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

TL;DR

This paper derives an explicit formula for inverting infinite hermitian positive definite Toeplitz matrices using the Szeg"o function, with applications to fractional Brownian motion and other Gaussian processes.

Contribution

It provides the first explicit general formula for inverting infinite Toeplitz matrices, linking it to the spectral density's Szeg"o function.

Findings

Explicit inverse formula in terms of Szeg"o function

Application to fractional Brownian motion

Examples for m-diagonal 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.