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

Document Type

Article

Publication Date

1-1-2024

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.

Publication Title

Electronic Communications in Probability

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