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By Dukk V.

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2 Since PfXs;n Xk;n Xm;n g ¼ 1 for any s < k < m, one has the evident inequality  r E Xk;n  r Á À r  E Xs;n  þXm;n  : Hence, if E|Xk,n|r = ∞, then at least one of equalities E|Xs,n|r = ∞ or E|Xm,n|r = ∞ is valid. It means that if E|Xk,n|r = ∞, then at least one of the following relations hold:  r  r  r EX1;n  ¼ E X2;n  ¼ . . ¼ E XkÀ1;n  ¼ 1 or  r  r  r EXkþ1;n  ¼ EXkþ2;n  ¼ . . 1 Let X’s have the Cauchy distribution with the density function f ð xÞ ¼ 1 : pð1 þ x2 Þ Show that for any r = 1, 2,.

F. F have positive derivative F′ for all x > x0. If the following relation is valid for some α > 0: xF 0 ðxÞ=ð1 À F ð xÞÞ ! a; ð2:4:7Þ as x → ∞, then F 2 D(H2,α). The centering, an, and normalizing, bn, constants can be taken to satisfy relations an = 0 and F(bn) = 1 − 1/n. f. F have positive derivative F′ for x in some interval (x1, x0) and F′(x) = 0 for x > x0. If ðx0 À xÞF 0 ðxÞ=ð1 À F ð xÞÞ ! a; x ! x0 ; ð2:4:8Þ then F 2 D(H1,α). The centering, an, and normalizing, bn, constants can be taken to satisfy relations an = x0 and F(x0 − bn) = 1 − 1/n.

One more definition—the definition of the so-called inter-record times DðnÞ; n ¼ 1; 2; . ; is closely connected with the record times. 2 Definitions of Record Values and Record Times 47 Dð1Þ ¼ Lð1Þ ¼ 1; DðnÞ ¼ LðnÞ À Lðn À 1Þ; n ¼ 2; 3; . : Indeed, one immediately has the next equality: LðnÞ ¼ Dð1Þ þ Dð2Þ þ Á Á Á þ DðnÞ; n ¼ 1; 2; . : Analogously it is possible to define inter-record times for lower records. The kth records are a natural extension of the classical records. The kth record times Lðn; kÞ and the kth record values X ðn; kÞ for any k ¼ 1; 2; .

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