Detecting Whether a Stochastic Process is Finitely Expressed in a Basis

TL;DR

This paper presents a hypothesis testing method to determine if a stochastic process's sample paths almost surely have a finite basis expansion, using finite, possibly noisy observations and combining classical irrationality tests with non-parametric covariance estimation.

Contribution

It introduces a novel testing scheme that guarantees finite decision errors asymptotically, enabling detection of finite versus infinite basis expansions in stochastic processes.

Findings

The scheme almost surely makes finitely many incorrect decisions.

It can handle noisy and discretely observed sample paths.

The approach combines Cover's irrationality test with covariance estimation.

Abstract

Is it possible to detect if the sample paths of a stochastic process almost surely admit a finite expansion with respect to some/any basis? The determination is to be made on the basis of a finite collection of discretely/noisily observed sample paths. We show that it is indeed possible to construct a hypothesis testing scheme that is almost surely guaranteed to make only finitely many incorrect decisions as more data are collected. Said differently, our scheme almost certainly detects whether the process has a finite or infinite basis expansion for all sufficiently large sample sizes. Our approach relies on Cover's classical test for the irrationality of a mean, combined with tools for the non-parametric estimation of covariance operators.