2 research outputs found
Occupancy distributions in Markov chains via Doeblin's ergodicity coefficient
We apply Doeblin's ergodicity coefficient as a computational tool to
approximate the occupancy distribution of a set of states in a homogeneous but
possibly non-stationary finite Markov chain. Our approximation is based on new
properties satisfied by this coefficient, which allow us to approximate a chain
of duration n by independent and short-lived realizations of an auxiliary
homogeneous Markov chain of duration of order ln(n). Our approximation may be
particularly useful when exact calculations via first-step methods or transfer
matrices are impractical, and asymptotic approximations may not be yet
reliable. Our findings may find applications to pattern problems in Markovian
and non-Markovian sequences that are treatable via embedding techniques.Comment: 12 pages, 2 table
Occupancy distributions in Markov chains via Doeblin's ergodicity coefficient
We state and prove new properties about Doeblin's ergodicity coefficient for finite Markov chains. We show that this coefficient satisfies a sub-multiplicative type inequality (analogous to the Markov-Dobrushin's ergodicity coefficient), and provide a novel but elementary proof of Doeblin's characterization of weak-ergodicity for non-homogeneous chains. Using Doeblin's coefficient, we illustrate how to approximate a homogeneous but possibly non-stationary Markov chain of duration by independent and short-lived realizations of an auxiliary chain of duration of order . This leads to approximations of occupancy distributions in homogeneous chains, which may be particularly useful when exact calculations via one-step methods or transfer matrices are impractical, and when asymptotic approximations may not be yet reliable. Our findings may find applications to pattern problems in Markovian and non-Markovian sequences that are treatable via embedding techniques