A certain generalization of the law of the iterated logarithm (Q1117569)

[For the entire collection of the original see Zbl 0626.00025.] Let \(X_ 1,...,X_ n\) be independent random variables and let \(S_ n=\sum^{n}_{i=1}X_ i\). For the sequence of random variables \(T_ n=\sum^{p}_{1}(S_{t_ j}-S_{t_{j-1}})\), where \(t_ 0=0<t_ 1<...<t_ p=n\), \(p\geq 1\), under certain conditions on \(t_ i\), \(i=\overline{1,n}\), we prove a series of general theorems of the type of the iterated logarithm laws.