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Large deviations of sums of independent random variables. (English) Zbl 0641.60032
The authors derive in a simple way relatively sharp upper and lower bounds for the probability of large deviations of sums of independent random variables of the form \(\sum^{\infty}_{n=1}r_ nX_ n\), where \[ P(X_ n=1)=P(X_ n=-1)=\quad and\quad \sum^{\infty}_{n=1}r^ 2_ n<\infty. \] In the case \(r_ n=n^{-3/4}\), for example, their result implies that \[ \exp (-c_ 1V\quad 4)\leq P(\sum^{\infty}_{n=1}X_ n/n^{3/4}\geq V)\leq \exp (-c_ 2V\quad 4) \] holds for all \(V\geq 1\) with suitable positive constants \(c_ 1\) and \(c_ 2\). The authors’ result can be applied to improve estimates of P. D. T. A. Elliott, Invent. Math. 21, 319-338 (1973; Zbl 0265.10022), for the limiting distribution of values of quadratic L-series.
Reviewer: A.Hildebrand

60F10 Large deviations
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
Zbl 0265.10022
Full Text: DOI EuDML