# Poisson Approximation to Weighted Sums of Independent Random Variables

Pratima Eknath Kadu

###### Abstract

This paper deals with Poisson approximation to weighted sums of double-indexed
independent random variables using Stein’s method. The obtained results
improved the existing bounds. As applications, limit theorems for sums of
weighted Bernoulli and geometric random variables are discussed.

Keywords: Poisson distribution; weighted sums; limit theorems; Stein’s method.
MSC 2020 Subject Classifications: Primary: 62E17, 60F05; Secondary: 62E20,
60E05.

##  1 Introduction and Preliminaries

Weighted sums of random variables (rvs) have been successfully applied in the
areas related to least-square estimators, non-parametric regression function
estimators, and Jackknife estimates, among many others. Such sums are also
useful in characterizing various statistics. The exact distributions of
weighted sums of rvs are not tractable in general, and therefore its
approximation is of interest to study their asymptotic behavior. Limit
theorems for weighted sums of rvs have been discussed in the literature, for
example, see Chow and Lai [7], Dolera [8], Lai and Robbins [14], Zhang [20],
and references therein.
Let $Y_{i,n},1\leq i\leq n$, be double-indexed independent rvs concentrated on
$\mathbb{Z}_{+}=\\{0,1,2,\ldots\\}$, the set of non-negative integers. Also,
define

$\displaystyle Z_{n}:=\sum_{i=1}^{n}c_{i}Y_{i,n},$ (1.1)

the weighted sums of double-indexed independent rvs, where
$c_{i}\in\mathbb{N}=\\{1,2,\ldots\\}$, the set of positive integers. We assume
$c_{i}=1$ for at least one $i$ so that $Z_{n}$ becomes $\mathbb{Z}_{+}$-valued
random variable (rv). Limit theorems for the weighted sums of rvs have been
discussed in the literature if the sum of weights is finite. For example, Chow
and Lai [7] are considered the finite sum of the square of weights, and Bhati
and Rattihalli [5] are considered the geometrically weighted sums. But if the
sums of weights are not finite then the study of limit behavior of such
distributions becomes challenging. Therefore, in this paper, we consider
natural weights and obtain limit theorems for $Z_{n}$ that helps to visualize
the behavior of $Z_{n}$.
Next, let $X\sim\mathcal{P}_{\lambda}$, the Poisson distribution, then the
probability mass function of $X$ is given by

$\displaystyle\mathbb{P}(X=k)=\frac{e^{-\lambda}{\lambda}^{k}}{k!},\quad
k\in\mathbb{Z}_{+}.$ (1.2)

We obtain error bound for X-approximation to $Z_{n}$. The total variance
distance is used as a distance metric. Our results are derived using Stein’s
method (Stein [17]) which can be followed in three steps. First, for a random
variable $Y$, find a Stein operator $\mathscr{A}$ such that
$\mathbb{E}(\mathscr{A}g(Y))=0$, for $g\in G_{Y}$, where $G_{Y}:=\\{g\in
G|g(0)=0,\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
g(k)=0\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ k\notin S(Y)\\}$,
$G:=\\{f|f:\mathbb{Z}_{+}\rightarrow\mathbb{R}\text{ is bounded}\\}$, and
$S(Y)$ is the support of random variable $Y$. Second, solve the following
Stein equation

$\mathscr{A}g(k)=f(k)-\mathbb{E}f(Y),\leavevmode\nobreak\
k\in\mathbb{Z}_{+}\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ f\in G.$
(1.3)

Finally, putting a random variable $Z$ in place of $k$ and taking expectation
and supremum, we have

$d_{TV}(Y,Z):=\sup_{f\in\mathcal{I}}\big{|}\mathbb{E}f(Y)-\mathbb{E}f(Z)\big{|}=\sup_{f\in\mathcal{I}}\big{|}\mathbb{E}[\mathscr{A}g(Z)]\big{|},$
(1.4)

where $\mathcal{I}=\\{{\bf 1}_{A}\leavevmode\nobreak\ |\leavevmode\nobreak\
A\subseteq\mathbb{Z}_{+}\\}$ and ${\bf 1}_{A}$ is the indicator function of
$A$. Now, for the random variable X defined in (1.2), the Stein operator is
given by

$\mathscr{A}g(k)=\lambda g(k+1)-kg(k),\quad{\rm for}\leavevmode\nobreak\
k\in\mathbb{Z}_{+}\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ g\in
G_{X}$ (1.5)

and the solution to the Stein equation (1.3), say $g_{f}$, satisfies

$\|g_{f}\|\leq\frac{1}{\max(1,\sqrt{\lambda})},\quad\text{and}\quad\|\Delta
g_{f}\|\leq\frac{1}{\max(1,\lambda)},\quad\text{for $f\in\mathcal{I}$, $g\in
G_{X}$}.$ (1.6)

For additional details, see Barbour and Hall [2], Barbour et. al. [3], Chen
et. al. [6], Brown and Xia [4], and Kumar et. al. [13]. For recent
developments, see Ley et. al. [16], Upadhye and Barman [18], Kumar [12], and
references therein.
This paper is organized as follows. In Section 2, we obtain the error bound
for Poisson approximation to $Z_{n}$. We show that our bound is better than
the existing bound by numerical comparisons. As applications, we also discuss
limit theorems for weighted sums of Bernoulli and geometric rvs.

##  2 Bounds For Poisson Approximation

In this section, we obtain error bound for Poisson approximation to weighted
sums of rvs and discuss some limit theorems. Recall that
$\mathcal{P}_{\lambda}$ follows Poisson distribution with parameter
$\lambda>0$ and $Z_{n}=\sum_{i=1}^{n}c_{i}Y_{i,n}$, where $Y_{i,n},1\leq i\leq
n$, are double-indexed rvs. The following theorem gives the upper bound for
the total variation distance between $Z_{n}$ and $\mathcal{P}_{\lambda}$.

######  Theorem 2.1.

Let $Z_{n}$ and $\mathcal{P}_{\lambda}$ be defined as in (1.1) and (1.2),
respectively. Then

$\displaystyle
d_{TV}(Z_{n},\mathcal{P}_{\lambda})\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{1}{\max(1,\lambda)}\sum_{i=1}^{n}\sum_{j=1}^{\infty}jc_{i}|c_{i}\mathbb{E}(Y_{i,n})p_{i,n}(jc_{i})-(jc_{i}+1)p_{i,n}(jc_{i}+1)|,$

where $p_{i,n}(k)=\mathbb{P}(Y_{i,n}=k)$.

###### Proof.

Replacing $k$ by $Z_{n}$ in (1.5) and taking expectation, we have

$\displaystyle\mathbb{E}[\mathscr{A}g(Z_{n})]$
$\displaystyle={\lambda}\mathbb{E}[g(Z_{n}+1)]-\mathbb{E}[Z_{n}g(Z_{n})]$
$\displaystyle=(\lambda-\mathbb{E}Z_{n})\mathbb{E}[g(Z_{n}+1)]+\sum_{i=1}^{n}c_{i}\mathbb{E}(Y_{i,n})\mathbb{E}[g(Z_{n}+1)]-\sum_{i=1}^{n}c_{i}\mathbb{E}\left[Y_{i,n}g(Z_{n})\right].$
(2.1)

Next, let $Z_{i,n}=Z_{n}-c_{i}Y_{i,n}$ then $Y_{i,n}$ and $Z_{i,n}$ are
independent rvs. Adding and subtracting
$\sum_{i=1}^{n}c_{i}\mathbb{E}\left[Y_{i,n}g(Z_{i,n}+1)\right]$ in (2.1), we
get

$\displaystyle\mathbb{E}[\mathscr{A}g(Z_{n})]$
$\displaystyle=(\lambda-\mathbb{E}Z_{n})\mathbb{E}[g(Z_{n}+1)]+\sum_{i=1}^{n}c_{i}\mathbb{E}(Y_{i,n})\mathbb{E}[g(Z_{n}+1)-g(Z_{i,n}+1)]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\sum_{i=1}^{n}c_{i}\mathbb{E}[Y_{i,n}(g(Z_{n})-g(Z_{i,n}+1))].$ (2.2)

Consider the following expression from the second term in (2.2)

$\displaystyle\mathbb{E}[g(Z_{n}+1)-g(Z_{i,n}+1)]$
$\displaystyle=\mathbb{E}[g(Z_{i,n}+c_{i}Y_{i,n}+1)-g(Z_{i,n}+1)]$
$\displaystyle=\sum_{k=0}^{\infty}{\mathbb{E}}[g(Z_{i,n}+k+1)-g(Z_{i,n}+1)]p_{i,n,c_{i}}^{*}(k),$
(2.3)

where

$\displaystyle
p_{i,n,c_{i}}^{*}(k)=\mathbb{P}(c_{i}Y_{i,n}=k)=\left\\{\begin{array}[]{ll}p_{i,n}\left(\frac{k}{c_{i}}\right),&\text{if
}k=jc_{i},j=0,1,2,\ldots,\\\ 0,&\text{otherwise}.\end{array}\right.$

Note that

$\displaystyle\sum_{l=1}^{k}\Delta g(Z_{i,n}+l)=g(Z_{i,n}+k+1)-g(Z_{i,n}+1).$
(2.4)

Substituting (2.4) in (2.3), we have

$\displaystyle\mathbb{E}[g(Z_{n}+1)-g(Z_{i,n}+1)]$
$\displaystyle=\sum_{k=1}^{\infty}\sum_{l=1}^{k}{\mathbb{E}}[\Delta
g(Z_{i,n}+l)]p_{i,n,c_{i}}^{*}(k).$ (2.5)

Therefore, the second term of (2.2) leads to

$\displaystyle\sum_{i=1}^{n}c_{i}\mathbb{E}(Y_{i,n})\mathbb{E}[g(Z_{n}+1)-g(Z_{i,n}+1)]$
$\displaystyle=\sum_{i=1}^{n}\sum_{k=1}^{\infty}\sum_{l=1}^{k}c_{i}\mathbb{E}(Y_{i,n}){\mathbb{E}}[\Delta
g(Z_{i,n}+l)]p_{i,n,c_{i}}^{*}(k).$ (2.6)

Similarly,

$\displaystyle\sum_{i=1}^{n}c_{i}\mathbb{E}[Y_{i,n}(g(Z_{n})-g(Z_{i,n}+1))]$
$\displaystyle=\sum_{i=1}^{n}\sum_{k=2}^{\infty}\sum_{l=1}^{k-1}k\mathbb{E}[\Delta
g(Z_{i,n}+l)]p_{i,n,c_{i}}^{*}(k)$
$\displaystyle=\sum_{i=1}^{n}\sum_{k=1}^{\infty}\sum_{l=1}^{k}(k+1)\mathbb{E}[\Delta
g(Z_{i,n}+l)]p_{i,n,c_{i}}^{*}(k+1).$ (2.7)

Substituting (2.6) and (2.7) in (2.2), we get

$\displaystyle\mathbb{E}[\mathscr{A}g(Z_{n})]$
$\displaystyle=(\lambda-\mathbb{E}Z_{n})\mathbb{E}[g(Z_{n}+1)]+\sum_{i=1}^{n}\sum_{k=1}^{\infty}\sum_{l=1}^{k}c_{i}\mathbb{E}(Y_{i,n}){\mathbb{E}}[\Delta
g(Z_{i,n}+l)]p_{i,n,c_{i}}^{*}(k)$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\sum_{i=1}^{n}\sum_{k=1}^{\infty}\sum_{l=1}^{k}(k+1)\mathbb{E}[\Delta
g(Z_{i,n}+l)]p_{i,n,c_{i}}^{*}(k+1).$

Hence,

$\displaystyle|\mathbb{E}[\mathscr{A}g(Z_{n})]|$
$\displaystyle\leq|\lambda-\mathbb{E}Z_{n}|\leavevmode\nobreak\ \|g\|+\|\Delta
g\|\sum_{i=1}^{n}\sum_{k=1}^{\infty}k|c_{i}\mathbb{E}(Y_{i,n})p_{i,n,c_{i}}^{*}(k)-(k+1)p_{i,n,c_{i}}^{*}(k+1)|$
$\displaystyle=|\lambda-\mathbb{E}Z_{n}|\leavevmode\nobreak\ \|g\|+\|\Delta
g\|\sum_{i=1}^{n}\sum_{j=1}^{\infty}jc_{i}|c_{i}\mathbb{E}(Y_{i,n})p_{i,n}(jc_{i})-(jc_{i}+1)p_{i,n}(jc_{i}+1)|.$

Using (1.6) the result follows. ∎

######  Remark 2.1.

Note that the bound given in Theorem 2.1 is an improvement over the bound
given in Corollary 2.1 of Kumar [12]. In particular, if $Y_{i,n}\sim
Ber(p_{i,n})$ then, from Theorem 2.1, we have

$\displaystyle d_{TV}(Z_{n},\mathcal{P}_{\lambda})$
$\displaystyle\leq\frac{\sum_{i=1}^{n}c_{i}^{2}{p}^{2}_{i,n}}{\max\left(1,\lambda\right)},$
(2.8)

where $\lambda=\sum_{i=1}^{n}c_{i}p_{i,n}$. Also, from Corollary 2.1 of Kumar
[12], we have

$\displaystyle d_{TV}(Z_{n},\text{PB}(N,p))$
$\displaystyle\leq\frac{\gamma}{\left\lfloor N\right\rfloor
pq}\sum_{i=1}^{n}c_{i}\left(\sum_{\ell=1}^{c_{i}-1}|q-\ell
q_{i,n}|p_{i,n}+qc_{i}p_{i,n}^{2}\right),$ (2.9)

where PB$(N,p)$ follows pseudo-binomial distribution with
$N=(1/p)\sum_{i=1}^{n}c_{i}p_{i,n}$ and
$q=1-p=\left(\sum_{i=1}^{n}c_{i}^{2}p_{i,n}q_{i,n}\right)/\left(\sum_{i=1}^{n}c_{i}p_{i,n}\right)$.
Also,
$\gamma\leq\sqrt{\frac{2}{\pi}}\big{(}\frac{1}{4}+\sum_{i=1}^{n}\gamma_{i}-\gamma^{*}\big{)}^{-1/2}$
with $\gamma_{i}=\min\\{1/2,1-d_{TV}(c_{i}Y_{i,n},c_{i}Y_{i,n}+1)\\}$ and
$\gamma^{*}=\max_{1\leq i\leq n}\gamma_{i}$. Note that the bound given in
(2.8) is better than the bound given in (2.9). Also, the bound given in (2.9)
seems to be invalid ($q\geq 1$) for small values of $p_{i,n}$. For example,
let the values of $p_{i,n}$ be as follows:

Table 1: The values of $p_{i,n}$ | $i$ | $p_{i,n}$ | $c_{i}$ | $i$ | $p_{i,n}$ | $c_{i}$ | $i$ | $p_{i,n}$ | $c_{i}$
---|---|---|---|---|---|---|---|---|---
Set 1 | 1-10 | 0.5 | 1 | 11-20 | 0.45 | 2 | 21-30 | 0.40 | 1
Set 2 | 1-10 | 0.05 | 1 | 11-20 | 0.04 | 2 | 21-30 | 0.04 | 3

Then, the following table gives the comparison between the bounds given in
(2.8) and (2.9).

$n$ | Set 1 | Set 2
---|---|---
From (2.8) |  From (2.9) |  From (2.8) |  From (2.9)
10 | 0.5 | 0.797885 | 0.025 | 0.797885
20 | 0.757143 | 1.60360 | 0.0684615 | Not Valid
30 | 0.677778 | 1.34907 | 0.0772727 | Not Valid

Observe that our bounds are better than the bounds given by Kumar [12].

######  Corollary 2.1.

Assume the conditions of Theorem 2.1 holds then, for $c_{i}=1,1\leq i\leq n$,
we have

$\displaystyle
d_{TV}(Z_{n},\mathcal{P}_{\lambda})\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{1}{\max(1,\lambda)}\sum_{i=1}^{n}\sum_{k=1}^{\infty}k|\mathbb{E}(Y_{i,n})p_{i,n}(k)-(k+1)p_{i,n}(k+1)|.$

######  Remarks 2.1.

  * (i)

The bound given in Corollary 2.1 is also obtained by Kumar et al. [13, Theorem
3.1]. Therefore, if $Y_{i,n}$ follows power series distribution then all
limiting results obtained by Kumar et al. [13] follows as special cases.

  * (ii)

Observe that if $Y_{i,n}\sim Po({\lambda}_{i,n})$ and
$\lambda=\sum_{i=1}^{n}\lambda_{i,n}$ then
$d_{TV}(Z_{n},\mathcal{P}_{\lambda})=0$, as expected.

  * (iii)

If $Y_{i,n}\sim Ber(p_{i,n})$ then it can be easily verified that

$\displaystyle d_{TV}(Z_{n},\mathcal{P}_{\lambda})$
$\displaystyle\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{\sum_{i=1}^{n}{p}^{2}_{i,n}}{\max\left(1,\lambda\right)}.$
(2.10)

Moreover, if $\lambda=\sum_{i=1}^{n}{p}_{i,n}$ then, we have

$\displaystyle d_{TV}(Z_{n},\mathcal{P}_{\lambda})$
$\displaystyle\leq\frac{\sum_{i=1}^{n}{p}^{2}_{i,n}}{\max\left(1,\lambda\right)},$

which is the bound also obtained by Barbour and Hall [2], and an improvement
over the bounds given by Le Cam [15] and Kerstan [11].

  * (iv)

If $Y_{i,n}\sim Geo(p_{i,n})$ then it can be easily verified that

$\displaystyle d_{TV}(Z_{n},\mathcal{P}_{\lambda})$
$\displaystyle\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{\sum_{i=1}^{n}\left(\frac{q_{i,n}}{p_{i,n}}\right)^{2}}{\max\left(1,\lambda\right)},$
(2.11)

where $q_{i,n}\leq 1/2$. Moreover, if
$\lambda=\sum_{i=1}^{n}\frac{q_{i,n}}{p_{i,n}}$ then, we have

$\displaystyle d_{TV}(Z_{n},\mathcal{P}_{\lambda})$
$\displaystyle\leq\frac{\sum_{i=1}^{n}\left(\frac{q_{i,n}}{p_{i,n}}\right)^{2}}{\max\left(1,\lambda\right)},$

which is the bound also obtained by Kadu [10] and Kumar et al. [13], and an
improvement over the bounds given by Vellaisamy and Upadhye [19].

From (2.10), if $\mathbb{E}Z_{n}=\sum_{i=1}^{n}p_{i,n}\to\lambda$ and
$\max_{1\leq i\leq n}p_{i,n}\to 0$ then
$Z_{n}\stackrel{{\scriptstyle\mathcal{L}}}{{\to}}\mathcal{P}_{\lambda}$.
Similarly, from (2.11), if $\mathbb{E}Z_{n}\to\lambda$ and $\max_{1\leq i\leq
n}q_{i,n}\to 0$ then
$Z_{n}\stackrel{{\scriptstyle\mathcal{L}}}{{\to}}\mathcal{P}_{\lambda}$. For
more details, see Examples 3.2 and 3.4 of Kumar et al. [13]. Now, we
generalize these two results for weighted sums considered in this paper. The
following examples gives the limiting results for weighted sums of Bernoulli
and geometric independent (non-identical) rvs.

######  Example 2.1.

Let $Y_{i,n}\sim Ber(p_{i,n})$ then, from Theorem 2.1, we have

$\displaystyle d_{TV}(Z_{n},\mathcal{P}_{\lambda})$
$\displaystyle\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{\sum_{i=1}^{n}c_{i}^{2}{p}^{2}_{i,n}}{\max\left(1,\lambda\right)}$
$\displaystyle\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{p_{n}^{*}\sum_{i=1}^{n}c_{i}{p}_{i,n}}{\max\left(1,\lambda\right)},$

where $p_{n}^{*}=\max_{1\leq i\leq n}c_{i}p_{i,n}$. Note that if
$\mathbb{E}Z_{n}=\sum_{i=1}^{n}c_{i}p_{i,n}\to\lambda$ and $p_{n}^{*}\to 0$
then $d_{TV}(Z_{n},\mathcal{P}_{\lambda})\to 0$ as $n\to\infty$. This shows
that $Z_{n}\stackrel{{\scriptstyle\mathcal{L}}}{{\to}}\mathcal{P}_{\lambda}$.

######  Example 2.2.

Let $Y_{i,n}\sim Geo(p_{i,n})$ then, from Theorem 2.1, we have

$\displaystyle d_{TV}(Z_{n},\mathcal{P}_{\lambda})$
$\displaystyle\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{1}{\max\left(1,\lambda\right)}\sum_{i=1}^{n}\sum_{j=1}^{\infty}jc_{i}q_{i,n}^{jc_{i}+1}(c_{i}+(jc_{i}+1)p_{i,n})$
$\displaystyle=\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{1}{\max\left(1,\lambda\right)}\sum_{i=1}^{n}\frac{c_{i}q_{i,n}^{c_{i}+1}}{(1-q_{i,n}^{c_{i}})^{3}}\left[c_{i}(1-q_{i,n}^{c_{i}})+p_{i,n}((c_{i}-1)q_{i,n}^{c_{i}}+c_{i}+1)\right]$
$\displaystyle\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{1}{\max\left(1,\lambda\right)}\sum_{i=1}^{n}\frac{c_{i}^{2}q_{i,n}^{c_{i}+1}}{(1-q_{i,n}^{c_{i}})^{3}}\left[3+q_{i,n}^{c_{i}}\right]$
$\displaystyle\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{4}{\max\left(1,\lambda\right)}\sum_{i=1}^{n}\frac{c_{i}^{2}q_{i,n}^{c_{i}+1}}{(1-q_{i,n}^{c_{i}})^{3}}$
$\displaystyle\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{4q_{n}^{*}}{\max\left(1,\lambda\right)}\sum_{i=1}^{n}\frac{c_{i}q_{i,n}}{(1-q_{i,n}^{c_{i}})^{3}},$

where $q_{n}^{*}=\max_{1\leq i\leq n}c_{i}q_{i,n}^{c_{i}}$. Note that if
$\sum_{i=1}^{n}c_{i}q_{i,n}\to\lambda$ and $q_{n}^{*}\to 0$ then
$d_{TV}(Z_{n},\mathcal{P}_{\lambda})\to 0$ as $n\to\infty$. This shows that
$Z_{n}\stackrel{{\scriptstyle\mathcal{L}}}{{\to}}\mathcal{P}_{\lambda}$.

######  Remark 2.2.

Observe that if $c_{i}=1$ then the conditions in Examples 2.1 and 2.2 reduced
to the standard conditions used in Examples 3.2 and 3.4 of Kumar et al. [13],
respectively.

Next, we discuss an application of our result to compound Poisson distribution
in the following example.

######  Example 2.3.

Let $c_{i}=i$, $Y_{i,n}\sim Poi(\lambda_{i,n})$ and
$Z_{n}=\sum_{i=1}^{n}iY_{i,n}$ then $Z_{\infty}$ follows compound Poisson
distribution. Also, from Theorem 2.1, we have

$\displaystyle d_{TV}(Z_{n},\mathcal{P}_{\lambda})$
$\displaystyle\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{1}{\max\left(1,\lambda\right)}\sum_{i=2}^{n}i(i-1)e^{-\lambda_{i,n}}\sum_{j=1}^{\infty}j\frac{\lambda_{i,n}^{ij+1}}{(ij)!}$
$\displaystyle\leq\frac{|\lambda-\mathbb{E}Z_{n}|}{\max(1,\sqrt{\lambda})}+\frac{1}{\max\left(1,\lambda\right)}\sum_{i=2}^{n}i(i-1)\lambda_{i,n}^{i+1}e^{\lambda_{i,n}^{i}-\lambda_{i,n}}.$

Note that if $i\lambda_{i,n}\searrow\leavevmode\nobreak\ 0$ then the above
bound is useful in practice. See Barbour et al. [1] and Gan and Xia [9] for
details.

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