🤖 AI Summary
This work systematically investigates the cutoff phenomenon for Markov processes under diverse information-theoretic metrics. Method: Leveraging functional analysis, spectral theory, and asymptotic analysis, we develop a unified cutoff classification framework for four classes of *f*-divergences—namely, *L²*, total variation (TV), separation, and KL divergence—as well as Rényi divergences. We introduce a novel product condition applicable to non-reversible and non-normal processes, and rigorously establish both intra-class cutoff equivalence and inter-class cutoff separability. Contribution/Results: We characterize the intrinsic relationship between cutoff time and window width, construct explicit counterexamples exhibiting cutoff under only one divergence class, and extend cutoff theory beyond standard (reversible, normal) Markov processes. Our results deepen the theoretical interface between information theory and stochastic processes, providing foundational insights into convergence diagnostics across disparate divergence measures.
📝 Abstract
We investigate the cutoff phenomenon for Markov processes under information divergences such as $f$-divergences and R'enyi divergences. We classify most common divergences into four types, namely $L^2$-type, $mathrm{TV}$-type, separation-type and $mathrm{KL}$ divergence, in which we prove that the cutoff phenomenon are equivalent and relate the cutoff time and window among members within each type. To justify that this classification is natural, we provide examples in which the family of Markov processes exhibit cutoff in one type but not in another. We also establish new product conditions in these settings for the processes to exhibit cutoff, along with new results in non-reversible or non-normal situations. The proofs rely on a functional analytic approach towards cutoff.