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Sharp large deviation estimates for Gaussian extrema

José M. Zapata

TL;DR

The paper derives sharp large-deviation asymptotics for the maximum of i.i.d. standard normal samples by normalizing the maximum with Fisher–Tippett constants and a logarithmic scale. It proves a full LDP for the rescaled maximum $Z_n$ over all Borel right-tail sets, with rate function $I(x)=x+x^2/4$, implying tail probabilities decay as $n^{-I_A}$. This provides more accurate right-tail approximations than the classical Gumbel limit, with practical implications for risk assessment and actuarial applications. The work contrasts with previous Gaussian-extrema results by delivering a tail-focused, all-Borel-set large-deviation framework.

Abstract

We establish sharp large-deviation asymptotic estimates for the maximum order statistic of i.i.d.\ standard normal random variables on all Borel subsets of the positive real line. This result yields more accurate tail approximations than the classical Gumbel limit.

Sharp large deviation estimates for Gaussian extrema

TL;DR

The paper derives sharp large-deviation asymptotics for the maximum of i.i.d. standard normal samples by normalizing the maximum with Fisher–Tippett constants and a logarithmic scale. It proves a full LDP for the rescaled maximum over all Borel right-tail sets, with rate function , implying tail probabilities decay as . This provides more accurate right-tail approximations than the classical Gumbel limit, with practical implications for risk assessment and actuarial applications. The work contrasts with previous Gaussian-extrema results by delivering a tail-focused, all-Borel-set large-deviation framework.

Abstract

We establish sharp large-deviation asymptotic estimates for the maximum order statistic of i.i.d.\ standard normal random variables on all Borel subsets of the positive real line. This result yields more accurate tail approximations than the classical Gumbel limit.

Paper Structure

This paper contains 2 sections, 4 theorems, 33 equations, 1 figure.

Table of Contents

  1. Main result
  2. Proof of the main result

Key Result

Theorem 1

Let $(Z_n)_{n\in\mathbb{N}}$ be defined by eq:normalized. Then, for every Borel set $A\subset [0,\infty)$,

Figures (1)

  • Figure 1: Comparison of $\mathbb{P}(Z_n > x)$ on a logarithmic scale: true probability (black), standard Gumbel approximation (blue), and large-deviation approximation (orange). For large values of $x$, the true probability becomes numerically indistinguishable from zero and its logarithm drops to $-\infty$. The large-deviation approximation accurately captures the extreme tail behavior, while the Gumbel approximation is only reliable for moderate deviations.

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 5