On universal estimates for binary renewal processes

TL;DR

This paper develops universal estimation methods for binary renewal processes, enabling prediction of future behavior without prior distribution knowledge, and explores the necessity of moment conditions for accurate estimation.

Contribution

The paper introduces new universal estimators for binary renewal processes, including expected renewal time and conditional distributions, with analysis of moment condition requirements.

Findings

Universal estimates for expected renewal time

Conditional distribution estimates for renewal times

Necessity of moment conditions demonstrated

Abstract

A binary renewal process is a stochastic process $\{X_n\}$ taking values in $\{0,1\}$ where the lengths of the runs of 1's between successive zeros are independent. After observing ${X_0,X_1,...,X_n}$ one would like to predict the future behavior, and the problem of universal estimators is to do so without any prior knowledge of the distribution. We prove a variety of results of this type, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal. Some of our results require a moment condition on the time to renewal and we show by an explicit construction how some moment condition is necessary.