Relative Stability for Strictly Stationary Sequences

Abstract

For a nonnegative strictly stationary random sequence satisfying the``minimal'' dependence condition necessary and sufficient conditions for the relative stability are found. As an application the well-known Khinchine stability result for i.i.d. random variables is proved for uniformly strong mixing sequences. AcademicPress AMS 1991 subject classification: 60F05; 60G10.

On Limit Theorems for Continued Fractions

“…Furthermore, we no longer have to assume that condition ( * ) on page 56 of [32] is satisfied. By Theorem 3 and the results in [35] we get that functionals of digits in continued fraction expansion satisfy the Raikov principle (Cf. Twierdzenie 4, §28, Ch.…”

“…[7]). Since n ln 2P[ On the other hand, by Theorem 4 in [35], the convergence in probability cannot be replaced by the almost sure one.…”

Marcinkiewicz laws with infinite moments

“…For positive random variables {X k } with the canonical normalization and r = 1 we may drop restrictions on ϕ 1 (cf. [61], p. 247). Theorem 1 and Theorem 2 yield in this case the following stability result (cf.…”

Marcinkiewicz laws with infinite moments

“…Theorem 1 and Theorem 2 yield in this case the following stability result (cf. [35], [19], [54], [45] and [61], [62]). …”

“…In view of the proof of Theorem 3 in [61] it suffices to prove that (iii) entails (i). The condition (iii) and (31) give…”