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ON UNIVERSAL ESTIMATES FOR BINARY RENEWAL PROCESSES

Gusztáv Morvai, Benjamin Weiss

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Abstract

A binary renewal process is a stochastic process {Xn} taking values in {0,1} where the lengths of the runs of 1’s between successive zeros are independent. After observing X0,X1,...,Xn 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. 1. Introduction. The

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What this paper is about

A binary renewal process is a stochastic process {Xn} taking values in {0,1} where the lengths of the runs of 1’s between successive zeros are independent. After observing X0,X1,...,Xn 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. 1. Introduction. The

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Available abstract

A binary renewal process is a stochastic process {Xn} taking values in {0,1} where the lengths of the runs of 1’s between successive zeros are independent. After observing X0,X1,...,Xn 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. 1. Introduction. The

Key concepts: Renewal theory, Estimator, Moment (physics), Binary number, Mathematics, Applied mathematics, Markov chain, Markov renewal process

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