads to

    $$\begin{aligned} d_{TV}({\mathcal {L}}(W^*),\text {NB}(r,{\bar{p}}))\le \frac{\gamma ^*}{r {\bar{q}}} \sum _{i=1}^{n}\left( {\bar{p}}(n_iq_i+1)\left| \frac{q_i}{p_i}-\frac{{\bar{q}}}{{\bar{p}}}\right| \frac{n_i(n_i+1)q_i^2}{p_i^2}+d_i\right) , \end{aligned}$$

    where \(\gamma ^*\le \sqrt{\frac{2}{\pi }}(\frac{1}{4}+\sum _{i=1}^{n}{\mathbb {P}}(\eta _i=\left\lfloor (n_i-1)q_i / p_i\right\rfloor )-\max _{1\le i\le n}{\mathbb {P}}(\eta _i=\left\lfloor (n_i-1)q_i / p_i\right\rfloor ))^{-1/2}\) and \(q_i=1-p_i\), which is an order improvement over the bound given in Theorem 3.1 of Teerapabolarn [21].

Example 2.3

(Compound Poisson Distribution) Let \(w_i=i\), \(\eta _i\sim \text {Po}(\lambda _i)\), the Poisson distribution, for \(i\in J=\{1,2,\ldots ,n\}\), and \(S_n=\sum _{i=1}^{n}i\eta _i\). The distribution of \(S_\infty \) is known as compound Poisson distribution. The mean and variance of \(S_n\) satisfy \(\sum _{i=1}^{n}i\lambda _i={\mathbb {E}}S_n< \mathrm {Var}(S_n)=\sum _{i=1}^{n}i^2\lambda _i\). Therefore, the negative binomial approximation is suitable in the view of the applicability of parameters. Hence, from (2.18), we have

$$\begin{aligned} d_{TV}({\mathcal {L}}(S_n),\text {NB}(r,{\bar{p}}))\le \sqrt{\frac{2}{\pi }}\frac{2}{r {\bar{q}}} \sum _{i=1}^{n}E_i, \end{aligned}$$

where \({\bar{q}}=\sum _{i=1}^{n}i(i-1)\lambda _i\big / \sum _{i=1}^{n}i^2\lambda _i\) and \(r=\left( \sum _{i=1}^{n}i\lambda _i\right) ^2\big / \sum _{i=1}^{n}i(i-1)\lambda _i\), and

$$\begin{aligned} E_i\le \left\{ \begin{array}{ll} {\bar{q}}\lambda _i^2(\lambda _i+1)+{\bar{q}}(i\lambda _i)^2, &{} \text {if } i=1;\\ \frac{1}{2}{\mathbb {E}}[i\eta _i(i\eta _i-1)({\bar{p}}(i\eta _i +i\lambda _i)+2)]+{\bar{q}}(i\lambda _i)^2, &{} \text {if } i\ge 2. \end{array}\right. \end{aligned}$$

Note that if \(i\lambda _i\) is decreasing in i, then the bound is useful in practice. For similar conditions, see Barbour et al. [1] and Gan and Xia [10].

3 Proofs

In this section, we prove the main results presented in Sect. 2.

Proof of Theorem 2.1

Consider the Stein operator (1.3), and taking expectation with respect to W, we have

$$\begin{aligned} {\mathbb {E}}[{\mathscr {A}}_Xg(W)]&=\alpha {\mathbb {E}}[g(W+1)]+\beta {\mathbb {E}}[Wg(W+1)]-{\mathbb {E}}[Wg(W)]\\&=(1-\beta )\left[ \frac{\alpha }{1-\beta } {\mathbb {E}}[g(W+1)]- {\mathbb {E}}[Wg(W)]\right] +\beta {\mathbb {E}}[W\Delta g(W)]. \end{aligned}$$

Using (2.2), the above expression leads to

$$\begin{aligned} {\mathbb {E}}[{\mathscr {A}}_Xg(W)]&=(1-\beta )\left[ \sum _{i\in J}\omega _i {\mathbb {E}}\eta _i {\mathbb {E}}[g(W+1)]- \sum _{i\in J}\omega _i {\mathbb {E}}[\eta _ig(W)]\right] +\beta {\mathbb {E}}[W\Delta g(W)]. \end{aligned}$$
(3.1)

Let \(W_i=W-\sum _{j\in A_i}\omega _i \eta _i=W-\eta _{A_i}^{*}\) so that \(\eta _i\) and \(W_i\) are independent random variables. Therefore,

$$\begin{aligned} {\mathbb {E}}[{\mathscr {A}}_Xg(W)]&=(1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\eta _i {\mathbb {E}}[g(W+1)-g(W_i+1)]\nonumber \\&~~~- (1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}[\eta _i(g(W)-g(W_i+1))]+\beta {\mathbb {E}}[W\Delta g(W)]\nonumber \\&=(1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\eta _i {\mathbb {E}}\Bigg [\sum _{j=1}^{\eta _{A_i}^{*}}\Delta g(W_i+j)\Bigg ]\nonumber \\&~~~- (1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\Bigg [\eta _i\sum _{j=1}^{\eta _{A_i}^{*}-1}\Delta g(W_i+j)\Bigg ]+\beta \sum _{i\in J}\omega _i {\mathbb {E}}[\eta _i\Delta g(W)]. \end{aligned}$$
(3.2)

Next, let \(W_i^*=W-\sum _{j\in B_i}\omega _i \eta _i=W-\eta _{B_i}^{*}\) so that \(\eta _i\) and \(\eta _{A_i}\) are independent of \(W_i^*\). Also, observe that

$$\begin{aligned}&(1-\beta )\left\{ \sum _{i\in J}\omega _i {\mathbb {E}}\eta _i {\mathbb {E}}\Bigg [\sum _{j=1}^{\eta _{A_i}^{*}}1\Bigg ]- \sum _{i\in J}\omega _i {\mathbb {E}}\Bigg [\eta _i\sum _{j=1}^{\eta _{A_i}^{*}-1}1\Bigg ]\right\} +\beta \sum _{i\in J}\omega _i {\mathbb {E}}\eta _i\nonumber \\&=(1-\beta )\left\{ \sum _{i\in J}\omega _i {\mathbb {E}}\eta _i {\mathbb {E}}[\eta _{A_i}^{*}]- \sum _{i\in J}\omega _i {\mathbb {E}}[\eta _i(\eta _{A_i}^{*}-1)]\right\} +\beta \sum _{i\in J}\omega _i {\mathbb {E}}\eta _i\nonumber \\&=(1-\beta )\left[ \frac{1}{1-\beta }\sum _{i\in J}\omega _i {\mathbb {E}}\eta _i-\sum _{i\in J}\sum _{j\in A_i}\omega _i\omega _j({\mathbb {E}}[\eta _i \eta _j]-{\mathbb {E}}\eta _i {\mathbb {E}}\eta _j)\right] \nonumber \\&=(1-\beta )\left[ \frac{1}{1-\beta }{\mathbb {E}}(W_n)- \mathrm {Var}(W_n)\right] =0. \end{aligned}$$
(3.3)

Multiply \({\mathbb {E}}[\Delta g(W+1)]\) in (3.3) and using the corresponding expression in (3.2), we get

$$\begin{aligned} {\mathbb {E}}[{\mathscr {A}}_Xg(W)]=&(1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\eta _i {\mathbb {E}}\Bigg [\sum _{j=1}^{\eta _{A_i}^{*}}(\Delta g(W_i+j)-\Delta g(W_i^*+1))\Bigg ]\nonumber \\&- (1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\Bigg [\eta _i\sum _{j=1}^{\eta _{A_i}^{*}-1}(\Delta g(W_i+j)-\Delta g(W_i^*+1))\Bigg ]\nonumber \\&+\beta \sum _{i\in J}\omega _i {\mathbb {E}}[\eta _i(\Delta g(W)-\Delta g(W_i^*+1))]\nonumber \\&-\sum _{i\in J}\omega _i\{(1-\beta )[{\mathbb {E}}(\eta _i) {\mathbb {E}}(\eta _{A_i}^{*})-{\mathbb {E}}(\eta _i(\eta _{A_i}^{*}\nonumber \\&-1))]+\beta {\mathbb {E}}(\eta _i)\}{\mathbb {E}}[\Delta g(W+1)-\Delta g(W_i^*+1)]\nonumber \\&=(1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\eta _i {\mathbb {E}}\Bigg [\sum _{j=1}^{\eta _{A_i}^{*}}\sum _{\ell =1}^{\eta _{B_i\backslash A_i}^*+j-1}\Delta ^2 g(W_i^*+\ell )\Bigg ]\nonumber \\&- (1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\Bigg [\eta _i\sum _{j=1}^{\eta _{A_i}^{*}-1}\sum _{\ell =1}^{\eta _{B_i\backslash A_i}^*+j-1}\Delta ^2 g(W_i^*+\ell )\Bigg ]\nonumber \\&+\beta \sum _{i\in J}\omega _i {\mathbb {E}}\Bigg [\eta _i\sum _{\ell =1}^{\eta _{B_i}^{*}-1}\Delta ^2 g(W_i^*+\ell )\Bigg ]\nonumber \\&-\sum _{i\in J}\omega _i\{(1-\beta )[{\mathbb {E}}(\eta _i) {\mathbb {E}}(\eta _{A_i}^{*})-{\mathbb {E}}(\eta _i(\eta _{A_i}^{*}-1))]\nonumber \\&+\beta {\mathbb {E}}(\eta _i)\}{\mathbb {E}}\left[ \sum _{\ell =1}^{\eta _{B_i}^{*}}\Delta ^2 g(W_i^*+\ell )\right] \nonumber \\ =&(1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\eta _i {\mathbb {E}}\Bigg [\sum _{j=1}^{\eta _{A_i}^{*}}\sum _{\ell =1}^{\eta _{B_i\backslash A_i}^*+j-1}{\mathbb {E}}[\Delta ^2 g(W_i^*+\ell )|\eta _{A_i},\eta _{B_i}]\Bigg ]\nonumber \\&- (1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\Bigg [\eta _i\sum _{j=1}^{\eta _{A_i}^{*}-1}\sum _{\ell =1}^{\eta _{B_i\backslash A_i}^*+j-1}{\mathbb {E}}[\Delta ^2 g(W_i^*+\ell )|\eta _i,\eta _{A_i},\eta _{B_i}]\Bigg ]\nonumber \\&+\beta \sum _{i\in J}\omega _i {\mathbb {E}}\Bigg [\eta _i\sum _{\ell =1}^{\eta _{B_i}^{*}-1}{\mathbb {E}}[\Delta ^2 g(W_i^*+\ell )|\eta _{B_i}]\Bigg ]\nonumber \\&-\sum _{i\in J}\omega _i\{(1-\beta )[{\mathbb {E}}(\eta _i) {\mathbb {E}}(\eta _{A_i}^{*})-{\mathbb {E}}(\eta _i(\eta _{A_i}^{*}-1))]\nonumber \\&+\beta {\mathbb {E}}(\eta _i)\}{\mathbb {E}}\left[ \sum _{\ell =1}^{\eta _{B_i}^{*}}{\mathbb {E}}[\Delta ^2 g(W_i^*+\ell )|\eta _{B_i}]\right] . \end{aligned}$$
(3.4)

Hence,

$$\begin{aligned} \left| {\mathbb {E}}[{\mathscr {A}}_Xg(W)]\right| \le&\Vert \Delta g\Vert \left\{ \frac{(1-\beta )}{2}\left[ \sum _{i\in J} \omega _i{\mathbb {E}}\eta _i {\mathbb {E}}[\eta _{A_i}^*(2\eta _{B_i}^{*}-\eta _{A_i}^{*}-1){\mathcal {D}}(W|\eta _{A_i},\eta _{B_i})]\right. \right. \nonumber \\&\left. +\sum _{i\in J}\omega _i{\mathbb {E}}[\eta _i(\eta _{A_i}^{*}-1)(2\eta _{B_i}^{*}-\eta _{A_i}^{*}-2){\mathcal {D}}(W|\eta _i,\eta _{A_i},\eta _{B_i})]\right] \nonumber \\&+\sum _{i\in J}\omega _i\bigr |(1-\beta )[{\mathbb {E}}(\eta _i) {\mathbb {E}}(\eta _{A_i}^{*})-{\mathbb {E}}(\eta _i(\eta _{A_i}^{*}-1))]\nonumber \\&+\beta {\mathbb {E}}(\eta _i)\bigr |{\mathbb {E}}[\eta _{B_i}^{*}{\mathcal {D}}(W|\eta _{B_i})]\nonumber \\&+\left. |\beta | \sum _{i\in J} \omega _i{\mathbb {E}}[\eta _i(\eta _{B_i}^{*}-1){\mathcal {D}}(W|\eta _{B_i})]\right\} . \end{aligned}$$
(3.5)

Using (1.2), the proof follows.

Proof of Theorem 2.2

Let \(A_i=B_i=\{i\}\), then \(\{\eta _i, i\in J\}\) becomes independent random variables, and \(W_i=W_i^*\) is independent of \(\eta _{A_i}=\eta _{B_i}=\eta _i\). Therefore, from (3.4), we have

$$\begin{aligned} {\mathbb {E}}[{\mathscr {A}}_Xg(W^*)]=&(1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\eta _i {\mathbb {E}}\Bigg [\sum _{j=1}^{\omega _i \eta _i}\sum _{\ell =1}^{j-1}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]\Bigg ]\\&- (1-\beta )\sum _{i\in J}\omega _i {\mathbb {E}}\Bigg [\eta _i\sum _{j=1}^{\omega _i \eta _i-1}\sum _{\ell =1}^{j-1}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]\Bigg ]\\&+\beta \sum _{i\in J}\omega _i {\mathbb {E}}\Bigg [\eta _i\sum _{\ell =1}^{\omega _i \eta _i-1}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]\Bigg ]\\&-\sum _{i\in J}\omega _i\{(1-\beta )[{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _{i}-1))]\\&+\beta {\mathbb {E}}(\eta _i)\}{\mathbb {E}}\left[ \sum _{\ell =1}^{\omega _i\eta _i}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]\right] \\ =&(1-\beta )\sum _{i\in J}\omega _i{\mathbb {E}}\eta _i \sum _{k=1}^{\infty }\sum _{j=1}^{\omega _i k}\sum _{\ell =1}^{j-1}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k)\\&-(1-\beta )\sum _{i\in J}\omega _i \sum _{k=1}^{\infty }\sum _{j=1}^{\omega _i k-1}\sum _{\ell =1}^{j-1}k{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k)\\&+\beta \sum _{i\in J}^{n}\omega _i \sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k-1}k {\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k)\\&-\sum _{i\in J}\omega _i\{(1-\beta )[{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _{i}-1))]\\&+\beta {\mathbb {E}}(\eta _i)\}\sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k)\\ =&\sum _{i\in J}\omega _i \sum _{k=1}^{\infty }\sum _{j=1}^{\omega _i k}\sum _{\ell =1}^{j-1}[(1-\beta ){\mathbb {E}}\eta _i+\beta k]{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k)\\&-\sum _{i\in J}\omega _i \sum _{k=1}^{\infty }\sum _{j=1}^{\omega _i k-1}\sum _{\ell =1}^{j-1}k{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k)\\&-\sum _{i\in J}\omega _i\{(1-\beta )[{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _{i}-1))]\\&+\beta {\mathbb {E}}(\eta _i)\}\sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k)\\ =&\sum _{i\in J}\omega _i \sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k-1}\ell [(1-\beta ){\mathbb {E}}\eta _i+\beta k] {\mathbb {E}}[\Delta ^2 g(W_i+\omega _i k-\ell )]p_i(k)\\&-\sum _{i\in J}\omega _i \sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k-2}\ell k {\mathbb {E}}[\Delta ^2 g(W_i+\omega _i k-\ell -1)]p_i(k)\\&-\sum _{i\in J}\omega _i\{(1-\beta )[{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _{i}-1))]\\&+\beta {\mathbb {E}}(\eta _i)\}\sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k). \end{aligned}$$

Case I: If \(\omega _i=1\), then

$$\begin{aligned} {\mathbb {E}}[{\mathscr {A}}_Xg(W^*)]=&\sum _{i\in J}\sum _{k=2}^{\infty }\sum _{\ell =1}^{k-1}\ell [(1-\beta ){\mathbb {E}}\eta _i +\beta k]{\mathbb {E}}[\Delta ^2 g(W_i+ k-\ell )]p_i(k)\nonumber \\&-\sum _{i\in J} \sum _{k=3}^{\infty }\sum _{\ell =1}^{k-2}\ell k {\mathbb {E}}[\Delta ^2 g(W_i+k-\ell -1)]p_i(k)\nonumber \\&-\sum _{i\in J}\{(1-\beta )[{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _{i}-1))]+\beta {\mathbb {E}}(\eta _i)\}\nonumber \\&\times \sum _{k=1}^{\infty }\sum _{\ell =1}^{k}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k)\nonumber \\ =&\sum _{i\in J}\sum _{k=2}^{\infty }\sum _{\ell =1}^{k-1}\ell [(1-\beta ){\mathbb {E}}\eta _i p_i(k)+\beta kp_i(k)\nonumber \\&-(k+1)p_i(k+1)]{\mathbb {E}}[\Delta ^2 g(W_i+ k-\ell )]\nonumber \\&-\sum _{i\in J}\{(1-\beta )[{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _{i}-1))]+\beta {\mathbb {E}}(\eta _i)\}\nonumber \\&\times \sum _{k=1}^{\infty }\sum _{\ell =1}^{k}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k). \end{aligned}$$
(3.6)

Case II: If \(\omega _i \ge 2\), then

$$\begin{aligned} {\mathbb {E}}[{\mathscr {A}}g(W)]=&\sum _{i\in J}\omega _i \sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k-1}\ell [(1-\beta ){\mathbb {E}}\eta _i+\beta k] {\mathbb {E}}[\Delta ^2 g(W_i+\omega _i k-\ell )]p_i(k)\nonumber \\&-\sum _{i\in J}\omega _i \sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k-2}\ell k {\mathbb {E}}[\Delta ^2 g(W_i+\omega _i k-\ell -1)]p_i(k)\nonumber \\&-\sum _{i\in J}\omega _i\{(1-\beta )[{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _{i}-1))]+\beta {\mathbb {E}}(\eta _i)\}\nonumber \\&\sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k)\nonumber \\ =&\sum _{i\in J}\omega _i \sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k-1}[(1-\beta )\ell {\mathbb {E}}\eta _i+\beta \ell k\nonumber \\&-(\ell -1)k]{\mathbb {E}}[\Delta ^2 g(W_i+\omega _i k-\ell )]p_i(k)\nonumber \\&-\sum _{i\in J}\omega _i\{(1-\beta )[{\mathbb {E}}(\eta _i)^2-{\mathbb {E}}(\eta _i(\eta _{i}-1))]+\beta {\mathbb {E}}(\eta _i)\}\nonumber \\&\times \sum _{k=1}^{\infty }\sum _{\ell =1}^{\omega _i k}{\mathbb {E}}[\Delta ^2 g(W_i+\ell )]p_i(k). \end{aligned}$$
(3.7)

It is shown that in Barbour and Xia [5] (see also Barbour and Čekanavičius [4], p. 517) \(|{\mathbb {E}}(\Delta ^2 g(W_i+\cdot ))|\le \gamma \Vert \Delta g\Vert \). Hence, from (3.6) and (3.7), we have

$$\begin{aligned} \left| {\mathbb {E}}[{\mathscr {A}}g(W)]\right| \le \gamma \Vert \Delta g\Vert \sum _{i\in J}\omega _i \left( \sum _{k=1}^{\infty }h_i(k)+d_i\right) . \end{aligned}$$
(3.8)

Using (1.2), the proof follows. \(\square \)