Asymptotics of the rate function in the large deviation principle for sums of independent identically distributed random variables

Authors

Iosif Pinelis, Michigan Technological UniversityFollow

Document Type

Article

Publication Date

3-15-2024

Department

Department of Mathematical Sciences

Abstract

Let Λ∗be the rate function in the large deviation principle for the sums X1 + · · · + Xn of independent identically distributed random variables X1, X2, …. It is shown that Λ∗(x) ∼ − ln P(X1 ≥ x) (as x → ∞) if and only if ln P(X1 ≥ x) ∼ L0(x) for some concave function L0. The main ingredient of the proof is the general, explicit expression of a suitable quasi-minimizer in t ≥ 0 of the Bernstein–Chernoff upper bound e−txEetX1 on P(X1 ≥ x), which is amenable to analysis and, at the same time, is close enough to a true minimizer.

Publisher's Statement

© 2024. Publisher’s version of record: https://doi.org/10.1214/24-ECP584

Publication Title

Electronic Communications in Probability

Recommended Citation

Pinelis, I. (2024). Asymptotics of the rate function in the large deviation principle for sums of independent identically distributed random variables. Electronic Communications in Probability, 29. http://doi.org/10.1214/24-ECP584
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p2/605

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Version

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