Inferring the residual waiting time for binary stationary time series

For a binary stationary time series define σn to be the number of consecutive ones up to the first zero encountered after time n, and consider the problem of estimating the conditional distribution and conditional expectation of σn after one has observed the first n outputs. We present a sequence of stopping times and universal estimators for these quantities which are pointwise consistent for all ergodic binary stationary processes. In case the process is a renewal process with zero the renewal state the stopping times along which we estimate have density one.