## 1 Introduction

Let $\left\{{X}_{n,j},j=1,2,\dots ,n;n=1,2,\dots \right\}$ be a row-wise triangular array of independent integer-valued random variables with success probabilities $P\left({X}_{n,j}=1\right)={p}_{n,j}$; $P\left({X}_{n,j}=0\right)=1-{p}_{n,j}-{q}_{n,j}$; ${p}_{n,j},{q}_{n,j}\in \left(0,1\right)$; ${p}_{n,j}+{q}_{n,j}\in \left(0,1\right)$; $j=1,2,\dots ,n$; $n=1,2,\dots$ . Set ${S}_{n}={\sum }_{j=1}^{n}{X}_{n,j}$ and ${\lambda }_{n}=E\left({S}_{n}\right)={\sum }_{j=1}^{n}{p}_{n,j}$. 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

(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

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

It is to be noticed that another inequality in Poisson approximation is usually expressed in terms of the total variation distance ${d}_{TV}\left({S}_{n},{Z}_{\lambda }\right)$

${d}_{TV}\left({S}_{n},{Z}_{\lambda }\right)\le \sum _{j=1}^{n}{p}_{n,j}^{2},$
(3)

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

${d}_{TV}\left(P,Q\right):=\frac{1}{2}\sum _{x\in {\mathbb{Z}}_{+}}|P\left(x\right)-Q\left(x\right)|.$
(4)

(For other surveys, see [14], 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 [111], 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 [1220], and [21].

The results obtained in this paper are extensions of known results in [1, 5, 911], 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 [1315, 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

$\left({A}_{X}f\right)\left(x\right)=E\left(f\left(X+x\right)\right)=\sum _{k=0}^{\mathrm{\infty }}f\left(x+k\right)P\left(X=k\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }f\in \mathbb{K},\mathrm{\forall }x\in {\mathbb{Z}}_{+},$
(5)

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

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, 1719], 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. 1.

${A}_{X}\left(f+g\right)={A}_{X}\left(f\right)+{A}_{X}\left(g\right)$.

2. 2.

${A}_{X}\left(\alpha f\right)=\alpha {A}_{X}\left(f\right)$.

3. 3.

$\parallel {A}_{X}\left(f\right)\parallel ⩽\parallel f\parallel$.

4. 4.

$\parallel {A}_{X}\left(f\right)+{A}_{Y}\left(f\right)\parallel ⩽\parallel {A}_{X}\left(f\right)\parallel +\parallel {A}_{Y}\left(f\right)\parallel$.

5. 5.

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

${A}_{X+Y}\left(f\right)={A}_{X}{A}_{Y}\left(f\right)={A}_{Y}{A}_{X}\left(f\right).$

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

$\begin{array}{rl}{A}_{X+Y}f\left(x\right)& =\sum _{l=0}^{\mathrm{\infty }}f\left(x+l\right)P\left(X+Y=l\right)=\sum _{r,k=0}^{\mathrm{\infty }}f\left(x+k+r\right)P\left(Y=k\right)P\left(X=r\right)\\ ={A}_{X}\left({A}_{Y}f\left(x\right)\right)={A}_{X}{A}_{Y}f\left(x\right).\end{array}$
1. 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 $f\in \mathbb{K}$, ${A}_{{S}_{n}}\left(f\right)={A}_{{X}_{1}}{A}_{{X}_{2}}\cdots {A}_{{X}_{n}}\left(f\right)$ is the operator associated with the partial sum ${S}_{n}={X}_{1}+{X}_{2}+\cdots +{X}_{n}$.

2. 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 $f\in \mathbb{K}$,

$\parallel {A}_{{\sum }_{k=1}^{n}{X}_{k}}\left(f\right)-{A}_{{\sum }_{k=1}^{n}{Y}_{k}}\left(f\right)\parallel ⩽\sum _{k=1}^{n}\parallel {A}_{{X}_{k}}\left(f\right)-{A}_{{Y}_{k}}\left(f\right)\parallel .$
(6)

Clearly,

$\begin{array}{c}{A}_{{X}_{1}}{A}_{{X}_{2}}\cdots {A}_{{X}_{n}}-{A}_{{Y}_{1}}{A}_{{Y}_{2}}\cdots {A}_{{Y}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{k=1}^{n}{A}_{{X}_{1}}{A}_{{X}_{2}}\cdots {A}_{{X}_{k-1}}\left({A}_{{X}_{k}}-{A}_{{Y}_{k}}\right){A}_{{Y}_{k+1}}\cdots {A}_{{Y}_{n}}.\hfill \end{array}$

Accordingly,

$\begin{array}{rcl}\parallel {A}_{{\sum }_{k=1}^{n}{X}_{k}}\left(f\right)-{A}_{{\sum }_{k=1}^{n}{Y}_{k}}\left(f\right)\parallel & ⩽& \sum _{k=1}^{n}\parallel {A}_{{X}_{1}}\dots {A}_{{X}_{k-1}}\left({A}_{{X}_{k}}-{A}_{{Y}_{k}}\right){A}_{{Y}_{k+1}}\cdots {A}_{{Y}_{n}}\left(f\right)\parallel \\ ⩽& \sum _{k=1}^{n}\parallel {A}_{{Y}_{k+1}}\cdots {A}_{{Y}_{n}}\left({A}_{{X}_{k}}-{A}_{{Y}_{k}}\right)\left(f\right)\parallel \\ ⩽& \sum _{k=1}^{n}\parallel {A}_{{X}_{k}}\left(f\right)-{A}_{{Y}_{k}}\left(f\right)\parallel .\end{array}$
1. 8.

$\parallel {A}_{X}^{n}\left(f\right)-{A}_{Y}^{n}\left(f\right)\parallel \le n\parallel {A}_{X}\left(f\right)-{A}_{Y}\left(f\right)\parallel$.

Lemma 2.1 The equation ${A}_{X}f\left(x\right)={A}_{Y}f\left(x\right)$ 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

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

Taking

then we recover

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

It follows that $P\left({X}_{n}⩽t\right)-P\left(X⩽t\right)\to 0$ 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 $\left(\mathrm{\Omega },\mathbb{A},\mathbb{P}\right)$ be a probability space and let $\mathbb{Z}\left(\mathrm{\Omega },\mathbb{A}\right)$ be a space of real-valued -measurable random variables $X:\mathrm{\Omega }\to \mathbb{R}$.

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

1. 1.

$P\left(X=Y\right)=1⇒d\left(X,Y\right)=0$;

2. 2.

$d\left(X,Y\right)=d\left(Y,X\right)$;

3. 3.

$d\left(X,Y\right)\le d\left(X,Z\right)+d\left(Z,Y\right)$.

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 ${d}_{TR}\left(X,Y;f\right)$ of two random variables X and Y with respect to the function $f\in \mathbb{K}$ is defined by

${d}_{TR}\left(X,Y;f\right):=\parallel {A}_{X}f-{A}_{Y}f\parallel =\underset{x\in {\mathbb{Z}}_{+}}{sup}|Ef\left(X+x\right)-Ef\left(Y+x\right)|.$
(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. 1.

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

2. (a)

For every $f\in \mathbb{K}$, the distance ${d}_{TR}\left(X,Y;f\right)=0$ if $P\left(X=Y\right)=1$.

3. (b)

${d}_{TR}\left(X,Y;f\right)={d}_{TR}\left(Y,X;f\right)$ for every $f\in \mathbb{K}$.

4. (c)

${d}_{TR}\left(X,Y;f\right)\le {d}_{TR}\left(X,Z;f\right)+{d}_{TR}\left(Z,Y;f\right)$ for every $f\in \mathbb{K}$.

1. 2.

If ${d}_{TR}\left(X,Y;f\right)=0$ for every $f\in \mathbb{K}$, then ${F}_{X}\equiv {F}_{Y}$.

2. 3.

Let $\left\{{X}_{n},n\ge 1\right\}$ be a sequence of random variables and let X be a random variable. The condition

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

1. 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 $f\in \mathbb{K}$,

${d}_{TR}\left(\sum _{j=1}^{n}{X}_{j},\sum _{j=1}^{n}{Y}_{j};f\right)\le \sum _{j=1}^{n}{d}_{TR}\left({X}_{j},{Y}_{j};f\right).$
(8)

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

${d}_{TR}\left(\sum _{j=1}^{n}{X}_{j},\sum _{j=1}^{n}{Y}_{j};f\right)\le n{d}_{TR}\left({X}_{1},{Y}_{1};f\right).$
1. 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 $\left\{{N}_{n},n\ge 1\right\}$ 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 $f\in \mathbb{K}$,

${d}_{TR}\left(\sum _{j=1}^{{N}_{n}}{X}_{j},\sum _{j=1}^{{N}_{n}}{Y}_{j};f\right)\le \sum _{k=1}^{\mathrm{\infty }}P\left({N}_{n}=k\right)\sum _{j=1}^{k}{d}_{TR}\left({X}_{j},{Y}_{j};f\right).$
(9)
2. 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 $\left\{{N}_{n},n\ge 1\right\}$ 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 $E\left({N}_{n}\right)<+\mathrm{\infty }$, $n\ge 1$. Then, for every $f\in \mathbb{K}$, we have

${d}_{TR}\left(\sum _{j=1}^{{N}_{n}}{X}_{j},\sum _{j=1}^{{N}_{n}}{Y}_{j};f\right)\le E\left({N}_{n}\right)\cdot {d}_{TR}\left({X}_{1},{Y}_{1};f\right).$

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

Then

${d}_{TR}\left(X,Y,{\chi }_{A}\right)={d}_{TV}\left(X,Y\right),$

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

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

For a deeper discussion of the total variation distance, we refer the reader to [14], and [5].

## 3 Main results

Let $\left\{{A}_{{X}_{n,j}},j=1,2,\dots ,n;n=1,2,\dots \right\}$ 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 $\left\{{A}_{{Z}_{{p}_{n,j}}},j=1,2,\dots ,n;n=1,2,\dots \right\}$ 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 $\left\{{X}_{n,j},j=1,2,\dots ,n;n=1,2,\dots \right\}$ be a row-wise triangular array of independent, integer-valued random variables with probabilities $P\left({X}_{n,j}=1\right)={p}_{n,j}$, $P\left({X}_{n,j}=0\right)=1-{p}_{n,j}-{q}_{n,j}$; ${p}_{n,j},{q}_{n,j}\in \left(0,1\right)$; ${p}_{n,j}+{q}_{n,j}\in \left(0,1\right)$; $j=1,2,\dots ,n$; $n=1,2,\dots$ . 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 $f\in \mathbb{K}$,

${d}_{TR}\left({S}_{n},{Z}_{{\lambda }_{n}};f\right)\le 2\parallel f\parallel \sum _{j=1}^{n}\left({p}_{n,j}^{2}+{q}_{n,j}\right).$

Proof Applying (8), we have

${d}_{TR}\left({S}_{n},{Z}_{{\lambda }_{n}},f\right)\le \sum _{j=1}^{n}{d}_{TR}\left({X}_{n,j},{Z}_{{p}_{n,j}};f\right)=\sum _{k=1}^{n}\parallel {A}_{{X}_{n,j}}\left(f\right)-{A}_{{Z}_{{p}_{n,j}}}\left(f\right)\parallel .$

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

$\begin{array}{rcl}{A}_{{X}_{nj}}f\left(x\right)-{A}_{{Z}_{{p}_{n,j}}}f\left(x\right)& =& \sum _{r=0}^{\mathrm{\infty }}f\left(x+r\right)\left(P\left({X}_{nj}=r\right)-P\left({Z}_{{\lambda }_{{p}_{n,j}}}=r\right)\right)\\ =& \sum _{r=0}^{\mathrm{\infty }}f\left(x+r\right)\left(P\left({X}_{nj}=r\right)-\frac{{e}^{-{p}_{n,j}}{p}_{n,j}^{r}}{r!}\right)\\ =& f\left(x\right)\left(1-{p}_{n,j}-{q}_{n,j}-{e}^{-{p}_{n,j}}\right)\\ +f\left(x+1\right)\left({p}_{n,j}-{p}_{n,j}{e}^{-{p}_{n,j}}\right)\\ +\sum _{r=2}^{\mathrm{\infty }}f\left(x+r\right)\left(P\left({X}_{n,j}=r\right)-\frac{{e}^{-{p}_{n,j}}{p}_{n,j}^{r}}{r!}\right).\end{array}$

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

$\begin{array}{c}|{A}_{{X}_{n,j}}f\left(x\right)-{A}_{{Z}_{{p}_{n,j}}}f\left(x\right)|\hfill \\ \phantom{\rule{1em}{0ex}}=|f\left(x\right)\left(1-{p}_{n,j}-{q}_{n,j}-{e}^{-{p}_{n,j}}\right)+f\left(x+1\right)\left({p}_{n,j}-{p}_{n,j}{e}^{-{p}_{n,j}}\right)\hfill \\ \phantom{\rule{2em}{0ex}}+\sum _{r=2}^{\mathrm{\infty }}f\left(x+r\right)\left(P\left({X}_{n,j}=r\right)-\frac{{e}^{-{p}_{n,j}}{p}_{n,j}^{r}}{r!}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}=|f\left(x\right)\left(1-{p}_{n,j}-{q}_{n,j}-{e}^{-{p}_{n,j}}\right)|+|f\left(x+1\right)\left({p}_{n,j}-{p}_{n,j}{e}^{-{p}_{n,j}}\right)|\hfill \\ \phantom{\rule{2em}{0ex}}+|\sum _{r=2}^{\mathrm{\infty }}f\left(x+r\right)\left(P\left({X}_{n,j}=r\right)-\frac{{e}^{-{p}_{n,j}}{p}_{n,j}^{r}}{r!}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le |f\left(x\right)\left(1-{p}_{n,j}-{q}_{n,j}-{e}^{-{p}_{n,j}}\right)|+|f\left(x+1\right)\left({p}_{n,j}-{p}_{n,j}{e}^{-{p}_{n,j}}\right)|\hfill \\ \phantom{\rule{2em}{0ex}}+|\sum _{r=2}^{\mathrm{\infty }}f\left(x+r\right)P\left({X}_{n,j}=r\right)|+|\sum _{r=2}^{\mathrm{\infty }}f\left(x+r\right)\frac{{e}^{-{p}_{n,j}}{p}_{n,j}^{r}}{r!}|\hfill \\ \phantom{\rule{1em}{0ex}}\le \left({e}^{-{p}_{n,j}}+{p}_{n,j}+{q}_{n,j}-1\right)\underset{x\in {\mathbb{Z}}_{+}}{sup}|f\left(x\right)|+\left({p}_{n,j}-{p}_{n,j}{e}^{-{p}_{n,j}}\right)\underset{x\in {\mathbb{Z}}_{+}}{sup}|f\left(x\right)|\hfill \\ \phantom{\rule{2em}{0ex}}+\underset{x\in {\mathbb{Z}}_{+}}{sup}|f\left(x\right)||\sum _{r=2}^{\mathrm{\infty }}P\left({X}_{n,k}=r\right)|+\underset{x\in {\mathbb{Z}}_{+}}{sup}|f\left(x\right)||\sum _{r=2}^{\mathrm{\infty }}\frac{{e}^{-{p}_{n,j}}{p}_{n,j}^{r}}{r!}|\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{x\in {\mathbb{Z}}_{+}}{sup}|f\left(x\right)|\left({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}}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=2\parallel f\parallel \left({p}_{n,j}-{p}_{n,j}{e}^{-{p}_{n,j}}+{q}_{n,j}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le 2\parallel f\parallel \left({p}_{n,j}^{2}+{q}_{n,j}\right).\hfill \end{array}$

One infers that

$\mathrm{\forall }f\in K,\phantom{\rule{1em}{0ex}}\parallel {A}_{{X}_{n,j}}\left(f\right)-{A}_{{Z}_{{p}_{n,j}}}\left(f\right)\parallel \le 2\parallel f\parallel \left({p}_{n,j}^{2}+{q}_{n,j}\right).$

Therefore, applying (8), we can assert that

${d}_{TR}\left({S}_{n},{Z}_{{\lambda }_{n}};f\right)\le 2\parallel f\parallel \sum _{j=1}^{n}\left({p}_{n,j}^{2}+{q}_{n,j}\right).$

This completes the proof. □

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

$|P\left({S}_{n}=r\right)-P\left({Z}_{{\lambda }_{n}}=r\right)|\le 2\sum _{j=1}^{n}\left({p}_{n,j}^{2}+{q}_{n,k}\right).$

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

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

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

Theorem 3.2 Let $\left\{{X}_{n,j},j=1,2,\dots ,n;n=1,2,\dots \right\}$ be a row-wise triangular array of independent, integer-valued random variables with probabilities $P\left({X}_{n,j}=1\right)={p}_{n,j}$, $P\left({X}_{n,j}=0\right)=1-{p}_{n,j}-{q}_{n,j}$; ${p}_{n,j},{q}_{n,j}\in \left(0,1\right)$; ${p}_{n,j}+{q}_{n,j}\in \left(0,1\right)$; $j=1,2,\dots ,n$; $n=1,2,\dots$ . 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 $f\in \mathbb{K}$,

${d}_{TR}\left({S}_{{N}_{n}},{Z}_{{\lambda }_{{N}_{n}}};f\right)\le 2\parallel f\parallel E\left(\sum _{j=1}^{{N}_{n}}\left({p}_{{N}_{n},j}^{2}+{q}_{{N}_{n},j}\right)\right).$

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

$\begin{array}{rcl}{d}_{TR}\left({S}_{{N}_{n}},{Z}_{{\lambda }_{{N}_{n}}};f\right)& \le & \sum _{m=1}^{\mathrm{\infty }}P\left({N}_{n}=m\right){d}_{TR}\left({S}_{m},{Z}_{{\lambda }_{m}};f\right)\\ \le & \sum _{m=1}^{\mathrm{\infty }}P\left({N}_{n}=m\right)2\parallel f\parallel \sum _{j=1}^{m}\left({p}_{{N}_{n},j}^{2}+{q}_{{N}_{n},j}\right)\\ =& 2\parallel f\parallel \sum _{m=1}^{\mathrm{\infty }}\left[P\left({N}_{n}=m\right)\sum _{j=1}^{m}\left({p}_{{N}_{n},j}^{2}+{q}_{{N}_{n},j}\right)\right]\\ =& 2\parallel f\parallel E\left(\sum _{j=1}^{{N}_{n}}\left({p}_{{N}_{n},j}^{2}+{q}_{{N}_{n},j}\right)\right).\end{array}$

Therefore,

${d}_{TR}\left({S}_{{N}_{n}},{Z}_{{\lambda }_{{N}_{n}}};f\right)\le 2\parallel f\parallel E\left(\sum _{j=1}^{{N}_{n}}\left({p}_{{N}_{n},j}^{2}+{q}_{{N}_{n},j}\right)\right).$

The proof is complete. □

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

$|P\left({S}_{{N}_{n}}=r\right)-P\left({Z}_{{\lambda }_{{N}_{n}}}=r\right)|\le 2E\left(\sum _{j=1}^{{N}_{n}}\left({p}_{{N}_{n},j}^{2}+{q}_{{N}_{n},j}\right)\right).$

Theorem 3.3 Let $\left\{{X}_{k,j}\right\}$ ($k=1,2,\dots$ ; $j=1,2,\dots$) be a double array of independent integer-valued random variables with probabilities $P\left({X}_{k,j}=1\right)={p}_{k,j}$, $P\left({X}_{k,j}=0\right)=1-{p}_{k,j}-{q}_{k,j}$, ${p}_{n,k}\in \left(0,1\right)$; $k=1,2,\dots$ ; $j=1,2,\dots$ . 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 $f\in \mathbb{K}$,

${d}_{TR}\left({S}_{nm},{Z}_{{\delta }_{n,m}},f\right)\le 2\parallel f\parallel \sum _{k=1}^{n}\sum _{j=1}^{m}\left({p}_{k,j}^{2}+{q}_{k,j}\right).$

Proof Applying the inequality in (8), we have

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

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

${d}_{TR}\left({S}_{k,j},{Z}_{{\lambda }_{k,j}},f\right)\le 2\parallel f\parallel \left({p}_{k,j}^{2}+{q}_{k,j}\right).$

Therefore,

${d}_{TR}\left({S}_{nm},{Z}_{{\delta }_{nm}},f\right)\le 2\parallel f\parallel \sum _{k=1}^{n}\sum _{j=1}^{m}\left({p}_{k,j}^{2}+{q}_{k,j}\right).$

This completes the proof. □

Theorem 3.4 Let $\left\{{X}_{k,j},k=1,2,\dots ;j=1,2,\dots \right\}$ be a double array of independent integer-valued random variables with $P\left({X}_{k,j}=1\right)={p}_{k,j}$; $P\left({X}_{k,j}=0\right)=1-{p}_{k,j}-{q}_{k,j}$; ${p}_{k,j},{q}_{k,j}\in \left(0,1\right)$; ${p}_{k,j}+{q}_{k,j}\in \left(0,1\right)$; $k=1,2,\dots$ ; $n=1,2,\dots$ . 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 ${\delta }_{{N}_{n}{M}_{m}}=E\left({S}_{{N}_{n}{M}_{m}}\right)={\sum }_{k=1}^{{N}_{n}}{\sum }_{j=1}^{{M}_{m}}{p}_{k,j}$. Then, for all functions $f\in \mathbb{K}$,

${d}_{TR}\left({S}_{{N}_{n}{M}_{m}},{Z}_{{\delta }_{{N}_{n}{M}_{m}}},f\right)\le 2\parallel f\parallel E\left(\sum _{k=1}^{{N}_{n}}\sum _{j=1}^{{M}_{n}}\left({p}_{k,j}^{2}+{q}_{k,j}\right)\right).$

Proof According to Definition 2.1, we have

$\begin{array}{rcl}\left({A}_{{S}_{{N}_{n}{M}_{m}}}f\right)\left(x\right)& :=& E\left(f\left({S}_{{N}_{n}{M}_{m}}+x\right)\right)\\ =& \sum _{n=1}^{\mathrm{\infty }}P\left({N}_{n}=n\right)\sum _{m=1}^{\mathrm{\infty }}P\left({M}_{n}=m\right)\left({A}_{{S}_{nm}}f\right)\left(x\right)\end{array}$

and

$\begin{array}{rcl}\left({A}_{{Z}_{{\delta }_{{N}_{n}{M}_{m}}}}f\right)\left(x\right)& :=& E\left(f\left({Z}_{{\delta }_{{N}_{n}{M}_{m}}}+x\right)\right)\\ =& \sum _{n=1}^{\mathrm{\infty }}P\left({N}_{n}=n\right)\sum _{m=1}^{\mathrm{\infty }}P\left({M}_{n}=m\right)\left({A}_{{Z}_{{\delta }_{nm}}}f\right)\left(x\right).\end{array}$

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

$\begin{array}{c}\parallel {A}_{{S}_{{N}_{n}{M}_{m}}}\left(f\right)-{A}_{{Z}_{{\delta }_{{N}_{n}{M}_{m}}}}\left(f\right)\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{n=1}^{\mathrm{\infty }}P\left({N}_{n}=n\right)\sum _{m=1}^{\mathrm{\infty }}P\left({M}_{n}=m\right)\parallel {A}_{{S}_{nm}}\left(f\right)-{A}_{{Z}_{{\delta }_{n,m}}}\left(f\right)\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le 2\parallel f\parallel \sum _{n=1}^{\mathrm{\infty }}P\left({N}_{n}=n\right)\sum _{m=1}^{\mathrm{\infty }}P\left({M}_{n}=m\right)\left(\sum _{k=1}^{n}\sum _{j=1}^{m}\left({p}_{k,j}^{2}+{q}_{k,j}\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}=2\parallel f\parallel \sum _{n=1}^{\mathrm{\infty }}P\left({N}_{n}=n\right)E\left(\sum _{k=1}^{n}\sum _{j=1}^{{M}_{m}}\left({p}_{k,j}^{2}+{q}_{k,j}\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}=2\parallel f\parallel E\left(\sum _{k=1}^{{N}_{n}}\sum _{j=1}^{{M}_{m}}\left({p}_{k,j}^{2}+{q}_{k,j}\right)\right).\hfill \end{array}$

Thus,

${d}_{TR}\left({S}_{{N}_{n}{M}_{m}},{Z}_{{\delta }_{{N}_{n},{M}_{m}}},f\right)\le 2\parallel f\parallel E\left(\sum _{k=1}^{{N}_{n}}\sum _{j=1}^{{M}_{n}}\left({p}_{k,j}^{2}+{q}_{k,j}\right)\right).$

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.