Let us consider a typical cluster sampling design: the entire population consists of different clusters, and the probability for each cluster to be selected into a sample is known. The sum of sample elements is then equal to $S={w_{1}}{S_{1}}+{w_{2}}{S_{2}}+\cdots +{w_{N}}{S_{N}}$. Here, ${S_{i}}$ is the sum of independent identically distributed (iid) random variables (rvs) from the i-th cluster. A similar situation arises in actuarial mathematics when the sum S models the discounted amount of the total net loss of a company, see, for example, [24]. Note that then ${S_{i}}$ may be the sum of dependent rvs. Of course, in actuarial models, ${w_{i}}$ are also typically random, which makes our research just a first step in this direction. In many papers, the limiting behavior of weighted sums is investigated with the emphasis on weights or tails of distributions, see, for example, [6, 16–18, 23, 25–30], and references therein. We, however, concentrate on the impact of $S-{w_{i}}{S_{i}}$ on ${w_{i}}{S_{i}}$. Our research is motivated by the following simple example. Let us assume that ${S_{i}}$ is in some sense close to ${Z_{i}}$, $i=1,2$. Then a natural approximation to ${w_{1}}{S_{1}}+{w_{2}}{S_{2}}$ is ${w_{1}}{Z_{1}}+{w_{2}}{Z_{2}}$. Suppose that we want to estimate the closeness of both sums in some metric $d(\cdot ,\cdot )$. The standard approach which works for the majority of metrics then gives The triangle inequality (1) is not always useful. For example, let ${S_{1}}$ and ${Z_{1}}$ have the same Poisson distribution with parameter n and let ${S_{2}}$ and ${Z_{2}}$ be Bernoulli variables with probabilities 1/3 and 1/4, respectively. Then (1) ensures the trivial order of approximation $O(1)$ only. Meanwhile, both S and Z can be treated as small (albeit different) perturbations to the same Poisson variable and, therefore, one can expect closeness of their distributions at least for large n. The ‘smoothing’ effect that other sums have on the approximation of ${w_{i}}{S_{i}}$ is already observed in [7] (see also references therein). For some general results involving the concentration functions, see, for example, [10, 20].

To make our goals more explicit, we need additional notation. Let $\mathbb{Z}$ denote the set of all integers. Let $\mathcal{F}$ (resp. ${\mathcal{F}_{Z}}$, resp. $\mathcal{M}$) denote the set of probability distributions (resp. distributions concentrated on integers, resp. finite signed measures) on $\mathbb{R}$. Let ${I_{a}}$ denote the distribution concentrated at real a and set $I={I_{0}}$ . Henceforth, the products and powers of measures are understood in the convolution sense. Further, for a measure M, we set ${M}^{0}=I$ and $\exp \{M\}={\sum _{k=0}^{\infty }}{M}^{k}/k!$. We denote by $\widehat{M}(t)$ the Fourier–Stieltjes transform of M. The real part of $\widehat{M}(t)$ is denoted by $Re\widehat{M}(t)$. Observe also that $\widehat{\exp \{M(t)\}}=\exp \{\widehat{M}(t)\}$. We also use $\mathcal{L}(\xi )$ to denote the distribution of ξ.

The Kolmogorov (uniform) norm $|M{|_{K}}$ and the total variation norm $\| M\| $ of M are defined by respectively. Here $M={M}^{+}-{M}^{-}$ is the Jordan–Hahn decomposition of M. Also, for any two measures M and V, $|M{|_{K}}\leqslant \| M\| $, $|MV{|_{K}}\leqslant \| M\| \cdot |V{|_{K}}$, $|\widehat{M}(t)|\leqslant \| M\| $, $\| \exp \{M\}\| \leqslant \exp \{\| M\| \}$. If $F\in \mathcal{F}$, then $|F{|_{K}}=\| F\| =\| \exp \{F-I\}\| =1$. Observe also that, if M is concentrated on integers, then

For $F\in \mathcal{F}$, $h\geqslant 0$, Lévy’s concentration function is defined by

All absolute positive constants are denoted by the same symbol C. Sometimes to avoid possible ambiguities, the constants C are supplied with indices. Also, the constants depending on parameter N are denoted by $C(N)$. We also assume usual conventions ${\sum _{j=a}^{b}}=0$ and ${\prod _{j=a}^{b}}=1$, if $b<a$. The notation Θ is used for any signed measure satisfying $\| \varTheta \| \leqslant 1$. The notation θ is used for any real or complex number satisfying $|\theta |\leqslant 1$.

The results of this section are partially inspired by a comprehensive analytic research of probability generating functions in [12] and the papers on mod-Poisson convergence, see [2, 13, 14], and references therein. Assumptions in the above-mentioned papers are made about the behavior of characteristic or probability generating functions. The inversion inequalities are then used to translate their differences to the differences of distributions. In principle, mod-Poisson convergence means that if an initial rv is a perturbation of some Poisson rv, then their distributions must be close. Formally, it is required for $\exp \{-{\tilde{\lambda }_{n}}({\mathrm{e}}^{\mathrm{i}t}-1)\}{f_{n}}(t)$ to have a limit for some sequence of Poisson parameters ${\tilde{\lambda }_{n}}$, as $n\to \infty $. Here, ${f_{n}}(t)$ is a characteristic function of an investigated rv. Division by a certain Poisson characteristic function is one of the crucial steps in the proof of Theorem 2.1 below, which makes it applicable to rvs satisfying the mod-Poisson convergence definition, provided they can be expressed as sums of independent rvs. Though we use factorial moments, similar to Section 7.1 in [2], our work is much more closer in spirit to [21], where general lemmas about the closeness of lattice measures are proved.

In this section, we consider a general case of independent non-identically distributed rvs, forming a triangular array (a scheme of series). Let ${S_{i}}={X_{i1}}+{X_{i2}}+\cdots +{X_{i{n_{i}}}}$, ${Z_{i}}={Z_{i1}}+{Z_{i2}}+\cdots +{Z_{i{n_{i}}}}$, $i=1,2,\dots ,N$. We assume that all the ${X_{ij}}$, ${Z_{ij}}$ are mutually independent and integer-valued. Observe that, in general, $S={\sum _{i=1}^{N}}{w_{i}}{S_{i}}$ and $Z={\sum _{i=1}^{N}}{w_{i}}{Z_{i}}$ are not integer-valued and, therefore, the standard methods of estimation of lattice rvs do not apply. Note also that, since any infinitely divisible distribution can be expressed as a sum of rvs, Poisson, compound Poisson and negative binomial rvs can be used as ${Z_{i}}$.

The distribution of ${X_{ij}}$ (resp. ${Z_{ij}}$) is denoted by ${F_{ij}}$ (resp. ${G_{ij}}$). The closeness of characteristic functions will be determined by the closeness of corresponding factorial moments. Though it is proposed in [2] to use standard factorial moments even for rvs taking negative values, we think that right-hand side and left-hand side factorial moments, already used in [21], are more natural characteristics. Let, for $k=1,2,\dots \hspace{0.1667em}$, and any $F\in {\mathcal{F}_{Z}}$, For the estimation of the remainder terms we also need the following notation: ${\beta _{k}^{\pm }}({F_{ij}},{G_{ij}})={\nu _{k}^{\pm }}({F_{ij}})+{\nu _{k}^{\pm }}({G_{ij}})$, ${\sigma _{ij}^{2}}=\max (\mathrm{Var}({X_{ij}}),\mathrm{Var}({Z_{ij}}))$, and For the last equality, see (1.9) and (5.15) in [5]. Next we formulate our assumptions. For some fixed integer $s\geqslant 1$, $i=1,\dots ,N,\hspace{2.5pt}j=1,\dots ,{n_{i}}$,

Now we are in position to formulate the main result of this section.

The factor ${({\sum _{i=1}^{n}}{\sum _{j=1}^{{n_{i}}}}{u_{ij}})}^{-1/2}$ estimates the impact of S on approximation of ${w_{i}}{S_{i}}$. The estimate (6) takes care of a possible symmetry of distributions.

If, in each sum ${S_{i}}$ and ${Z_{i}}$, all the rvs are identically distributed, then we can get rid of the factor containing variances. We say that condition (ID) is satisfied if, for each $i=1,2,\dots ,N$, all rvs ${X_{ij}}$ and ${Z_{ij}}$ ($j=1,\dots ,{n_{i}}$) are iid with distributions ${F_{i}}$ and ${G_{i}}$, respectively. Observe, that if condition (ID) is satisfied, then the characteristic functions of S and Z are respectively equal to We also use notation ${u_{i}}$ instead of ${u_{ij}}$, since now ${u_{i1}}={u_{i2}}=\cdots ={u_{i{n_{i}}}}$.

How does Theorem 2.1 compare to the known results? In [4], compound Poisson-type approximations to non-negative iid rvs in each sum were considered under the additional Franken-type condition: see [8]. Similar assumptions were used in [7, 21]. Observe that Franken’s condition requires almost all probabilistic mass to be concentrated at 0 and 1. Indeed, then ${\nu _{1}^{+}}({F_{j}})<1$ and ${F_{j}}\{1\}\geqslant {\sum _{k=3}^{\infty }}k(k-2){F_{j}}\{k\}$. Meanwhile, Theorems 2.1 and 2.2 hold under much milder assumptions and, as demonstrated in the example below, can be useful even if (8) is not satisfied. Therefore, even for the case of one sum when $N=1$, our results are new.

Example. Let $N=2$, ${w_{1}}=1$, ${w_{2}}=\sqrt{2}$, and ${F_{j}}$ and ${G_{j}}$ be defined by ${F_{j}}\{0\}=0.375$, ${F_{j}}\{1\}=0.5$, ${F_{j}}\{4\}=0.125$, ${G_{j}}\{0\}=0.45$, ${G_{j}}\{1\}=0.25$, ${G_{j}}\{2\}=0.25$, ${G_{j}}\{5\}=0.05$, $(j=1,2)$. We assume that ${n_{2}}=n$ and ${n_{1}}=⌈\sqrt{n}\hspace{0.1667em}⌉$ is the smallest integer greater or equal to $\sqrt{n}$. Then ${\nu _{k}^{+}}({F_{j}})={\nu _{k}^{+}}({G_{j}})$, $k=1,2,3$, ${\beta _{4}^{+}}({F_{j}},{G_{j}})=9$, ${u_{j}}=3/8$, $(j=1,2)$. Therefore, by Theorem 2.2 In this case, Franken’s condition (8) is not satisfied, since ${\nu _{1}^{+}}({F_{j}})-{\nu _{2}^{+}}({F_{j}})-{({\nu _{1}^{+}}({F_{j}}))}^{2}<0$.

Next we apply Theorem 2.2 to the negative binomial distribution. For real $r>0$ and $0<\tilde{p}<1$, let $\xi \sim \mathrm{NB}(r,\tilde{p})$ denote the distribution with Here $\tilde{q}=1-\tilde{p}$. Note that r is not necessarily an integer.

Let ${X_{1j}}$ be concentrated on non-negative integers (${\nu _{k}^{-}}({F_{j}})=0$). We approximate ${S_{i}}$ by ${Z_{i}}\sim \mathrm{NB}({r_{i}},{p_{i}})$ with so that $\mathrm{E}{S_{i}}={r_{i}}{\tilde{q}_{i}}/{\tilde{p}_{i}}$ and $\mathrm{Var}{S_{i}}={r_{i}}{\tilde{q}_{i}}/{\tilde{p}_{i}^{2}}$. Observe that

We already mentioned that it is not always natural to assume independence of rvs. In this section, we still assume that $S={w_{1}}{S_{1}}+{w_{2}}{S_{2}}+\cdots +{w_{N}}{S_{N}}$ with mutually independent ${S_{i}}$. On the other hand, we assume that each ${S_{i}}$ has a Markov Binomial (MB) distribution, that is, ${S_{i}}$ is a sum of Markov dependent Bernoulli variables. Such a sum S has a slightly more realistic interpretation in actuarial mathematics. Assume, for example, that we have N insurance policy holders, i-th of whom can get ill during an insurance period and be paid a claim ${w_{i}}$. The health of the policy holder depends on the state of her/his health in the previous period. Therefore, we have a natural two state (healthy, ill) Markov chain. Then ${S_{i}}$ is an aggregate claim for ith insurance policy holder after ${n_{i}}$ periods, meanwhile S is an aggregate claim of all holders. Limit behavior of the MB distribution is a popular topic among mathematicians, discussed in numerous papers, see, for example, [3, 9, 11], and references therein.

Let $0,{\xi _{i1}},\dots ,{\xi _{i{n_{i}}}},\dots \hspace{0.1667em}$ , ($i=1,2,\dots ,N$) be a Markov chain with the transition probabilities The distribution of ${S_{i}}={\xi _{i1}}+\cdots +{\xi _{i{n_{i}}}}$ $({n_{i}}\in \mathbb{N})$ is called the Markov binomial distribution with parameters ${p_{i}},{q_{i}},{\overline{p}_{i}},{\overline{q}_{i}},{n_{i}}$. The definition of a MB rv slightly differs from paper to paper. We use the one from [3]. Note that the Markov chain, considered above, is not necessarily stationary. Furthermore, the distribution of ${w_{i}}{S_{i}}$ is denoted by ${H_{in}}=\mathcal{L}({w_{i}}{S_{i}})$. For approximation of ${H_{in}}$ we use the signed compound Poisson (CP) measure with matching mean and variance. Such signed CP approximations usually outperform both the normal and CP approximations, see, for example, [1, 3, 20]. Let Observe that ${\widehat{Y}_{i}}(t)+1$ is the characteristic function of the geometric distribution. Let ${Y_{i}}$ be a measure corresponding to ${\widehat{Y}_{i}}(t)$. For approximation of ${H_{in}}$ we use the signed CP measure ${D_{in}}$ The CP limit occurs when $n{\overline{q}_{i}}\to \tilde{\lambda }$, see, for example, [3]. Therefore, we assume ${\overline{q}_{i}}$ to be small, though not necessarily vanishing. Let, for some fixed integer ${k_{0}}\geqslant 2$, In principle, the first assumption in (12) can be dropped, but then exponentially vanishing remainder terms appear in all results, making them very complicated.

Lemma 4.1 is a version of Le Cam’s smoothing inequality, see Lemma 9.3 in [5] and Lemma 3 on p. 402 in [15].

Lemma 4.2 contains well-known properties of Levy’s concentration function, see, for example, Chapter 1 in [19] or Section 1.5 in [5].

Expansion in left-hand and right-hand factorial moments for Fourier–Stieltjes transforms is given in [21]. Here we need its analogue for distributions.

The first estimate in (19) is given in [2] p. 884, the second estimate in (19) is trivial. For the proof of (20), see p. 81 in [5].

Lemma 4.6 is a well-known inversion inequality for lattice distributions. Its proof can be found, for example, in [5], Lemma 5.1.

The proof is similar to the one given in [22]. Let ${A_{i}}=\exp \{{n_{i}}{\gamma _{i}}{Y_{i}}/30\}$. From Lemma 4.7, it follows that Here we have added index to ${\varTheta _{i}}$ emphasizing that they might be different for different i. As usual, we assume that the convolution ${\prod _{k=N+1}^{N}}={\prod _{k=1}^{0}}=I$. Let also denote by ${\sum _{i}^{\ast }}$ summation over all indices $\{{j_{1}},{j_{2}},\dots ,{j_{i-1}}\in \{0,1\}\}$. Taking into account Lemma 4.7 and the properties of the Kolmogorov and total variation norms given in the Introduction, we get

Both summands on the right-hand side of (31) are estimated similarly. Observe that Next we apply Lemma 4.1 with $W={Y_{i}}$ and $h=\max {w_{i}}/\pi $ and V with By Lemma 4.7 Observe that Therefore, using Lemma 4.2, we prove Next observe that by Lemma 4.7, The last estimate, (32) and the trivial inequality $1/(ab)<2/(a+b)$, valid for any $a,b\geqslant 1$, allow us to obtain The estimation of the second sum in (31) is almost identical and, therefore, omitted.  □