zbMATH — the first resource for mathematics

On large deviations of sums of independent random variables. (English) Zbl 1125.60024
Summary: Extensions of some limit theorems are proved for tail probabilities of sums of independent identically distributed random variables satisfying the one-sided or two-sided Cramér’s condition. The large deviation \(x\)-region under consideration is broader than in the classical Cramér’s theorem, and the estimate of the remainder is uniform with respect to \(x\). The corresponding asymptotic expansion with arbitrarily many summands is also obtained.

60F10 Large deviations
60G50 Sums of independent random variables; random walks
62E20 Asymptotic distribution theory in statistics
Full Text: DOI
[1] Gnedenko B. V., Limit Distributions for Sums of Independent Random Variables., 2. ed. (1968)
[2] DOI: 10.1007/BF00534343 · Zbl 0416.60030 · doi:10.1007/BF00534343
[3] Jensen J. L., Saddlepoint Approximations (1995) · Zbl 1274.62008
[4] DOI: 10.1137/1110033 · Zbl 0235.60028 · doi:10.1137/1110033
[5] Petrov V. V., Sums of Independent Random Variables (1975) · Zbl 0322.60043
[6] Petrov V. V., Limit Theorems of Probability Theory (1995)
[7] Weber N. C., Theor. Stochastic Process. 3 pp 468– (1997)
[8] DOI: 10.1007/s00440-005-0494-8 · Zbl 1122.62009 · doi:10.1007/s00440-005-0494-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.