Precise asymptotics for a series of T. L. Lai

Abstract:

Let $X,~X_{1}, X_{2},...$ be i.i.d. random variables with $EX=0$, and set $S_{n}=X_{1}+...+X_{n}$. We prove that, for $1<p<3/2,$ \begin{equation*} \lim _{\varepsilon \searrow \sigma \sqrt {2p-2}}\sqrt {\varepsilon ^{2}-\sigma ^{2}(2p-2)}\sum _{n\geq 2}n^{p-2}P(|S_{n}|\geq \varepsilon \sqrt {n\log n} )=\sigma \sqrt {\frac {2}{p-1}}, \end{equation*} under the assumption that $EX^{2}=\sigma ^{2}$ and $E[\left \vert X\right \vert ^{2p}(\log ^{+}\left \vert X\right \vert )^{-p}]<\infty .$ Necessary and sufficient conditions for the convergence of the sum above were established by Lai (1974).