The persistence exponents of Gaussian random fields connected by the Lamperti transform

TL;DR

This paper investigates the persistence exponents of Gaussian random fields, specifically the fractional Brownian sheet and its Lamperti transform, establishing their existence and interrelation under certain conditions.

Contribution

It proves the existence of persistence exponents for both fields and derives a relation between them, extending previous work on Brownian sheets.

Findings

Persistence exponents exist for the fields X and Y.

The relation between the exponents of X and Y is established.

Conditions for the existence of the exponents are identified.

Abstract

The (fractional) Brownian sheet is a simplest example of a Gaussian random field X whose covariance is the tensor product of a finite number (d) of nonnegative correlation functions of self-similar Gaussian processes. Let Y be the homogeneous Gaussian field obtained by applying to X the Lamperti transform, which involves the exponential change of time and the amplitude normalization to have unit variance. Under some assumptions, we prove the existence of the persistence exponents for both fields, X and Y, and find the relation between them. The exponent for X is the limit of ln1/P{X(t)<1, t in [0,T]^d}/(lnT)^d, T>>1. In terms of Y it has the form lim ln1/P{Y(t)<0,t in TG}/T^d:=Q, T>>1, where G is a suitable d-simplex and T is a similarity coefficient; G can be selected in form [0,c]^d if d=2 . The exponent Q exists for any continuous Gaussian homogeneous field Y with non-negative covariance when G is bounded region with a regular boundary. The problem of the existence and connection of exponents for X and Y was raised by Li and Shao [Ann. Probab. 32:1, 2004] and originally concerned the Brownian sheet.