Recursive estimation of time-average variance constants

TL;DR

This paper introduces a recursive algorithm for estimating the time-average variance constant of stationary processes, significantly reducing memory and computational complexity while ensuring convergence under certain conditions.

Contribution

A novel recursive method for TAVC estimation that improves efficiency and allows real-time updates, unlike traditional batch-based approaches.

Findings

Recursive TAVC estimator converges under short-range dependence.

Memory complexity is reduced to O(1), computational complexity scales linearly.

Convergence rates and almost sure convergence are established.

Abstract

For statistical inference of means of stationary processes, one needs to estimate their time-average variance constants (TAVC) or long-run variances. For a stationary process, its TAVC is the sum of all its covariances and it is a multiple of the spectral density at zero. The classical TAVC estimate which is based on batched means does not allow recursive updates and the required memory complexity is O(n). We propose a faster algorithm which recursively computes the TAVC, thus having memory complexity of order O(1) and the computational complexity scales linearly in $n$. Under short-range dependence conditions, we establish moment and almost sure convergence of the recursive TAVC estimate. Convergence rates are also obtained.