Asymptotic Probabilities of Attaining the Maximum in Heterogeneous Gaussian Samples

TL;DR

This paper analyzes the asymptotic probabilities of maximum attainment in heterogeneous Gaussian samples, identifying precise conditions for non-degenerate limits and extending results to multiple groups.

Contribution

It provides a complete asymptotic classification of maximum comparison probabilities in heterogeneous Gaussian samples, including a generalized integral representation for multiple groups.

Findings

Non-degenerate limit exists under specific growth conditions of sample sizes.

Outside the critical regime, the probability converges to 0 or 1.

Extended analysis to multiple Gaussian groups with integral representations.

Abstract

We study asymptotic probabilities of attaining the maximum in heterogeneous Gaussian samples. In the two-group setting, the first sample has variance $1$ and size $n_1$, while the second has variance $\sigma^2>1$ and size $n_2$. We investigate the probability that the maximum of the standard-variance group exceeds that of the high-variance group. Using the classical extreme-value normalization for Gaussian maxima together with a second-order comparison of the centering terms, we show that this probability admits a non-degenerate limit if and only if $n_1\sim C n_2^{\sigma^2}(\log n_2)^{-(\sigma^2-1)/2}$ as $n_1,n_2\to\infty$ for some $C\in(0,\infty)$. In that regime, the limit admits an integral representation. Outside the critical regime, the comparison necessarily degenerates to $0$ or $1$. We then extend the analysis to finitely many independent Gaussian groups and obtain a generalized integral representation for the limiting winning probabilities. The results provide a complete asymptotic classification for this maximum-comparison problem