A law of the single logarithm for moving averages of random variables under exponential moment conditions (Q2714366)

A law of the single logarithm for increments of partial sums of independent, identically distributed random variables \(\{X_k\}_{k=1}^{\infty}\) is proved. It is assumed, that \(EX_1=0 \), \(E X_1^2=\sigma^2 <\infty \), and \(E \exp(tg_p(X_1))<\infty \) for all \(t\) in a neighbourhood of 0 with \(g_p(x)=\text{sgn} x \cdot |x|^{1/p}\), \(p\geq 1\). The main result states, that NEWLINE\[NEWLINE \limsup_{n\to \infty} {1\over {c_n}} \sum_{k=n+1}^{n+a_n} X_k=s\quad \text{a.s.}, NEWLINE\]NEWLINE where \(a_n=[(\log n)^p]\), \(c_n=(\log n)^{(p+1)/2}\). In the special case, when NEWLINE\[NEWLINEt_2=\sup \{t\geq 0: E \exp(tg_p(X_1)) \}=\infty,NEWLINE\]NEWLINE the limit \(s=\sqrt{2}\sigma\). In the general case \(s\) is expressed through \(\sigma\), \(t_2\) and \(g_p(x)\). It is also shown, that the moment conditions imposed are the weakest possible ones. The results proved generalize the results of \textit{A. de Acosta} [Ann. Probab. 11, 78-101 (1983; Zbl 0504.60033)].