Is the Brownian bridge a good noise model on the boundary of a circle?

TL;DR

This paper evaluates the suitability of the Brownian bridge as a noise model on a circle's boundary, examining process classes, their Fourier expansions, and properties of parameter estimation.

Contribution

It introduces conditions for good noise models on a circle, analyzes process classes, and studies the asymptotic behavior of maximum likelihood estimates.

Findings

Identifies classes of processes fitting the noise model conditions

Provides Fourier series expansions for these processes

Proves asymptotic properties of parameter estimates

Abstract

In this paper we study periodical stochastic processes, and we define the conditions that are needed by a model to be a good noise model on the circumference. The classes of processes that fit the required conditions are studied together with their expansion in random Fourier series in order to provide results about their path regularity. Finally, we discuss a simple and flexible parametric model with prescribed regularity that is used in applications, and we prove the asymptotic properties of the maximum likelihood estimates of model parameters.