On the inverse problem of fractional Brownian motion and the inverse of infinite Toeplitz matrices

Abstract

The inverse problem of fractional Brownian motion and other Gaussian processes with stationary increments involves inverting an infinite hermitian positively definite Toeplitz matrix (a matrix that has equal elements along its diagonals). The problem of inverting Toeplitz matrices is interesting on its own and has various applications in physics, signal processing, statistics, etc. A large body of literature has emerged to study this question since the seminal work of Szeg\"o on Toeplitz forms in 1920's. In this paper we obtain, for the first time, an explicit general formula for the inverse of infinite hermitian positive definite Toeplitz matrices. Our formula is explicitly given in terms of the Szeg\"o function associated to the spectral density of the matrix. These results are applied to the fractional Brownian motion and to $m$-diagonal Toeplitz matrices and we provide explicit examples.