Journal of the
Korean Mathematical Society JKMS

Let $\{X,X_n;n\geq1\}$ be a $\beta$-mixing sequence of identical non-negative random variables with super-heavy tailed distributions and $S_n=X_1+X_2+\cdots+X_n$. For $\varepsilon>0$, $b>1$ and appropriate values of $x$, we obtain the logarithmic asymptotics behaviors for the tail probabilities $\mathbb{P}(S_n>e^{\varepsilon n^{x}})$ and $\mathbb{P}(S_n>e^{\varepsilon b^{n}})$. Moreover, our results are applied to the log-Pareto distribution and the distribution for the super-Petersburg game.

Keywords: Large deviations, super-heavy tailed, $beta$-mixing sequence

MSC numbers: Primary 60F10

Supported by: This work is supported by National Natural Science Foundation of China (NSFC-11971154).