Asymptotic Expansions for Distributions of Sums of Independent Random Vectors
We consider the asymptotic behavior of the convolution \({P}^{{_\ast}n}(\sqrt{n}A)\) of a k-dimensional probability distribution P(A) as \(n \rightarrow \infty \) for A from the σ-algebra \({\mathfrak{M}}^{k}\) of Borel subsets of Euclidian space R k or f
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p Abstract We consider the asymptotic behavior of the convolution P n . nA/ of a k-dimensional probability distribution P .A/ as n ! 1 for A from the algebra Mk of Borel subsets of Euclidian space Rk or from its subclasses (often appearing in mathematical statistics). We will deal with two questions: construction of asymptotic expansions and estimating the remainder terms by using necessary and sufficient p conditions. The most widely and deeply investigated cases are those p where P n . nA/ are approximated by the k-dimensional normal laws ˚ n .A n/ or by the accompanying ones en.P E0 / . In this and other papers, estimating the remainder terms, we extensively use the method developed in the candidate thesis of Yu. V. Prokhorov (Limit theorems for sums of independent random variables. Candidate Thesis, Moscow, 1952) (adviser A. N. Kolmogorov) and there obtained necessary and sufficient conditions (see also Prokhorov (Dokl Akad Nauk SSSR 83(6):797–800 (1952) (in Russian); 105:645–647, 1955 (in Russian)). Keywords Probability distributions in Rk • Convolutions • Bergstr¨om identity • Appell polynomials • Chebyshev–Cramer asymptotic expansion
Mathematics Subject Classification (2010): 60F05, 60F99
A. Bikelis () Faculty of Informatics, Vytautas Magnus University, Vileikos st. 8, 44404 Kaunas, Lithuania e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 6, © Springer-Verlag Berlin Heidelberg 2013
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1 Introduction We first present three theorems from the thesis of Yu. V. Prokhorov [10]. Let 1 ; 2 ; : : : ; n ; : : : be a sequence of independent identically distributed random variables with distribution function F .x/ D P f1 < xg. Theorem P4. Let F .x/ satisfy one of the following two conditions: 1. F .x/ is a discrete distribution function; 2. There exists an integer n0 such that F n0 .x/ has an absolutely continuous component. Then there exists a sequence fGn .x/g of infinitely divisible distribution functions such that kF n .x/ Gn .x/k ! 0 as n ! 1; where k k stands for the total variation. Theorem P5. In order that kF n .xBn C An / G.x/k ! 0;
n ! 1;
for some appropriately chosen constants Bn > 0 and An and a stable distribution function G.x/, the following conditions are necessary and sufficient: 1. F n .xBn C An / ! G.x/; 2. There exists n0 such that
n ! 1; Z
1
1
where pn0 .x/ D
x 2 R1 ;
pn0 .x/ dx > 0;
d F no . dx .x/
Theorem P6. Suppose that 1 takes only the values m D 0; ˙1; : : : and that the stable distribution function G.x/ has a density g.x/. Then ˇˇ X ˇˇ ˇP f1 C C n D mg 1 g m An ˇ ! 1 ˇ ˇ Bn Bn m if and only if the following two conditions are satisfied: 1. F n .xBn C An / ! G.x/; n ! 1; x 2 R1 ; 2. The maximal step of the distribution of 1 equals 1. In the case where G.x/ D ˚.x/ is the standard Gaussian distribution function, the following statement is proved.
Asymptotic Expansions for Distribu
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