Small deviations of modified sums of independent random variables

L. V. Rozovskii

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 13, Tome 361 (2008), pp. 109-122

Résumé

Let $S_n=X_1+\dots+X_n$, $n\ge1$, $S_0=0$, where $X_1,X_2,\dots$ are independent identically distributed random variables such that the distributions of $S_n/B_n$ converge weakly to nondegenerate distribution $F_\alpha$ as $n\to\infty$ for some positive $B_n$. We study the asymptotic behavior of sums such as $$ \sum_{n\ge1}f_n\,\mathbf P\Bigl(\frac1{B_n}R^*_n\le\frac r{\phi_n}\Bigr),\qquad r\nearrow\infty, $$ where $$ R^*_n=\max_{0\le k\le n}(S_k+d(k/n)\,S_n)-\min_{0\le k\le n}(S_k+d(k/n)\,S_n), $$ a function $d(t)$ is continuous on $[0,1]$ and has a power decrease at zero point $$ f_n\ge0,\qquad\sum_{n\ge1}f_n=\infty,\qquad\phi_n\nearrow\infty. $$ Bibl. – 13 titles.