Jackknife estimation for Markov processes with no mixing constraints

The jackknife resampling procedure is a technique to reduce the bias of a statistic. As with other resampling techniques, the jackknife procedure is motivated by and is well understood in the i.i.d. regime. However, analysis of the procedure when samples have memory is limited, and is predominantly restricted to cases with strong mixing or memory constraints. In this paper, we analyze a natural jackknife resampling procedure for Markov sources with no mixing assumptions. For the problem to be well posed without mixing assumptions, we instead adopt a physically motivated continuity condition that ensures that the information a bit in the past provides about the current bit, conditioned on all bits in between, diminishes with the amount of history we have. We analyze the jackknife estimate of the variance of conditional probability estimates given arbitrary contexts, and show that the bias of this jackknife procedure can be bounded by a small constant.