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

By Jurii V. Linnik, Iosif V. Ostrovskii

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F. 's of every order p, 1 < p <; 00 ,andprescribed type o, 0 <; o <; oo. f. f. 4). f. ip(t; F) is of order p and type o. 'Spectively, If p = oo, then we take { O, x~O, 1- exp { - xx In+ x}, x > 0, II. F. 'S 40 . {0, x- O. 2. REMARK. 4, one can derive fonnulas (Ramachandran [1]) relating the quantities . - lnk+ 2 M(r; cp) I , p1i=hm T-+oo Il r k = 1, 2, ... = ln+x (m = 1, 2, ... ). 's such that M(r, ip) increases more rapidly than expk r for fixed k = 1, 2, ...

F. 'l't,,(t), a ;;;;ii*· considered in Example 2 of §3. Since cp°' (t) = exp {ex [2eit ·:_ e2 it"+ 3e8it + 3e it 4 7) }, the function 'l'a(t) indeed has no zeros. f. f. f. ;;;; b, a< b), possesses an analytic continuation beyond the strip to a region in which it has zeros. · EXAMPLE 1. For w = a + ib, a, b > 0, set 00 cp p= 2 J· e-ax (1-cos bx) dx (t) = (1- it/w) (1-it/W) (1- it/a)2 I ' 1 ·-i-X2 0 J ' I 00 N (x) = -2 e-au (1-cos bu) u-1 du, O

We claim that b 1 ;;;r. b. Indeed, if b 1 < b, it is easy to see that there exists a disk ltl < b 2 , b 1 < b 2 < b, in which 1{'2 (t) may be analytically continued. , we conclude that 1{'2 (t) can be analytically continued into the strip IIm ti < b2 • But then 1{'2 (t) can be analytically continued into the half-plane Im t < b2 , contradicting the choice of the number b 1 Thus, the function 1{'2 (t) may be continued analytically into Im t < b. Since 1{'1 (t) is analytic in Im t > 0 and continuous in Im t ;;;r.

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