Asymptotic behavior of the partial sum of the series of large deviations probabilities

I. V. Hrusceva

Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 225-236

Résumé

Let $\{X_k\}_{k=1}^\infty$ be a sequence of independent symmetric random variables with a characteristic functions $f_k(t)$, $S_n=\sum_{k=1}^n X_k$. The asymptotic behavior of the sum $\sum_{n=1}^N\Prob\{|S_n|>n\varepsilon\}$ is investigated (for an arbitrary $\varepsilon>0$)) in the asumption that $f_k(t)$ belongs to the domain of attraction of the stable law with the index $\alpha$ ($0\alpha\leq2$).