Total variation asymptotics for sums of independent integer random variables. (English) Zbl 1018.60049

The authors derive an asymptotic expansion for the probability \(P[W_n\in A]\) on an arbitrary subset \(A\in \mathbb{Z}\), where \(W_n:=\sum^n_{j=1} Z_j\) is a sum of independent integer-valued random variables. After an Introduction, in Section 2 the authors establish properties of the solution of the Stein equation for certain signed measures on the integers. Section 3 deals with the simplest case, that of approximation by a centered Poisson distribution. For a large class of integer-valued random variables, the centered Poisson approximation already extends both the classical Poisson and the normal approximation. In Section 4 they move on to second-order expansions, concentrating on the case when the approximations are probability measures, as is relevant to Kolmogorov’s problem.
The main asymptotic expansion is proved in Section 5. Firstly it is supposed that the random variables \(Z_i\) are (integrably) “centered” in such a way that all the partial sums \(S_{rs} := \sum^s_{i=r}Z_i\) have second factorial cumulant satisfying \[ |k_2(S_{rs})|= |\text{Var} S_{rs}-ES_{rs}|\leq 1,\quad 1\leq r,s \leq n,\tag{1} \] as is clearly possible. Then, the following assumptions are made \[ \sigma^2_i :=\text{Var }Z_i\geq 2\quad\text{and}\quad 1 - d_{TV}(\ell(Z_i),\ell(Z_i+1))\geq \nu_*\tag{2} \] for all \(1\leq i\leq n\), where \(0 <\nu_*\leq 1/2\). The main result is given (Theorem 5): If \(Z_i\), \(1 \leq i\leq n\), are independent integer-valued random variables with finite \((r+1)\)th moments, which also satisfy (1) and (2), and if \(W = \sum^n_{i=1} Z_i\), then \(\|\ell(W)-\nu_r\|\leq 3Kn^{-r/2} (1+\tau_4)\), where \(K:=K (r, \tau_0, \tau_1,\tau_2,\tau_3,\nu_*) := \max \{C_1+C_2+C_3 + C_4, C_7,C_6(n_0)^{r/2}\}\). It is a very good and interesting paper.


60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
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[1] ARAK, T. V. (1981). On the convergence rate in Kolmogorov’s uniform limit theorem. Theory Probab. Appl. 26 219-239, 437-451. · Zbl 0495.60037
[2] ARAK, T. V. and ZAITSEV, A. Yu. (1988). Uniform limit theorems for sums of independent random variables. Proc. Steklov Inst. Math. 174. · Zbl 0659.60070
[3] BARBOUR, A. D., CHEN, L. H. Y. and LOH, W.-L. (1992). Compound Poisson approximation for nonnegative random variables via Stein’s method. Ann. Probab. 20 1843-1866. · Zbl 0765.60015
[4] BARBOUR, A. D., HOLST, L. and JANSON, S. (1992). Poisson Approximation. Oxford Univ. Press. · Zbl 0765.60015
[5] BARBOUR, A. D. and UTEV, S. (1999). Compound Poisson approximation in total variation. Stochastic Process. Appl. 82 89-125. · Zbl 0991.62008
[6] BARBOUR, A. D. and XIA, A. (1999). Poisson perturbations. ESAIM Probab. Statist. 3 131-150. · Zbl 0949.62015
[7] BOROVKOV, K. A. and PFEIFER, D. (1996). On improvements of the order of approximation in the Poisson limit theorem. J. Appl. Probab. 33 146-155. JSTOR: · Zbl 0852.60025
[8] CEKANAVI CIUS, V. (1997). Asymptotic expansions in the exponent: a compound Poisson approach. Adv. in Appl. Probab. 29 374-387. JSTOR: · Zbl 0895.60029
[9] CEKANAVI CIUS, V. (1998). Poisson approximations for sequences of random variables. Statist. Probab. Lett. 39 101-107. · Zbl 0917.60025
[10] CEKANAVI CIUS, V. and VAITKUS, P. (1999). A centred Poisson approximation via Stein’s method. Preprint 99-22, Dept. Math. Informatics, Vilnius Univ.
[11] CEKANAVI CIUS, V. and MIKALAUSKAS, M. (1999). Signed Poisson approximations for Markov chains. Stochastic Process. Appl. 82 205-227. · Zbl 0997.60073
[12] CHIHARA, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. · Zbl 0389.33008
[13] HIPP, C. (1986). Improved approximations for the aggregate claims distribution in the individual model. Astin Bull. 16 89-100.
[14] IBRAGIMOV, I. A. and LINNIK, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Grøningen. · Zbl 0219.60027
[15] KENDALL, M. G. and STUART, A. (1963). The Advanced Theory of Statistics 1, 2nd ed. Griffin, London.
[16] KRUOPIS, J. (1986). Precision of approximations of the generalized binomial distribution by convolutions of Poisson measures. Lithuanian Math. J. 26 37-49. · Zbl 0631.60019
[17] MESHALKIN, L. D. (1961). On the approximation of distributions of sums by infinitely divisible laws. Teor. Veroyatnost. i Primenen. 6 257-275. · Zbl 0099.13201
[18] PETROV, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin. · Zbl 0322.60043
[19] PETROV, V. V. (1995). Limit Theorems of Probability Theory. Oxford Univ. Press. · Zbl 0826.60001
[20] PIPIRAS, V. (1970). Asymptotic expansions for distribution functions of sums of independent lattice random variables. Litovsk. Mat. Sb. 10 517-536 (in Russian). · Zbl 0234.60023
[21] PRESMAN, E. L. (1983). Approximation of binomial distributions by infinitely divisible ones. Theory Probab. Appl. 28 393-403. · Zbl 0513.60023
[22] ROSENTHAL, H. P. (1970). On the subspaces of Lp (p &gt; 2) spanned by sequences of independent random variables. Israel J. Math. 8 273-303. · Zbl 0213.19303
[23] TSAREGRADSKII, I. P. (1958). On uniform approximation of the binomial distribution by infinitely divisible distributions. Theory Probab. Appl. 3 434-438. · Zbl 0100.14101
[24] ZAITSEV, A. Yu. (1991). An example of a distribution whose set of n-fold convolutions is uniformly separated from the set of infinitely divisible laws in the sense of the variation distance. Theory Probab. Appl. 36 419-425. · Zbl 0774.60020
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