The converse of the generalized Hartmann-Wintner theorem. (Russian, English) Zbl 1085.60020

Dil’man, S. V.

Vestn. Mosk. Univ., Ser. I 2003, No. 6, 56-58 (2003); translation in Mosc. Univ. Math. Bull. 58, No. 6, 37-39 (2003).

The author obtains an analogue of the Strassen theorem [V. Strassen, Z. Wahrscheinlichkeitstheorie Verw. Geb. 4, 265–268 (1966; Zbl 0141.16501)] as follows. Let \(\{X\}_{n=1}^\infty\) be a sequence of independent similarly distributed random values and the numerical sequence \(\{\varphi(n)\}_{n=1}^\infty\) \((0<\varphi(n)\nearrow\infty)\) satisfies the condition \[ \liminf_{n\to\infty}\frac{\varphi(n)}{\sqrt{2\text{LL}n}} = 1,\quad \text{LL}n = \log\log\,n. \] Then the correlations \[ \limsup_{n\to\infty}\frac{S(n)}{\sqrt{n}\varphi(n)} = 1\quad\text{almost sure},\quad \liminf_{n\to\infty}\frac{S(n)}{\sqrt{n}\varphi(n)} = -1\quad\text{almost sure} \] imply that \( \mathbf E X_1 = 0\), \(\mathbf E X_1^2 = 1. \)

Reviewer: A. A. Martynyuk (Kyïv)

MSC:

60F17	Functional limit theorems; invariance principles
60F05	Central limit and other weak theorems