Precise asymptotics for products of sums and U-statistics

Abstract

Let $\{X,X_{i},i\geq 1\}$ be a sequence of independent and identically distributed positive random variables with $E(X)=\mu >0$, $\operatorname{Var}(X)<\infty$. Put $S_{n}=\sum_{i=1}^{n}X_{i}$ and let $g(x)$ be a positive and differentiable function defined on $(0,+\infty)$ satisfying some mild conditions. We prove that, for any $s>1$, \[\lim_{\varepsilon\rightarrow0}\varepsilon^{1/s}\sum_{n=1}^{\infty}g'(n)P\Biggl\{\Biggl|\log\Biggl(\prod_{j=1}^{n}\frac{S_{j}}{j\mu}\Biggr)\Biggr|\geq\varepsilon\sqrt{n}g^{s}(n)\Biggr\}=E|N|^{1/s},\] where $N$ is a standard normal random variable. This result was also extended to product of U-statistics.