General laws of precise asymptotics for sums of random variables (Q2898901)

Let \(X_1, X_2,\dots\) be i.i.d. random variables with \(EX_1=0\) and \(0 < EX_1^2 < \infty\), and set \(S_{n}=X_1+ \dots +X_{n}\). Let \(0\leq \delta <1\), and let h be a positive, strictly increasing and differentiable function on \([n_0,\infty[\), satisfying also three more regularity conditions. The author obtains two general precise asymptotics describing the behaviour of the series NEWLINE\[NEWLINE\sum_{n\geq n_0}h^{\delta}(n)h'(n)P(|S_{n}|\geq \varepsilon \sqrt(nh(n)))NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sum_{n\geq n_0}h^{\delta}(n)h'(n)P(\max_{1\leq k\leq n}|S_{k}|\geq \varepsilon \sqrt(nh(n)))NEWLINE\]NEWLINE as \(\varepsilon \searrow 0\).