On the upper limit of a random sequence and the law of the iterated logarithm (Q2722157)

Let \(\Phi_{c}\) be a set of functions \(\varphi\) such that each \(\varphi\) is positive and does not decrease on the set \(x>x_{0}\) with some \(x_{0}\) and the series \(\sum 1/(n\varphi(n))\) converges. The author proves the following result. Let \((Y_{n}; n=1,2,\dots)\) be a sequence of random variables, let \((a_{n}; n=1,2,\dots)\) be a non-decreasing sequence of positive constants which satisfy the condition \(a_{n}\to\infty,\) and let for any \(x\) from some non-degenerated interval \(1<x<1+\beta\) the inequality NEWLINE\[NEWLINEP\Bigl(\max_{1\leq k\leq n}Y_{k}\geq xa_{n}\Bigr)\leq (L/\varphi(a_{n}))NEWLINE\]NEWLINE hold true for some constant \(L,\) some function \(\varphi\in \Phi_{c},\) and for all sufficiently large \(n.\) Then \(\lim\sup(Y_{n}/a_{n})\leq 1\) almost sure.