The effect that weighted summands have on each other in approximations of S=w1S1+w2S2+⋯+wNSN is investigated. Here, Si’s are sums of integer-valued random variables, and wi denote weights, i=1,…,N. Two cases are considered: the general case of independent random variables when their closeness is ensured by the matching of factorial moments and the case when the Si has the Markov Binomial distribution. The Kolmogorov metric is used to estimate the accuracy of approximation.
Characteristic functionconcentration functionfactorial momentsKolmogorov metricMarkov Binomial distributionweighted random variables60F0560J10Introduction
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
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.
Sums of independent rvs
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, 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}\]]]>νk+(Fij)=νk+(Gij),νk−(Fij)=νk−(Gij),k=1,2,…,sβs+1+(Fij,Gij)+βs+1−(Fij,Gij)<∞.
Now we are in position to formulate the main result of this section.
Let assumptions (2)–(4) hold. Then|L(S)−L(Z)|K⩽C(N,s)maxjwjminjwj(∑i=1N∑l=1niuil)−1/2∏l=1N(1+∑k=1nlσlk2/∑k=1nlulk)×∑i=1N∑j=1ni[βs+1+(Fij,Gij)+βs+1−(Fij,Gij)](∑k=1niuik)−s/2.If, in addition, s is even andβs+2+(Fij,Gij)+βs+2−(Fij,Gij)<∞, then|L(S)−L(Z)|K⩽C(N,s)maxjwjminjwj(∑i=1N∑l=1niuil)−1/2∏l=1N(1+∑k=1nlσlk2/∑k=1nlulk)×∑i=1N∑j=1ni(∑k=1niuik)−s/2(|βs+1+(Fij,Gij)−βs+1−(Fij,Gij)|+[βs+2+(Fij,Gij)+βs+2−(Fij,Gij)+βs+1−(Fij,Gij)](∑k=1niuik)−1/2).
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.
Let the assumptions (2)–(4) and the condition (ID) hold. Then|L(S)−L(Z)|K⩽C(N,s)maxjwjminjwj(∑i=1Nniui)−1/2×∑i=1Nβs+1+(Fi,Gi)+βs+1−(Fi,Gi)nis/2−1uis/2.
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:
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>00$]]> 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
Gˆj(t)=(p˜j1−q˜jeit)rj/nj.
Let assumptions of Theorem2.2hold withX1jconcentrated on non-negative integers and letEX1j3<∞,(j=1,…,N). LetGjbe defined by (9). Then|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.Hereu˜k=1−12max(‖(I1−I)Fk‖,(rkln1p˜k)−1/2).
(ii) Letνk+(Fj)≍C,wj≍C. Then the accuracy of approximation in (10) is of the orderO((n1+⋯+nN)−1/2).
Sums of Markov Binomial rvs
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 DinDin=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,
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.
LetHin=L(wiSi)and letDinbe defined by (11),i=1,…,N. Let the conditions stated in (12) be satisfied. Then|∏i=1NHin−∏i=1NDin|K⩽C(N,k0)maxwiminwi·∑i=1Nq‾i(pi+q‾i)∑k=1Nmax(nkq‾k,1).
Let allq‾i⩾C,i=1,…,N. Then, obviously, the right-hand side of (13) is majorized byC(N,k0)maxwiminwi·1maxnk.Therefore, even in this case, the result is comparable with the Berry–Esseen theorem.
Auxiliary results
Leth>00$]]>,W∈M,W{R}=0,U∈Fand|Uˆ(t)|⩽CVˆ(t), for|t|⩽1/hand some symmetric distribution V having non-negative characteristic function. Then|WU|K⩽C∫|t|⩽1/h|Wˆ(t)Uˆ(t)t|dt+C‖W‖Q(U,h)⩽C(sup|t|⩽1/h|Wˆ(t)||t|·1h+‖W‖)Q(V,h).
Lemma 4.1 is a version of Le Cam’s smoothing inequality, see Lemma 9.3 in [5] and Lemma 3 on p. 402 in [15].
LetF∈F,h>00$]]>anda>00$]]>. ThenQ(F,h)⩽(9695)2h∫|t|⩽1/h|Fˆ(t)|dt,Q(F,h)⩽(1+(ha))Q(F,a),Q(exp{a(F−I)},h)⩽CaF{|x|>h}.h\}}}.\end{aligned}\]]]>If, in addition,Fˆ(t)⩾0, thenh∫|t|⩽1/h|Fˆ(t)|dt⩽CQ(F,h).
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.
LetF∈FZand, for somes⩾1,νs+1+(F)+νs+1−(F)<∞. ThenF=I+∑m=1sνm+(F)m!(I1−I)m+∑m=1sνm−(F)m!(I−1−I)m+νs+1+(F)+νs+1−(F)(s+1)!(I1−I)s+1Θ.
For measures, concentrated on non-negative integers, (18) is given in [5], Lemma 2.1. Observe that distribution F can be expressed as a mixture F=p+F++p−F− of distributions F+, F− concentrated on non-negative and negative integers, respectively. Then Lemma 2.1 from [5] can be applied in turn to F+ and to F− (with I−1). The remainder terms can be combined, since (I−1−I)=I−1(I−I1)=(I1−I)Θ. □
LetF,G∈FZand, for somes⩾1,νj+(F)=νj+(G),νj−(F)=νj−(G),(j=1,2,…,s). Ifβs+1+(F,G)+βs+1(F,G)<∞, thenF−G=βs+1+(F,G)+βs+1−(F,G)(s+1)!(I1−I)s+1Θ.If, in addition,βs+2+(F,G)+βs+2(F,G)<∞and s is even, thenF−G=βs+1+(F,G)−βs+1−(F,G)(s+1)!(I1−I)s+1+[βs+2+(F,G)+βs+2−(F,G)+βs+1−(F,G)](I1−I)s+2ΘC(s).
Observe that
(I1−I)s+1+(I−1−I)s+1=(I1−I)s+1−(I−1)s+1(I1−I)s+1=(I1−I)s+1I−1(I1−I)∑j=1s+1(I−1)s+1−j=(I1−I)s+2Θ(s+1).
The lemma now follows from (18). □
LetF∈FZwith meanμ(F)and varianceσ2(F), both finite. Then, for all|t|⩽π,|Fˆ(t)|⩽1−(1−‖(I1−I)F‖/2)t24π⩽exp{−(1−‖(I1−I)F‖/2)πsin2t2},|(Fˆ(t)e−itμ(F))′|⩽π2σ2(F)|sin(t/2)|.
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].
LetM∈Mbe concentrated onZ,∑k∈Z|k||M{k}|<∞. Then, for anya∈R,b>00$]]>the following inequality holds‖M‖⩽(1+bπ)1/2(12π∫−ππ(|Mˆ(t)|2+1b2|(e−itaMˆ(t))′|2)dt)1/2.
Lemma 4.6 is a well-known inversion inequality for lattice distributions. Its proof can be found, for example, in [5], Lemma 5.1.
LetHin=L(wiSi)and letDinbe defined by (11),i=1,…,N. Let conditions (12) hold. Then, fori=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.
The statements follow from Lemma 5.4, Lemma 5.1 and the relations given on pp. 1131–1132 in [3]. The estimate for e−Cini follows from the first assumption in (12) and the following simple estimate
e−Cini⩽e−Cini/2e−Ciniq‾i/2⩽C(k0)nik021+Ciniq‾1⩽C(k0)q‾imin(1,Ci)(1+niq‾i)⩽C(k0)q‾imin(1,Ci)max(niq‾i,1).
□
Proofs for sums of independent rvs
Let Fij,w (resp. Gij,w) denote the distribution of wiXij (resp. wiZij). Note that Fˆij,w(t)=Fˆij(wit). By the triangle inequality
|L(S)−L(Z)|K=|∏i=1NL(wiSi)−∏i=1NL(wiZi)|K⩽∑i=1N|(L(wiSi)−L(wiZi))∏l=1i−1L(wlSl)∏l=i+1NL(wlZl)|K.
Similarly,
L(wiSi)−L(wiZi)=∏j=1niFij,w−∏j=1niGij,w=∑j=1ni(Fij,w−Gij,w)∏k=1j−1Fik,w∏k=j+1niGik,w.
For the sake of brevity, let
Eij:=∏k=1j−1Fik,w∏k=j+1niGik,w,Ti:=∏l=1i−1L(wlSl)∏l=i+1NL(wlZl)=∏l=1i−1∏m=1nlFlm,w∏l=i+1N∏m=1nlGlm,w.
Then, combining both equations given above with Lemma 4.4 , we get
|L(S)−L(Z)|K⩽C(s)∑i=1N∑j=1ni[βs+1+(Fij,Gij)+βs+1−(Fij,Gij)]|(Iwi−I)s+1EijTi|K.
Let |t|⩽π/maxiwi. Then it follows from (19) that
|Eˆij(t)Tˆi(t)|⩽euijsin2(twi/2)/πexp{−1π∑l=1N∑m=1nlulmsin2twl2}.
Observe that euijsin2(twi/2)/π⩽e1/π=C. Next, let
L:=18π∑l=1N∑m=1nlulm[(Iwl−I)+(I−wl−I)].
It is not difficult to check, that exp{L} is a CP distribution with non-negative characteristic function. Also, by the definition of exponential measure, exp{−L}, which can be called the inverse to exp{L}, is a signed measure with finite variation. We have
|(Iwi−I)s+1EijTi|K=|(Iwi−I)s+1EijTiexp{−L}exp{L}|K.
Next step is similar to the definition of mod-Poisson convergence. We apply Lemma 4.1 with h=maxwi/π and U1=exp{L} and W1=(Iwi−I)s+1EijTiexp{−L}. By Lemma 4.2,
Q(exp{L},h)⩽Cmaxwiminwi·Q(exp{L},minwi/2)⩽Cmaxwiminwi(∑l=1N∑m=1nlulm)−1/2.
From (22) and (23), it follows that
|Wˆ1(t)t|·1h⩽C(s)|sin(twi/2)|s+1h|t|exp{−12π∑l=1N∑m=1nlulmsin2twl2}⩽C(s)wih|sin(twi/2)|sexp{−12π∑m=1niuimsin2(twi/2)}⩽C(s)(∑m=1niuim)−s/2.
It remains to estimate ‖W1‖. Let
Φlm,w:=Flm,wexp{18πulm[(Iwl−I)+(I−wl−I)]},Ψlm,w:=Glm,wexp{18πulm[(Iwl−I)+(I−wl−I)]}
Then by the properties of the total variation norm,
‖W1‖⩽‖exp{18uij[(Iwi−I)+(I−wi−I)]}‖×‖(Iwi−I)s+1∏k=1j−1Φik,w∏k=j+1niΨik,w‖×∏l=1i−1‖∏m=1nlΦlm,w‖∏l=i+1N‖∏m=1nlΨlm,w‖.
The first norm in (27) is bounded by exp{18uij[‖Iwi−I‖+‖I−wi−I‖]}⩽exp{1/2}. The total variation norm is invariant with respect to scale. Therefore, without loss of generality, we can switch to wl=1. In this case, we use the notations Φik,Ψik. Then, again employing the inverse CP measures, we get
‖(Iwi−I)s+1∏k=1j−1Φik,w∏k=j+1niΨik,w‖=‖(I1−I)s+1∏k=1j−1Φik∏k=j+1niΨik‖=‖(I1−I)s+1∏k=1j−1Φik∏k=j+1niΨikexp{uij(I1−I)}exp{uij(I−I1)}‖⩽e2‖(I1−I)s+1exp{uij(I1−I)}∏k=1j−1Φik∏k=j+1niΨik‖.
We apply Lemma 4.6 with a=uij+∑k≠iniμik, b=1, where μik=ν1+(Fik)+ν1−(Fik) is the mean of Fik and, due to assumption (3), of Gik. Let
Δˆ(t):=(eit−1)s+1exp{uij(eit−1−it)}∏k=1j−1Φˆik(t)e−itμik∏k=j+1niΨˆike−itμik.
It follows from (19) that
|Δ(t)|⩽C(s)|sin(t/2)|s+1exp{−12π∑m=1niuimsin2(t/2)}⩽C(s)(∑m=1niuim)−s/2.
For the estimation of |Δ′(t)|, observe that by (19) and (20)
|(Φˆik(t)e−itμik)′|⩽|Fˆik(t)e−itμikuikπsin(t/2)e(uik/2π)sin2(t/2)|+|(Fˆik(t)e−itμik)′e(uik/2π)sin2(t/2)|⩽C(s)(uik+σik2)|sin(t/2)|⩽C(s)(uik+σik2)|sin(t/2)|exp{−uikπsin2(t/2)}e1/π.
The same bound holds for |(Ψˆik(t)exp{−itμik})′|. The direct calculation shows that
|((eit−1)s+1exp{uij(eit−1−it)})′|⩽C(s)|sin(t/2)|sexp{−1πuijsin2(t/2)}.
Taking into account of the previous two estimates, it is not difficult to prove that
|Δ′(t)|⩽C(s)|sin(t/2)|sexp{−1π∑k=1niuiksin2(t/2)}×(1+sin2(t/2)∑k=1,k≠jni(uik+σik2))⩽C(s)(∑k=1niuik)−s/2(1+∑k=1niσik2/∑k=1niuik).
From Lemma 4.6, it follows that
‖(Iwi−I)s+1∏k=1j−1Φik,w∏k=j+1niΨik,w‖⩽C(s)(∑k=1niuik)−s/2(1+∑k=1niσik2/∑k=1niuik).
The remaining two norms in (27) can be estimated similarly:
‖∏m=1nlΦlm,w‖,‖∏m=1nlΨlm,w‖⩽C(1+∑m=1nlσlm2/∑m=1nlulm).
Substituting (28), (29) into (27), we obtain
‖W1‖⩽C(N,s)(∑m=1niuim)−s/2∏l=1N(1+∑k=1nlσlk2/∑k=1nlulk).
Combining (30) with (25), (26) and (24), we get
|(Iwi−I)s+1EijTi|K⩽C(N,s)maxjwjminjwj(∑i=1N∑k=1niuik)−1/2×(∑m=1niuim)−s/2∏l=1N(1+∑k=1nlσlk2/∑k=1nlulk).
Substituting the last estimate into (21) we complete the proof of (5). The proof of (6) is very similar and, therefore, omitted. □
We outline only the differences from the proof of Theorem 2.1. No use of convolution with the inverse Poisson measure is required, since we have powers of Fini, which can be used for Levy’s concentration function. Let ⌊a⌋ denote an integer part of a and let a(k):=⌊(k−1)/2⌋, b(k):=⌊(ni−k)/2⌋. Then, as in the proof of Theorem 2.1, we obtain
|L(S)−L(Z)|K⩽C(s)∑i=1N∑k=1ni(βs+1+(Fi,Gi)+βs+1−(Fi,Gi))×|(Iwi−I)s+1Fiwa(k)Giwb(k)Fiwa(k)Giwb(k)∏j=1i−1Fjwnj∏j=i+1NGjwnj|K.
Here Fiw and Giw denote the distributions of wiXij and wiZij, respectively. We can apply Lemma 4.1 to the Kolmogorov norm given above, taking W=(Iwi−I)s+1Fiwa(k)Giwb(k). The remaining distribution is used in Levy’s concentration function. The Fourier–Stieltjes transform Wˆ(t)/t is estimated exactly as in the proof of Theorem 2.1. The total variation of any distribution is equal to 1, therefore ‖W‖⩽‖Iwi−I‖⩽2 and we can avoid application of Lemma 4.6. □
As proved in [1], p. 144,
12‖Gk(I1−I)‖⩽(pkν1+(Fk)qkln1pk)−1/2.
Observe that ν1+(Fj)=ν1+(Gj) and ν2+(Fj)=ν2+(Gj). It remains to find ν3+(Gj) and apply Theorem 2.2. □
Proof of Theorem 3.1
The proof is similar to the one given in [22]. Let Ai=exp{niγiYi/30}. From Lemma 4.7, it follows that
Hin=AiΘiC+e−CiniΘiC,Din=AiΘiC,i=1,2,…,N.
Here we have added index to Θi emphasizing that they might be different for different i. As usual, we assume that the convolution ∏k=N+1N=∏k=10=I. Let also denote by ∑i∗ summation over all indices {j1,j2,…,ji−1∈{0,1}}. Taking into account Lemma 4.7 and the properties of the Kolmogorov and total variation norms given in the Introduction, we get
|∏i=1NHin−∏i=1NDin|K⩽∑i=1N|(Hin−Din)∏k=1i−1Hkn∏k=i+1NDkn|K⩽∑i=1N|(Hin−Din)∑i∗∏k=1i−1AkjkΘkC×∏k=i+1NAkΘkC∏k=1i−1e−(1−jk)nkCkΘkC|K⩽C(N)∑i=1Nq‾i(pi+q‾i)∑i∗|Yiexp{niγiYi/60}∏k=1i−1Akjk∏k=i+1NAk|K×∏k=1i−1e−(1−jk)nkCk+C∑i=1N(pi+q‾i)e−Cini×∑i∗|(Iwi−I)∏k=1i−1Akjk∏k=i+1NAk|K∏k=1i−1e−(1−jk)nkCk.
Both summands on the right-hand side of (31) are estimated similarly. Observe that
|Yiexp{niγiYi/60}∏k=1i−1Akjk∏k=i+1NAkK|=|Yiexp{niγiYi60+130∑k=1i−1jknkγkYk+130∑k=i+1NnkγkYk}|K.
Next we apply Lemma 4.1 with W=Yi and h=maxwi/π and V with
Vˆ(t)=exp{−190[∑k=1i−1jkmax(nkq‾k,1)sin2(twk/2)+∑k=iNmax(nkq‾k,1)sin2(twk/2)]}.
By Lemma 4.7|Yˆi(t)|t1h+‖Yi‖⩽C.
Observe that
|exp{niγi60Yˆi(t)+130∑k=1i−1jknkγkYˆk(t)+130∑k=i+1NγkYˆk(t)}|⩽exp{−niγisin2(twi/2)45−245∑k=1i−1jknkγksin2(twk/2)−245∑k=i+1Nnkγksin2(twk/2)}⩽exp{−190[∑k=1i−1jknkq‾ksin2(twk/2)+∑k=iNnkq‾ksin2(twk/2)]}⩽eN/90exp{−190[∑k=1i−1jk(nkq‾k+1)sin2(twk/2)+∑k=iN(nkq‾k+1)sin2(twk/2)]}⩽eN/90exp{−190[∑k=1i−1jkmax(nkq‾k,1)sin2(twk/2)+∑k=iNmax(nkq‾k,1)sin2(twk/2)]}=eN/90Vˆ(t).
Therefore, using Lemma 4.2, we prove
|Yiexp{niγiYi/60}∏k=1i−1Akjk∏k=i+1NAk|K⩽C(N)Q(V,maxiwi/h)⩽C(N)(maxwiminwi)Q(V,minwi/2)⩽C(N)(maxwiminwi)(∑k=1i−1jkmax(nkq‾k,1)+∑k=i+1Nmax(nkq‾k,1))−1/2.
Next observe that by Lemma 4.7,
|∏k=1i−1e−(1−jk)nkCk|=Cexp{−∑k=1i−1(1−jk)Cknk}⩽C(k0,N)max(1,∑k=1i−1(1−jk)max(nkq‾k,1)).
The last estimate, (32) and the trivial inequality 1/(ab)<2/(a+b), valid for any a,b⩾1, allow us to obtain
∑i=1Nq‾i(pi+q‾i)∑i∗|Yiexp{niγiYi/60}∏k=1i−1Akjk∏k=i+1NAk|K∏k=1i−1e−(1−jk)nkCk⩽C(k0,N)maxwjminwj·∑i=1Nq‾i(pi+q‾i)∑k=1Nmax(nkq‾k,1).
The estimation of the second sum in (31) is almost identical and, therefore, omitted. □
Acknowledgement
The main part of the work was accomplished during the first author’s stay at the Department of Mathematics, IIT Bombay, during January, 2018. The first author would like to thank the members of the Department for their hospitality. We are grateful to the referees for useful remarks.
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