On bounds in Poisson approximation for integer-valued independent random variables

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.

1 Introduction

Let $\{X_{n,j},j=1,2,\dots ,n;n=1,2,\dots \}$ be a row-wise triangular array of independent integer-valued random variables with success probabilities $P(X_{n,j}=1)=p_{n,j}$; $P(X_{n,j}=0)=1-p_{n,j}-q_{n,j}$; $p_{n,j},q_{n,j}\in (0,1)$; $p_{n,j}+q_{n,j}\in (0,1)$; $j=1,2,\dots ,n$; $n=1,2,\dots$ . Set $S_n={\sum }_{j=1}^n{X}_{n,j}$ and ${\lambda }_n=E(S_n)={\sum }_{j=1}^n{p}_{n,j}$. Suppose that ${lim}_{n\to \mathrm{\infty }}{\lambda }_n=\lambda$ ($0<\lambda <+\mathrm{\infty }$). We will denote by $Z_{\lambda }$ the Poisson random variable with mean λ. It has long been known that in the case of all $q_{n,j}=0$ ($j=1,2,\dots ,n$; $n=1,2,\dots$), the partial sum $S_n$ is said to be a Poisson-binomial random variable, and the probability distributions of $S_n$, $n=1,2,\dots$ , are usually approximated by the distribution of $Z_{\lambda }$. Specially, under the assumptions that ${lim}_{n\to \mathrm{\infty }}{max}_{1\le j\le n}{p}_{n,j}=0$, the well-known Poisson approximation theorem states that

$$S_n\stackrel{d}{\to }{Z}_{\lambda }\text{as }n\to \mathrm{\infty }.$$

(1)

Here, and from now on, the notation $\stackrel{d}{\to }$ means the convergence in distribution. It is to be noticed that, for the information on the quality of the Poisson approximation, Le Cam (1960) [1] established the remarkable inequality

$$\sum _{k=0}^{\mathrm{\infty }}|P(S_n=k)-P(Z_{\lambda }=k)|\le 2\sum _{j=1}^n{p}_{n,j}^2.$$

(2)

It is to be noticed that another inequality in Poisson approximation is usually expressed in terms of the total variation distance $d_{TV}(S_n,Z_{\lambda })$

$$d_{TV}(S_n,Z_{\lambda })\le \sum _{j=1}^n{p}_{n,j}^2,$$

(3)

where for the distributions P and Q on ${\mathbb{Z}}_{+}=\{0,1,2,\dots \}$, the total variation distance between P and Q will be defined as follows:

$$d_{TV}(P,Q):=\frac{1}{2}\sum _{x\in {\mathbb{Z}}_{+}}|P(x)-Q(x)|.$$

(4)

(For other surveys, see [1–4], and [5].)

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 [1–11], and [12].

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 [13, 14], 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 [12–20], and [21].

The results obtained in this paper are extensions of known results in [1, 5, 9–11], and [4]. The present paper is also a continuation of [12].

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 3 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 [13–15, 18, 19], and [20], for the complete bibliography). Based on Renyi’s definition ([14], Chapter 8, Section 12, p.523), we redefine the Trotter-Renyi operator as follows.

Definition 2.1 The operator $A_X$ associated with a discrete random variable X is called the Trotter-Renyi operator, defined by

$$(A_{X}f)(x)=E(f(X+x))=\sum _{k=0}^{\mathrm{\infty }}f(x+k)P(X=k),\mathrm{\forall }f\in \mathbb{K},\mathrm{\forall }x\in {\mathbb{Z}}_{+},$$

(5)

where by is denoted the class of all real-valued bounded functions f on the set of all non-negative integers ${\mathbb{Z}}_{+}:=\{0,1,2,\dots \}$. The norm of the function $f\in \mathbb{K}$ is defined by $\parallel f\parallel ={sup}_{x\in {\mathbb{Z}}_{+}}|f(x)|$.

It is to be noticed that Renyi’s operator defined in [14] actually is a discrete form of Trotter’s operator (we refer the readers to [13, 15, 17–19], and [20], 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 $f,g\in \mathbb{K}$ and for $\alpha \in \mathbb{R}$:

• 1. $A_X(f+g)=A_X(f)+A_X(g)$.

• 2. $A_X(\alpha f)=\alpha A_X(f)$.

• 3. $\parallel A_X(f)\parallel ⩽\parallel f\parallel$.

• 4. $\parallel A_X(f)+A_Y(f)\parallel ⩽\parallel A_X(f)\parallel +\parallel A_Y(f)\parallel$.

• 5. Suppose that $A_X$, $A_Y$ are operators associated with two independent random variables X and Y. Then, for all $f\in \mathbb{K}$,

  $$A_{X+Y}(f)=A_X{A}_Y(f)=A_Y{A}_X(f).$$

In fact, for all $x\in {\mathbb{Z}}_{+}$

$$\begin{array}{rl}{A}_{X+Y}f(x)& =\sum _{l=0}^{\mathrm{\infty }}f(x+l)P(X+Y=l)=\sum _{r,k=0}^{\mathrm{\infty }}f(x+k+r)P(Y=k)P(X=r)\\ =A_X(A_{Y}f(x))=A_X{A}_{Y}f(x).\end{array}$$

• 6. Suppose that $A_{X_1},A_{X_2},\dots ,A_{X_n}$ are the operators associated with the independent random variables $X_1,X_2,\dots ,X_n$. Then, for all $f\in \mathbb{K}$, $A_{S_n}(f)=A_{X_1}{A}_{X_2}\cdots A_{X_n}(f)$ is the operator associated with the partial sum $S_n=X_1+X_2+\cdots +X_n$.

• 7. Suppose that $A_{X_1},A_{X_2},\dots ,A_{X_n}$ and $A_{Y_1},A_{Y_2},\dots ,A_{Y_n}$ are operators associated with independent random variables $X_1,X_2,\dots ,X_n$ and $Y_1,Y_2,\dots ,Y_n$. Moreover, assume that all $X_i$ and $Y_j$ are independent for $i,j=1,2,\dots ,n$. Then, for every $f\in \mathbb{K}$,

  $$\parallel A_{{\sum }_{k=1}^n{X}_k}(f)-A_{{\sum }_{k=1}^n{Y}_k}(f)\parallel ⩽\sum _{k=1}^n\parallel A_{X_k}(f)-A_{Y_k}(f)\parallel .$$

  (6)

Clearly,

$$\begin{array}{c}{A}_{X_1}{A}_{X_2}\cdots A_{X_n}-A_{Y_1}{A}_{Y_2}\cdots A_{Y_n}\hfill \\ =\sum _{k=1}^n{A}_{X_1}{A}_{X_2}\cdots A_{X_{k-1}}(A_{X_k}-A_{Y_k})A_{Y_{k+1}}\cdots A_{Y_n}.\hfill \end{array}$$

Accordingly,

$$\begin{array}{rcl}\parallel A_{{\sum }_{k=1}^n{X}_k}(f)-A_{{\sum }_{k=1}^n{Y}_k}(f)\parallel & ⩽& \sum _{k=1}^n\parallel A_{X_1}\dots A_{X_{k-1}}(A_{X_k}-A_{Y_k})A_{Y_{k+1}}\cdots A_{Y_n}(f)\parallel \\ ⩽& \sum _{k=1}^n\parallel A_{Y_{k+1}}\cdots A_{Y_n}(A_{X_k}-A_{Y_k})(f)\parallel \\ ⩽& \sum _{k=1}^n\parallel A_{X_k}(f)-A_{Y_k}(f)\parallel .\end{array}$$

• 8. $\parallel A_X^n(f)-A_Y^n(f)\parallel \le n\parallel A_X(f)-A_Y(f)\parallel$.

Lemma 2.1 The equation $A_{X}f(x)=A_{Y}f(x)$ for $f\in \mathbb{K}$, $x\in {\mathbb{Z}}_{+}$ shows that X and Y are identically distributed random variables.

Let $A_{X_1},A_{X_2},\dots ,A_{X_n},\dots$ be a sequence of Trotter-Renyi’s operators associated with the independent discrete random variables $X_1,X_2,\dots ,X_n,\dots$ , and assume that $A_X$ is a Trotter-Renyi operator associated with the discrete random variable X. The following lemma states one of the most important properties of the Trotter-Renyi operator.

Lemma 2.2 A sufficient condition for a sequence of random variables $X_1,X_2,\dots ,X_n,\dots$ to converge in distribution to a random variable X is that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel A_{X_n}(f)-A_X(f)\parallel =0,\mathit{\text{for all }}f\in \mathbb{K}.$$

Proof Since ${lim}_{n\to \mathrm{\infty }}\parallel A_{X_n}(f)-A_X(f)\parallel =0$, for all $f\in \mathbb{K}$, we conclude that

$$\underset{n\to \mathrm{\infty }}{lim}|\sum _{k=0}^{\mathrm{\infty }}f(x+k)(P(X_n=k)-P(X=k))|=0,\text{for all }f\in \mathbb{K}\text{ and for all }x\in {\mathbb{Z}}_{+}.$$

Taking

$$f(x)=\{\begin{array}{cc}1,\hfill & \text{if }0⩽x⩽t,\hfill \\ 0,\hfill & \text{if }x>t,\hfill \end{array}$$

then we recover

$$\underset{n\to \mathrm{\infty }}{lim}\left|\sum _{k=0}^t(P(X_n=k)-P(X=k))\right|=0.$$

It follows that $P(X_n⩽t)-P(X⩽t)\to 0$ as $n\to +\mathrm{\infty }$. We infer that $X_n\stackrel{d}{\to }X$ as $n\to +\mathrm{\infty }$. □

Before stating the definition of the Trotter-Renyi distance we firstly need the definition of a probability metric. Let $(\mathrm{\Omega },\mathbb{A},\mathbb{P})$ be a probability space and let $\mathbb{Z}(\mathrm{\Omega },\mathbb{A})$ be a space of real-valued -measurable random variables $X:\mathrm{\Omega }\to \mathbb{R}$.

Definition 2.2 A functional $d(X,Y):\mathbb{Z}(\mathrm{\Omega },\mathbb{A})\times \mathbb{Z}(\mathrm{\Omega },\mathbb{A})\to [0,\mathrm{\infty })$ is said to be a probability metric in $\mathbb{Z}(\mathrm{\Omega },\mathbb{A})$ if it possesses for the random variables $X,Y,Z\in \mathbb{Z}(\mathrm{\Omega },\mathbb{A})$ the following properties (see [2, 22] and [18] for more details):

• 1. $P(X=Y)=1\Rightarrow d(X,Y)=0$;

• 2. $d(X,Y)=d(Y,X)$;

• 3. $d(X,Y)\le d(X,Z)+d(Z,Y)$.

We now return to the definition of a probability distance based on the Trotter-Renyi operator (see [18, 19], and [21]).

Definition 2.3 The Trotter-Renyi distance $d_{TR}(X,Y;f)$ of two random variables X and Y with respect to the function $f\in \mathbb{K}$ is defined by

$$d_{TR}(X,Y;f):=\parallel A_{X}f-A_{Y}f\parallel =\underset{x\in {\mathbb{Z}}_{+}}{sup}|Ef(X+x)-Ef(Y+x)|.$$

(7)

Based on the properties of the Trotter-Renyi operator, some properties of the Trotter-Renyi distance are summarized in the following (see [13, 14, 18, 19], and [21] for more details) and we shall omit the proofs.

• 1. It is easy to see that $d_{TR}(X,Y;f)$ is a probability metric, i.e. for the random variables X, Y, and Z the following properties are possessed:

• (a) For every $f\in \mathbb{K}$, the distance $d_{TR}(X,Y;f)=0$ if $P(X=Y)=1$.

• (b) $d_{TR}(X,Y;f)=d_{TR}(Y,X;f)$ for every $f\in \mathbb{K}$.

• (c) $d_{TR}(X,Y;f)\le d_{TR}(X,Z;f)+d_{TR}(Z,Y;f)$ for every $f\in \mathbb{K}$.

• 2. If $d_{TR}(X,Y;f)=0$ for every $f\in \mathbb{K}$, then $F_X\equiv F_Y$.

• 3. Let $\{X_n,n\ge 1\}$ be a sequence of random variables and let X be a random variable. The condition

  $$\underset{n\to +\mathrm{\infty }}{lim}{d}_{TR}(X_n,X;f)=0,\text{for all }f\in \mathbb{K},$$

implies that $X_n\stackrel{d}{\to }X$ as $n\to \mathrm{\infty }$.

• 4. Suppose that $X_1,X_2,\dots ,X_n$; $Y_1,Y_2,\dots ,Y_n$ are independent random variables (in each group). Then, for every $f\in \mathbb{K}$,

  $$d_{TR}(\sum _{j=1}^n{X}_j,\sum _{j=1}^n{Y}_j;f)\le \sum _{j=1}^n{d}_{TR}(X_j,Y_j;f).$$

  (8)

Moreover, if the random variables are identically (in each group), then we have

$$d_{TR}(\sum _{j=1}^n{X}_j,\sum _{j=1}^n{Y}_j;f)\le n{d}_{TR}(X_1,Y_1;f).$$

• 5. Suppose that $X_1,X_2,\dots ,X_n$; $Y_1,Y_2,\dots ,Y_n$ are independent random variables (in each group). Let $\{N_n,n\ge 1\}$ be a sequence of positive integer-valued random variables that are independent of $X_1,X_2,\dots ,X_n$ and $Y_1,Y_2,\dots ,Y_n$. Then, for every $f\in \mathbb{K}$,

  $$d_{TR}(\sum _{j=1}^{N_n}{X}_j,\sum _{j=1}^{N_n}{Y}_j;f)\le \sum _{k=1}^{\mathrm{\infty }}P(N_n=k)\sum _{j=1}^k{d}_{TR}(X_j,Y_j;f).$$

  (9)

• 6. Suppose that $X_1,X_2,\dots ,X_n$; $Y_1,Y_2,\dots ,Y_n$ are independent identically distributed random variables (in each group). Let $\{N_n,n\ge 1\}$ be a sequence of positive integer-valued random variables that are independent of $X_1,X_2,\dots ,X_n$ and $Y_1,Y_2,\dots ,Y_n$. Moreover, suppose that $E(N_n)<+\mathrm{\infty }$, $n\ge 1$. Then, for every $f\in \mathbb{K}$, we have

  $$d_{TR}(\sum _{j=1}^{N_n}{X}_j,\sum _{j=1}^{N_n}{Y}_j;f)\le E(N_n)\cdot d_{TR}(X_1,Y_1;f).$$

Finally, we emphasize that the Trotter-Renyi distance in (7) and the total variation distance in (4) have a close relationship if the function f is chosen as an indicator function of a set $A\in {\mathbb{Z}}_{+}$, namely

$$f(x)={\chi }_A(x)=\{\begin{array}{cc}1,\hfill & \text{if }x\in A,\hfill \\ 0,\hfill & \text{if }x\notin A.\hfill \end{array}$$

Then

$$d_{TR}(X,Y,{\chi }_A)=d_{TV}(X,Y),$$

where we denote by $d_{TV}(X,Y)$ the total variation distance between two integer-valued random variables X and Y, defined as follows:

$$d_{TV}(X,Y)=\underset{A\subseteq {\mathbb{Z}}_{+}}{sup}|P(X\in A)-P(Y\in A)|=\frac{1}{2}\sum _{k\in {\mathbb{Z}}_{+}}|P(X=k)-P(Y=k)|.$$

For a deeper discussion of the total variation distance, we refer the reader to [1–4], and [5].

3 Main results

Let $\{A_{X_{n,j}},j=1,2,\dots ,n;n=1,2,\dots \}$ be a sequence of operators associated with the integer-valued random variables $X_{n,j}$, $j=1,2,\dots ,n$; $n=1,2,\dots$ , and let $\{A_{Z_{p_{n,j}}},j=1,2,\dots ,n;n=1,2,\dots \}$ be a sequence of operators associated with the Poisson random variables with parameters $p_{n,j}$, $j=1,2,\dots ,n$; $n=1,2,\dots$ . Since $Z_{{\lambda }_n}$ is a Poisson random variable with positive parameter ${\lambda }_n={\sum }_{j=1}^n{p}_{n,j}$, we can write $Z_{{\lambda }_n}\stackrel{d}{=}{\sum }_{j=1}^n{Z}_{p_{n,j}}$, where $Z_{p_{n,1}},Z_{p_{n,2}},\dots ,Z_{p_{n,n}}$ are independent Poisson random variables with positive parameters $p_{n,1},p_{n,2},\dots ,p_{n,n}$, and the notation $\stackrel{d}{=}$ denotes coincidence of distributions.

Theorem 3.1 Let $\{X_{n,j},j=1,2,\dots ,n;n=1,2,\dots \}$ be a row-wise triangular array of independent, integer-valued random variables with probabilities $P(X_{n,j}=1)=p_{n,j}$, $P(X_{n,j}=0)=1-p_{n,j}-q_{n,j}$; $p_{n,j},q_{n,j}\in (0,1)$; $p_{n,j}+q_{n,j}\in (0,1)$; $j=1,2,\dots ,n$; $n=1,2,\dots$ . Let us write $S_n={\sum }_{j=1}^n{X}_{n,j}$ and ${\lambda }_n={\sum }_{j=1}^n{p}_{n,j}$. We will denote by $Z_{{\lambda }_n}$ the Poisson random variable with parameter ${\lambda }_n$. Then, for all functions $f\in \mathbb{K}$,

$$d_{TR}(S_n,Z_{{\lambda }_n};f)\le 2\parallel f\parallel \sum _{j=1}^n(p_{n,j}^2+q_{n,j}).$$

Proof Applying (8), we have

$$d_{TR}(S_n,Z_{{\lambda }_n},f)\le \sum _{j=1}^n{d}_{TR}(X_{n,j},Z_{p_{n,j}};f)=\sum _{k=1}^n\parallel A_{X_{n,j}}(f)-A_{Z_{p_{n,j}}}(f)\parallel .$$

Moreover, for all $f\in \mathbb{K}$, for all $x\in {\mathbb{Z}}_{+}$ and $r\in \{0,1,\dots ,n\}$ we conclude that

$$\begin{array}{rcl}{A}_{X_{nj}}f(x)-A_{Z_{p_{n,j}}}f(x)& =& \sum _{r=0}^{\mathrm{\infty }}f(x+r)(P(X_{nj}=r)-P(Z_{{\lambda }_{p_{n,j}}}=r))\\ =& \sum _{r=0}^{\mathrm{\infty }}f(x+r)(P(X_{nj}=r)-\frac{e^{-p_{n,j}}{p}_{n,j}^r}{r!})\\ =& f(x)(1-p_{n,j}-q_{n,j}-e^{-p_{n,j}})\\ +f(x+1)(p_{n,j}-p_{n,j}{e}^{-p_{n,j}})\\ +\sum _{r=2}^{\mathrm{\infty }}f(x+r)(P(X_{n,j}=r)-\frac{e^{-p_{n,j}}{p}_{n,j}^r}{r!}).\end{array}$$

Therefore, for all functions $f\in K$, and for all $x\in {\mathbb{Z}}_{+}$, we have

$$\begin{array}{c}|A_{X_{n,j}}f(x)-A_{Z_{p_{n,j}}}f(x)|\hfill \\ =|f(x)(1-p_{n,j}-q_{n,j}-e^{-p_{n,j}})+f(x+1)(p_{n,j}-p_{n,j}{e}^{-p_{n,j}})\hfill \\ +\sum _{r=2}^{\mathrm{\infty }}f(x+r)(P(X_{n,j}=r)-\frac{e^{-p_{n,j}}{p}_{n,j}^r}{r!})|\hfill \\ =|f(x)(1-p_{n,j}-q_{n,j}-e^{-p_{n,j}})|+|f(x+1)(p_{n,j}-p_{n,j}{e}^{-p_{n,j}})|\hfill \\ +|\sum _{r=2}^{\mathrm{\infty }}f(x+r)(P(X_{n,j}=r)-\frac{e^{-p_{n,j}}{p}_{n,j}^r}{r!})|\hfill \\ \le |f(x)(1-p_{n,j}-q_{n,j}-e^{-p_{n,j}})|+|f(x+1)(p_{n,j}-p_{n,j}{e}^{-p_{n,j}})|\hfill \\ +|\sum _{r=2}^{\mathrm{\infty }}f(x+r)P(X_{n,j}=r)|+|\sum _{r=2}^{\mathrm{\infty }}f(x+r)\frac{e^{-p_{n,j}}{p}_{n,j}^r}{r!}|\hfill \\ \le (e^{-p_{n,j}}+p_{n,j}+q_{n,j}-1)\underset{x\in {\mathbb{Z}}_{+}}{sup}|f(x)|+(p_{n,j}-p_{n,j}{e}^{-p_{n,j}})\underset{x\in {\mathbb{Z}}_{+}}{sup}|f(x)|\hfill \\ +\underset{x\in {\mathbb{Z}}_{+}}{sup}|f(x)||\sum _{r=2}^{\mathrm{\infty }}P(X_{n,k}=r)|+\underset{x\in {\mathbb{Z}}_{+}}{sup}|f(x)|\left|\sum _{r=2}^{\mathrm{\infty }}\frac{e^{-p_{n,j}}{p}_{n,j}^r}{r!}\right|\hfill \\ =\underset{x\in {\mathbb{Z}}_{+}}{sup}|f(x)|(e^{-p_{n,j}}+p_{n,j}+q_{n,j}-1+p_{n,j}-p_{n,j}{e}^{-p_{n,j}}+q_{n,j}+1-e^{-p_{n,j}}-p_{n,j}{e}^{-p_{n,j}})\hfill \\ =2\parallel f\parallel (p_{n,j}-p_{n,j}{e}^{-p_{n,j}}+q_{n,j})\hfill \\ \le 2\parallel f\parallel (p_{n,j}^2+q_{n,j}).\hfill \end{array}$$

One infers that

$$\mathrm{\forall }f\in K,\parallel A_{X_{n,j}}(f)-A_{Z_{p_{n,j}}}(f)\parallel \le 2\parallel f\parallel (p_{n,j}^2+q_{n,j}).$$

Therefore, applying (8), we can assert that

$$d_{TR}(S_n,Z_{{\lambda }_n};f)\le 2\parallel f\parallel \sum _{j=1}^n(p_{n,j}^2+q_{n,j}).$$

This completes the proof. □

Corollary 3.1 Under the assumptions of Theorem  3.1, let $r\in \{0,1,\dots ,n\}$, we have

$$|P(S_n=r)-P(Z_{{\lambda }_n}=r)|\le 2\sum _{j=1}^n(p_{n,j}^2+q_{n,k}).$$

Remark 3.1 We consider Corollary 3.1 and assume that the following conditions are satisfied:

$$\begin{array}{rl}(\text{i})& \underset{n\to \mathrm{\infty }}{lim}\sum _{j=1}^n{q}_{n,j}=0,\\ (\text{ii})& \underset{n\to \mathrm{\infty }}{lim}\underset{1\le k\le n}{max}{p}_{n,j}=0,\\ (\text{iii})& \underset{n\to \mathrm{\infty }}{lim}{\lambda }_n=\underset{n\to \mathrm{\infty }}{lim}\sum _{j=1}^n{p}_{n,j}=\lambda (0<\lambda <+\mathrm{\infty }).\end{array}$$

Then $S_n\stackrel{d}{\to }{Z}_{\lambda }$ as $n\to \mathrm{\infty }$.

Theorem 3.2 Let $\{X_{n,j},j=1,2,\dots ,n;n=1,2,\dots \}$ be a row-wise triangular array of independent, integer-valued random variables with probabilities $P(X_{n,j}=1)=p_{n,j}$, $P(X_{n,j}=0)=1-p_{n,j}-q_{n,j}$; $p_{n,j},q_{n,j}\in (0,1)$; $p_{n,j}+q_{n,j}\in (0,1)$; $j=1,2,\dots ,n$; $n=1,2,\dots$ . Moreover, we suppose that $N_n$, $n=1,2,\dots$ are positive integer-valued random variables, independent of all $X_{n,j}$, $j=1,2,\dots ,n$; $n=1,2,\dots$ . Let us write $S_{N_n}={\sum }_{j=1}^{N_n}{X}_{n,j}$ and ${\lambda }_{N_n}={\sum }_{j=1}^{N_n}{p}_{n,j}$. We will denote by $Z_{{\lambda }_{N_n}}$ the Poisson random variable with parameter ${\lambda }_{N_n}$. Then, for all functions $f\in \mathbb{K}$,

$$d_{TR}(S_{N_n},Z_{{\lambda }_{N_n}};f)\le 2\parallel f\parallel E\left(\sum _{j=1}^{N_n}(p_{N_n,j}^2+q_{N_n,j})\right).$$

Proof According to Theorem 3.1 and (9), for all functions $f\in \mathbb{K}$, and for all $x\in {\mathbb{Z}}_{+}$, we have

$$\begin{array}{rcl}{d}_{TR}(S_{N_n},Z_{{\lambda }_{N_n}};f)& \le & \sum _{m=1}^{\mathrm{\infty }}P(N_n=m)d_{TR}(S_m,Z_{{\lambda }_m};f)\\ \le & \sum _{m=1}^{\mathrm{\infty }}P(N_n=m)2\parallel f\parallel \sum _{j=1}^m(p_{N_n,j}^2+q_{N_n,j})\\ =& 2\parallel f\parallel \sum _{m=1}^{\mathrm{\infty }}[P(N_n=m)\sum _{j=1}^m(p_{N_n,j}^2+q_{N_n,j})]\\ =& 2\parallel f\parallel E\left(\sum _{j=1}^{N_n}(p_{N_n,j}^2+q_{N_n,j})\right).\end{array}$$

Therefore,

$$d_{TR}(S_{N_n},Z_{{\lambda }_{N_n}};f)\le 2\parallel f\parallel E\left(\sum _{j=1}^{N_n}(p_{N_n,j}^2+q_{N_n,j})\right).$$

The proof is complete. □

Corollary 3.2 According to Theorem  3.2, let $r\in \{0,1,\dots ,n\}$, we have

$$|P(S_{N_n}=r)-P(Z_{{\lambda }_{N_n}}=r)|\le 2E\left(\sum _{j=1}^{N_n}(p_{N_n,j}^2+q_{N_n,j})\right).$$

Theorem 3.3 Let $\{X_{k,j}\}$ ($k=1,2,\dots$ ; $j=1,2,\dots$) be a double array of independent integer-valued random variables with probabilities $P(X_{k,j}=1)=p_{k,j}$, $P(X_{k,j}=0)=1-p_{k,j}-q_{k,j}$, $p_{n,k}\in (0,1)$; $k=1,2,\dots$ ; $j=1,2,\dots$ . Assume that for every $k=1,2,\dots$ the random variables $X_{k,1},X_{k,2},\dots$ , are independent, and for every $j=1,2,\dots$ the random variables $X_{1,j},X_{2,j},\dots$ are independent. Set $S_{nm}={\sum }_{k=1}^n{\sum }_{j=1}^m{X}_{k,j}$. Let us denote by $Z_{{\delta }_{n,m}}$ the Poisson random variable with mean ${\delta }_{n,m}={\sum }_{k=1}^n{\sum }_{j=1}^m{p}_{k,j}$. Then, for all $f\in \mathbb{K}$,

$$d_{TR}(S_{nm},Z_{{\delta }_{n,m}},f)\le 2\parallel f\parallel \sum _{k=1}^n\sum _{j=1}^m(p_{k,j}^2+q_{k,j}).$$

Proof Applying the inequality in (8), we have

$$\begin{array}{rcl}{d}_{TR}(S_{nm},Z_{{\delta }_{nm}},f)& \le & \sum _{k=1}^n{d}_{TR}(S_{km},Z_{{\mu }_{k,m}},f)\\ \le & \sum _{k=1}^n\sum _{j=1}^m{d}_{TR}(S_{k,j},Z_{{\lambda }_{k,j}},f).\end{array}$$

According to Theorem 3.1, for all functions $f\in \mathbb{K}$, and for all $x\in {\mathbb{Z}}_{+}$, we conclude that

$$d_{TR}(S_{k,j},Z_{{\lambda }_{k,j}},f)\le 2\parallel f\parallel (p_{k,j}^2+q_{k,j}).$$

Therefore,

$$d_{TR}(S_{nm},Z_{{\delta }_{nm}},f)\le 2\parallel f\parallel \sum _{k=1}^n\sum _{j=1}^m(p_{k,j}^2+q_{k,j}).$$

This completes the proof. □

Theorem 3.4 Let $\{X_{k,j},k=1,2,\dots ;j=1,2,\dots \}$ be a double array of independent integer-valued random variables with $P(X_{k,j}=1)=p_{k,j}$; $P(X_{k,j}=0)=1-p_{k,j}-q_{k,j}$; $p_{k,j},q_{k,j}\in (0,1)$; $p_{k,j}+q_{k,j}\in (0,1)$; $k=1,2,\dots$ ; $n=1,2,\dots$ . Assume that for every $k=1,2,\dots$ the random variables $X_{k,1},X_{k,2},\dots$ , are independent, and for every $j=1,2,\dots$ the random variables $X_{1,j},X_{2,j},\dots$ are independent. Set $S_{nm}={\sum }_{k=1}^n{\sum }_{j=1}^m{X}_{k,j}$. Suppose that $N_n$, $M_m$ are non-negative integer-valued random variables independent of all $X_{n,m}$, $n\ge 1$; $m\ge 1$. Let us denote by $Z_{{\delta }_{N_n{M}_m}}$ the Poisson random variable with mean ${\delta }_{N_n{M}_m}=E(S_{N_n{M}_m})={\sum }_{k=1}^{N_n}{\sum }_{j=1}^{M_m}{p}_{k,j}$. Then, for all functions $f\in \mathbb{K}$,

$$d_{TR}(S_{N_n{M}_m},Z_{{\delta }_{N_n{M}_m}},f)\le 2\parallel f\parallel E\left(\sum _{k=1}^{N_n}\sum _{j=1}^{M_n}(p_{k,j}^2+q_{k,j})\right).$$

Proof According to Definition 2.1, we have

$$\begin{array}{rcl}(A_{S_{N_n{M}_m}}f)(x)& :=& E(f(S_{N_n{M}_m}+x))\\ =& \sum _{n=1}^{\mathrm{\infty }}P(N_n=n)\sum _{m=1}^{\mathrm{\infty }}P(M_n=m)(A_{S_{nm}}f)(x)\end{array}$$

and

$$\begin{array}{rcl}(A_{Z_{{\delta }_{N_n{M}_m}}}f)(x)& :=& E(f(Z_{{\delta }_{N_n{M}_m}}+x))\\ =& \sum _{n=1}^{\mathrm{\infty }}P(N_n=n)\sum _{m=1}^{\mathrm{\infty }}P(M_n=m)(A_{Z_{{\delta }_{nm}}}f)(x).\end{array}$$

Therefore, for all functions $f\in \mathbb{K}$, and for all $x\in {\mathbb{Z}}_{+}$, we have

$$\begin{array}{c}\parallel A_{S_{N_n{M}_m}}(f)-A_{Z_{{\delta }_{N_n{M}_m}}}(f)\parallel \hfill \\ \le \sum _{n=1}^{\mathrm{\infty }}P(N_n=n)\sum _{m=1}^{\mathrm{\infty }}P(M_n=m)\parallel A_{S_{nm}}(f)-A_{Z_{{\delta }_{n,m}}}(f)\parallel \hfill \\ \le 2\parallel f\parallel \sum _{n=1}^{\mathrm{\infty }}P(N_n=n)\sum _{m=1}^{\mathrm{\infty }}P(M_n=m)\left(\sum _{k=1}^n\sum _{j=1}^m(p_{k,j}^2+q_{k,j})\right)\hfill \\ =2\parallel f\parallel \sum _{n=1}^{\mathrm{\infty }}P(N_n=n)E\left(\sum _{k=1}^n\sum _{j=1}^{M_m}(p_{k,j}^2+q_{k,j})\right)\hfill \\ =2\parallel f\parallel E\left(\sum _{k=1}^{N_n}\sum _{j=1}^{M_m}(p_{k,j}^2+q_{k,j})\right).\hfill \end{array}$$

Thus,

$$d_{TR}(S_{N_n{M}_m},Z_{{\delta }_{N_n,M_m}},f)\le 2\parallel f\parallel E\left(\sum _{k=1}^{N_n}\sum _{j=1}^{M_n}(p_{k,j}^2+q_{k,j})\right).$$

The proof is straightforward. □

Remark 3.2 In the case of all probabilities $q_{n,j}=0$, $j=1,2,\dots ,n$; $n=1,2,\dots$ the partial sum $S_n={\sum }_{j=1}^n{X}_{n,j}$ will become a Poisson-binomial random variable, and one concludes that the results of Theorems 3.1, 3.2, 3.3, and 3.4 are extensions of results in [12] (see [12] 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.