Statistical estimation of ergodic Markov chain kernel over discrete state space

Geoffrey Wolfer, Aryeh Kontorovich

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate the statistical complexity of estimating the parameters of a discrete-state Markov chain kernel from a single long sequence of state observations. In the finite case, we characterize (modulo logarithmic factors) the minimax sample complexity of estimation with respect to the operator infinity norm, while in the countably infinite case, we analyze the problem with respect to a natural entry-wise norm derived from total variation. We show that in both cases, the sample complexity is governed by the mixing properties of the unknown chain, for which, in the finite-state case, there are known finite-sample estimators with fully empirical confidence intervals.

Original languageEnglish
Pages (from-to)532-553
Number of pages22
JournalBernoulli
Volume27
Issue number1
DOIs
StatePublished - 1 Feb 2021

Keywords

  • Discrete state space
  • Ergodic Markov chain
  • Minimax theory

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