Parameter estimations for the Gaussian process with drift at discrete observation

TL;DR

This paper proves the bounded growth of second moments and positive definiteness of covariance matrices for Gaussian processes with drift, establishing strong consistency and asymptotic normality of estimators under broad conditions.

Contribution

It provides rigorous proofs for the growth bounds and positive definiteness, and derives the asymptotic properties of maximum likelihood estimators for such Gaussian processes.

Findings

Second moment growth is bounded by a power function

Covariance matrix is strictly positive definite

MLEs are strongly consistent and asymptotically normal

Abstract

This paper first strictly proved that the growth of the second moment of a large class of Gaussian processes is not greater than power function and the covariance matrix is strictly positive definite. Under these two conditions, the maximum likelihood estimators of the mean and variance of such classes of drift Gaussian process have strong consistency under broader growth of t_n. At the same time, the asymptotic normality of binary random vectors and the Berry-Ess\'{e}en bound of estimators are obtained by using the Stein's method via Malliavian calculus.