On the probability distribution of the local times of diagonally operator-self-similar Gaussian fields with stationary increments

TL;DR

This paper investigates the distributional properties of local times of diagonally operator-self-similar Gaussian fields with stationary increments, revealing asymptotic behaviors and conditions for finiteness of related limits.

Contribution

It establishes the existence and positivity of certain asymptotic limits for the moments and tail probabilities of local times under weak conditions, extending to intersection local times.

Findings

Existence of a positive parameter λ governing asymptotic limits.

Limits of normalized moments and tail probabilities are finite or infinite under conditions.

Application to intersection local times of Gaussian fields with similar properties.

Abstract

In this paper we study the local times of vector-valued Gaussian fields that are `diagonally operator-self-similar' and whose increments are stationary. Denoting the local time of such a Gaussian field around the spatial origin and over the temporal unit hypercube by $Z$, we show that there exists $\lambda\in(0,1)$ such that under some quite weak conditions, $\lim_{n\rightarrow +\infty}\frac{\sqrt[n]{\mathbb{E}(Z^n)}}{n^\lambda}$ and $\lim_{x\rightarrow +\infty}\frac{-\log \mathbb{P}(Z>x)}{x^{\frac{1}{\lambda}}}$ both exist and are strictly positive (possibly $+\infty$). Moreover, we show that if the underlying Gaussian field is `strongly locally nondeterministic', the above limits will be finite as well. These results are then applied to establish similar statements for the intersection local times of diagonally operator-self-similar Gaussian fields with stationary increments.