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=w1S1+w2S2+⋯+wNSN . Here, Si 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 Si may be the sum of dependent rvs. Of course, in actuarial models, wi 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−wiSi on wiSi . Our research is motivated by the following simple example. Let us assume that Si is in some sense close to Zi , i=1,2 . Then a natural approximation to w1S1+w2S2 is w1Z1+w2Z2 . Suppose that we want to estimate the closeness of both sums in some metric d(·,·) . The standard approach which works for the majority of metrics then gives (1) d(w1S1+w2S2,w1Z1+w2Z2)⩽d(w1S1,w1Z1)+d(w2S2,w2Z2). The triangle inequality (1) is not always useful. For example, let S1 and Z1 have the same Poisson distribution with parameter n and let S2 and Z2 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 wiSi 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 Z denote the set of all integers. Let F (resp.  FZ , resp.  M ) denote the set of probability distributions (resp. distributions concentrated on integers, resp. finite signed measures) on R . Let Ia denote the distribution concentrated at real a and set I=I0 . Henceforth, the products and powers of measures are understood in the convolution sense. Further, for a measure M, we set M0=I and exp{M}=∑k=0∞Mk/k! . We denote by Mˆ(t) the Fourier–Stieltjes transform of M. The real part of Mˆ(t) is denoted by ReMˆ(t) . Observe also that exp{M(t)}ˆ=exp{Mˆ(t)} . We also use L(ξ) to denote the distribution of ξ.

The Kolmogorov (uniform) norm |M|K and the total variation norm ‖M‖ of M are defined by |M|K=supx∈R|M((−∞,x])|,‖M‖=M+{R}+M−{R}, respectively. Here M=M+−M− is the Jordan–Hahn decomposition of M. Also, for any two measures M and V, |M|K⩽‖M‖ , |MV|K⩽‖M‖·|V|K , |Mˆ(t)|⩽‖M‖ , ‖exp{M}‖⩽exp{‖M‖} . If F∈F , then |F|K=‖F‖=‖exp{F−I}‖=1 . Observe also that, if M is concentrated on integers, then M= ∑k=−∞∞M{k}Ik,Mˆ(t)= ∑k=−∞∞eitkM{k},‖M‖= ∑k=−∞∞|M{k}|.

For F∈F , h⩾0 , Lévy’s concentration function is defined by Q(F,h)=supxF{[x,x+h]}.

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 ∑j=ab=0 and ∏j=ab=1 , if b<a . The notation Θ is used for any signed measure satisfying ‖Θ‖⩽1 . The notation θ is used for any real or complex number satisfying |θ|⩽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{−λ˜n(eit−1)}fn(t) to have a limit for some sequence of Poisson parameters λ˜n , as n→∞ . Here, fn(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 Si=Xi1+Xi2+⋯+Xini , Zi=Zi1+Zi2+⋯+Zini , i=1,2,…,N . We assume that all the Xij , Zij are mutually independent and integer-valued. Observe that, in general, S=∑i=1NwiSi and Z=∑i=1NwiZi 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 Zi .

The distribution of Xij (resp. Zij ) is denoted by Fij (resp. Gij ). 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,… , and any F∈FZ , νk+(Fij)= ∑m=k∞m(m−1)⋯(m−k+1)Fij{m},νk−(Fij)= ∑m=k∞m(m−1)⋯(m−k+1)Fij{−m}. For the estimation of the remainder terms we also need the following notation: βk±(Fij,Gij)=νk±(Fij)+νk±(Gij) , σij2=max(Var(Xij),Var(Zij)) , and uij=min{1−12‖Fij(I1−I)‖;1−12‖Gij(I1−I)‖}=min{ ∑k=−∞∞min(Fij{k},Fij{k−1}); ∑k=−∞∞min(Gij{k},Gij{k−1})}. For the last equality, see (1.9) and (5.15) in [5]. Next we formulate our assumptions. For some fixed integer s⩾1 , i=1,…,N,j=1,…,ni , (2) uij>0, ∑j=1niuij⩾1,ni⩾1,wi>0, 0,\hspace{2em}{\sum \limits_{j=1}^{{n_{i}}}}{u_{ij}}\geqslant 1,\hspace{2em}{n_{i}}\geqslant 1,\hspace{2em}{w_{i}}>0,\end{aligned}\]]]> (3) νk+(Fij)=νk+(Gij),νk−(Fij)=νk−(Gij),k=1,2,…,s (4) βs+1+(Fij,Gij)+βs+1−(Fij,Gij)<∞.

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

The factor (∑i=1n∑j=1niuij)−1/2 estimates the impact of S on approximation of wiSi . The estimate (6) takes care of a possible symmetry of distributions.

If, in each sum Si and Zi , 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,…,N , all rvs Xij and Zij ( j=1,…,ni ) are iid with distributions Fi and Gi , respectively. Observe, that if condition (ID) is satisfied, then the characteristic functions of S and Z are respectively equal to ∏i=1NFˆini(wit), ∏i=1NGˆini(wit). We also use notation ui instead of uij , since now ui1=ui2=⋯=uini .

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: (8) ν1+(Fj)−(ν1+(Fj))2−ν2+(Fj)>0, 0,\]]]> 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 ν1+(Fj)<1 and Fj{1}⩾∑k=3∞k(k−2)Fj{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 , w1=1 , w2=2 , and Fj and Gj be defined by Fj{0}=0.375 , Fj{1}=0.5 , Fj{4}=0.125 , Gj{0}=0.45 , Gj{1}=0.25 , Gj{2}=0.25 , Gj{5}=0.05 , (j=1,2) . We assume that n2=n and n1=⌈n⌉ is the smallest integer greater or equal to n . Then νk+(Fj)=νk+(Gj) , k=1,2,3 , β4+(Fj,Gj)=9 , uj=3/8 , (j=1,2) . Therefore, by Theorem 2.2 |L(S)−L(Z)|K⩽Cn1+n2(1n1+1n2)=O(n−1). In this case, Franken’s condition (8) is not satisfied, since ν1+(Fj)−ν2+(Fj)−(ν1+(Fj))2<0 .

Next we apply Theorem 2.2 to the negative binomial distribution. For real r>0 0$]]> and 0<p˜<1 , let ξ∼NB(r,p˜) denote the distribution with P(ξ=k)=r+k−1kp˜rq˜k,k=0,1,…. Here q˜=1−p˜ . Note that r is not necessarily an integer.

Let X1j be concentrated on non-negative integers ( νk−(Fj)=0 ). We approximate Si by Zi∼NB(ri,pi) with ri=(ESi)2VarSi−ESi,p˜i=ESiVarSi, so that ESi=riq˜i/p˜i and VarSi=riq˜i/p˜i2 . Observe that (9) Gˆj(t)=(p˜j1−q˜jeit)rj/nj. Corollary 2.1.

Let assumptions of Theorem 2.2 hold with X1j concentrated on non-negative integers and let EX1j3<∞ , (j=1,…,N) . Let Gj be defined by (9). Then (10) |L(S)−L(Z)|K⩽Cmaxjwjminjwj( ∑i=1Nniu˜i)−1/2× ∑k=1N[ν3+(Fk)+ν1+(Fk)ν2+(Fk)+(ν1+(Fk))3+(ν2+(Fk)−(ν1+(Fk))2)2ν1+(Fk)]u˜k−1. Here u˜k=1−12max(‖(I1−I)Fk‖,(rkln1p˜k)−1/2).

We already mentioned that it is not always natural to assume independence of rvs. In this section, we still assume that S=w1S1+w2S2+⋯+wNSN with mutually independent Si . On the other hand, we assume that each Si has a Markov Binomial (MB) distribution, that is, Si 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 wi . 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 Si is an aggregate claim for ith insurance policy holder after ni 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,ξi1,…,ξini,… , ( i=1,2,…,N ) be a Markov chain with the transition probabilities P(ξik=1|ξi,k−1=1)=pi,P(ξik=0|ξi,k−1=1)=qi,P(ξi,k=1|ξi,k−1=0)=q‾i,P(ξik=0|ξi,k−1=0)=p‾i,pi+qi=q‾i+p‾i=1,pi,q‾i∈(0,1),k∈N. The distribution of Si=ξi1+⋯+ξini (ni∈N) is called the Markov binomial distribution with parameters pi,qi,p‾i,q‾i,ni . 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 wiSi is denoted by Hin=L(wiSi) . For approximation of Hin 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 γi=qiq‾iqi+q‾i,Yˆi(t)=qieiwit1−pieiwit−1. Observe that Yˆi(t)+1 is the characteristic function of the geometric distribution. Let Yi be a measure corresponding to Yˆi(t) . For approximation of Hin we use the signed CP measure Din (11) Din=exp{(γi(q‾i−pi)qi+q‾i+niγi)Yi−ni(qiq‾i2(qi+q‾i)2(pi+qiqi+q‾i)+γi22)Yi2}. The CP limit occurs when nq‾i→λ˜ , see, for example, [3]. Therefore, we assume q‾i to be small, though not necessarily vanishing. Let, for some fixed integer k0⩾2 , (12) q‾i⩾1nk0,0<pi⩽12,q‾i⩽130,wi>0,ni⩾1,i=1,…,N. \hspace{0.1667em}0,\hspace{2em}{n_{i}}\hspace{0.1667em}\geqslant \hspace{0.1667em}1,\hspace{1em}i\hspace{0.1667em}=\hspace{0.1667em}1,\dots ,N.\]]]> In principle, the first assumption in (12) can be dropped, but then exponentially vanishing remainder terms appear in all results, making them very complicated. Theorem 3.1.

Let Hin=L(wiSi) and let Din be defined by (11), i=1,…,N . Let the conditions stated in (12) be satisfied. Then (13) | ∏i=1NHin− ∏i=1NDin|K⩽C(N,k0)maxwiminwi·∑i=1Nq‾i(pi+q‾i)∑k=1Nmax(nkq‾k,1).

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. Lemma 4.7.

Let Hin=L(wiSi) and let Din be defined by (11), i=1,…,N . Let conditions (12) hold. Then, for i=1,2,…,N , Hin−Din=q‾i(pi+q‾i)Yiexp{niγiYi/60}ΘC+(pi+q‾i)(Iwi−I)ΘCe−Cini,Hin=exp{niγiYi/30}ΘC+(pi+q‾i)(Iwi−I)ΘCe−Cini,Din=exp{niγiYi/30}ΘC,e−Cini⩽C(k0)q‾imax(niq‾i,1),|Yˆi(t)|⩽4|sin(twi/2)|,ReYˆi(t)⩾−43sin2(twi/2),q‾i2⩽γi⩽q‾i.