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Let X, X
1, X
2,... be a sequence of independent and identically distributed random variables with common distribution function F. Denote by F
n
the distribution function of centered and normed sum S
n
. Let F belong to the domain of attraction of the standard normal law , that is, lim F
n
(x)= (x), as n , uniformly in x . We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx
––1 ln(x), x > r, where 2, , c > 0, and r > 1. We write the classical asymptotic expansion (in powers of n
–1/2) and then add new terms of orders n
–/2 ln
n, n
–/2 ln-1
n, etc., where 0. 相似文献
3.
Let W
n be an n × n random symmetric sparse matrix with independent identically distributed entries such that the values 1 and 0 are taken with probabilities p/n and 1-p/n, respectively; here
is independent of n. We show that the limit of the expected spectral distribution functions of W
n has a discrete part. Moreover, the set of positive probability points is dense in (- +). In particular, the points
, and 0 belong to this set. 相似文献
4.
Multidimensional stable laws G
admit a well-known Lévy–LePage series representation
where
1,
2,... are the successive times of jumps of a standard Poisson process, and X
1, X
2,... denote i.i.d. random vectors, independent of
1,
2,.... We present (asymptotically) optimal bounds for the total variation distance between a stable law and the distribution of a partial sum of the Lévy–LePage series. In the one-dimensional case similar results were obtained earlier by Bentkus, Götze, and Paulauskas. 相似文献
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