Sharp large deviation estimates for Gaussian extrema

Key Points

• The aim is to present a sharp large deviation principle for the maximum order statistic of i.i.d. standard normal variables.
• Focused on i.i.d. standard normal random variables.
• Analyzed maximum order statistics on Borel subsets of the positive real line.
• Utilized large deviation techniques for mathematical proofs.
• Established a sharper approximation for the tail behavior than classical Gumbel limit.
• Demonstrated improved accuracy in tail approximations for Gaussian maxima.