Sharp large deviation results for sums of independent random variables

Abstract

We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein's condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds improve the inequalities of Bennett and Hoeffding by adding missing factors in the spirit of Talagrand (1995). We also complete Talagrand's inequality by giving a lower bound of the same form, leading to an equality. As a consequence, we obtain large deviation expansions similar to those of Cramér (1938), Bahadur-Rao (1960) and Sakhanenko (1991). We also show that our bound can be used to improve a recent inequality of Pinelis (2014). KeywordsBernstein's inequality, sharp large deviations, Cramér large deviations, expansion of BahadurRao, sums of independent random variables, Bennett's inequality, Hoeffding's inequality MSC(2010) 60F10, 60F05, 60E15, 60G50 Citation: Fan X Q, Grama I, Liu Q S. Sharp large deviation results for sums of independent random variables.

Non-uniform Berry–Esseen bounds for martingales with applications to statistical estimation

“…Let (ξ i , F i ) i=0,...,n be a sequence of martingale differences satisfying condition (A1) and S = (S k , F k ) k=0,...,n be the corresponding martingale defined by (14). For any real number λ with |λ| < ǫ −1 , define the exponential multiplicative martingale Z(λ) = (Z k (λ), F k ) k=0,...,n , where…”

Sharp large deviations for sums of bounded from above random variables

“…Such type large deviations have attracted a lot of attentions. We refer to Bercu and Rouault [4], Joutard [11], Fan, Grama and Liu [7] for more such type results. We are in position to prove Theorem 2.2.…”

Approximation Methods in Probability Theory

“…Further improvements are obtained in [81]. For more advanced Hoeffding's type moment inequalities, see [18,60,107]. For Jensen's type inequalities, see [76] and the references therein.…”