Relax aperiodicity and stationarity assumptions in strong lumpability equivalence

Determine whether the equivalence between strong lumpability of a Markov chain and the entropy condition H(Zt+1|Zt) = H(Zt+1|Xt) can be established without assuming aperiodicity and whether the stationarity requirement can be relaxed, or provide counterexamples demonstrating the necessity of these assumptions.

Background

The paper connects informational/causal closure to strong lumpability via an entropy characterization that currently holds under stationarity, irreducibility, and aperiodicity. These technical conditions constrain the scope of the results.

Clarifying whether aperiodicity and stationarity are essential would refine the theoretical foundations and potentially expand applicability to broader classes of Markov processes.