Can the tail for maximum of continuous random field be significantly more heavy than maximum of tails?

TL;DR

This paper constructs a continuous random process with light finite-dimensional tails but a heavy tail for its maximum, using advanced functional space embeddings to compare tail behaviors.

Contribution

It introduces a novel example demonstrating the discrepancy between finite-dimensional and maximum distribution tails in continuous processes, utilizing Orlicz and Grand Lebesgue Spaces.

Findings

Finite-dimensional tails can be light while maximum tail is heavy.

Embedding results in Orlicz and GLS are effective for tail comparison.

The example challenges assumptions about tail behavior in stochastic processes.

Abstract

We construct an example of a continuous centered random process with light tails of finite-dimensional distribution but with (relatively) heavy tail of maximum distribution. The apparatus for tails comparison are embedding results for Orlicz and Grand Lebesgue Spaces (GLS).