On universal estimates for binary renewal processes

Details On universal estimates for binary renewal processes On universal estimates for binary renewal processes

2008-11-13

arxiv.org/abs/0811.2076v1

Annals of Applied Probability 2008, Vol. 18, No. 5, 1970-1992

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AAP512 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/07-AAP512

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.

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