Read e-book online Decomposition of Random Variables and Vectors PDF

By Jurii V. Linnik, Iosif V. Ostrovskii

ISBN-10: 0821815989

ISBN-13: 9780821815984

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Example text

5) implies the existence of nk(F). Now suppose that m k(F) exists. 6). 'D is that if the moment mk(F), k ;;;ii 1, exists, then Wp(x) = o(x-k), x - + oo. § 2. Analyticity conditions for characteristic functions Let G be a region of the complex t-plane whose closure contains some real neighborhood r of the point t = 0. We shall assume that r is either int~rior to G or on its boundary; in the latter case, we stipulate that r contains no points of accumulation of nonreal points of the boundary of G.

5) ( rextF == P- 0 1 -oo We have SUFFICIENCY. 00 J (eitx_1- 1 ~x2 ) In ip(t, F) =iPt+ 1 ~/1 dG(x), -oo 0. 3 we have lext F = -h+ (F), where h+ (F) = lim r-1 In cp (ir; F).

With lext F = - oo. Suppose now that 0 0. f. 2. II. 'S 52 C1 - > 0, Cz > 0, C = Ct-+Cz Ct Cz X =(Ct +c2) - 1 r (1-a) COS n2a. 4 that lext F is finite if and only if c 1 = 0 and 0 < a < 1. f. of a stable distribution for which lext F is finite is cp (t; F) =exp { iPt- x where Im {J It la {1- i tan n; • sign t}} , = 0, K;;;, 0 and 0

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Decomposition of Random Variables and Vectors by Jurii V. Linnik, Iosif V. Ostrovskii


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