Asymptotic probability distribution of distances between local extrema of error terms of a moving average process

TL;DR

This paper analyzes the probability distribution of distances between local maxima in the error terms of a moving average process, revealing that the mean distance is 4 and the variance approaches 4 asymptotically.

Contribution

It provides the first detailed analysis of the asymptotic distribution of distances between local extrema in MA(q) processes, including mean and variance results.

Findings

Mean distance between local maxima is 4.

Asymptotic variance of distances is 4.

Distribution characteristics are derived for any q > 0.

Abstract

Consider error terms x(i) of a moving average process MA(q), where x(i)=e(i) + e(i-1)+...+e(i-q) and e(i) - independent identically distributed (i.i.d.) random variables. We recognize a term x(i) as a local maximum if the following condition holds true: x(i-1) < x(i) > x(i+1). If the local maximum x(i) is followed by the next local maxiumum x(k), then d=k-i is the distance between local maxima. The distances d(j) themselves are random vriables. In this paper we study the probability distribution of distances d(j). Particularly, we show that for any q>0 mean distance E[d(j)]=4 and asymptotically the variance is also equal to 4.