Beta Distribution of Long Memory Sequences

TL;DR

This paper investigates three long memory models and demonstrates that they produce sequences with nonstationary generalized beta marginal distributions, revealing a relationship between variance, range, and beta distribution parameters.

Contribution

It introduces a matrix distribution transformation that maps normal components of long memory models onto the beta distribution, providing new insights into their distributional properties.

Findings

Sequences have nonstationary generalized beta marginals

Variance/range ratio is stationary and linked to beta shape parameter

A matrix transformation maps normal components to beta distribution

Abstract

Three long memory models, ARFIMA, Timmer and Konig 1995, and a circular convolution model based on Wold's representation theorem are examined. Each model is shown to produce sequences with nonstationary generalized beta marginal distributions. It is demonstrated that the variance divided by the squared range of the sequence is stationary, and is a function of the shape parameter of the resulting symmetric beta distribution. Using the Wold model, a simple matrix distribution transformation is given that maps the normal components of the long memory model onto the beta distribution.