A certain generalization of the law of the iterated logarithm

Abstract

Let X₁ ,..., X_n be independent random variables and let$S_n = \sum\limits_{i = 1}^n {X_i }$. For the sequence of random variables

$$T_n = \sum\limits_1^p {(S_{t_j } - S_{t_{j - 1} } )} ,$$

where t₀=0<t₁<...<t_p=n, p⩾1, under certain conditions on t_i,$i = \overline {1,n}$, one proves a series of general theorems of the type of the iterated logarithm laws.