Occupation counts of rarely switching Markov chains: stratified local laws, support selection and the boundary layer
Abstract
A Markov chain with exit rates qi runs for N steps with transition matrix I+(T/N)Q. On each irreducible face that carries start mass, we prove a local limit theorem for the occupation counts with an explicit constant and relative error O(1/T), uniformly for 1≤ T≤ε N with εmaxiqi<1 and on compact subsets of the open face. Up to O(1), minus the log-probability of counts n with support F is |F|−1 times log(N/√T) plus N times the Donsker and Varadhan functional of I+(T/N)Q at n/N. Every admissible face has an explicit power of T in its largest probability for large T≤ N1/3, and a constant given by a finite-dimensional variational formula. For T=τlog N the face of the mode minimizes τλF+|F|−1 at first order, with λF the exit rate. If an admissible face has Δ≥1 more states than another and an exit rate smaller by δ>0, we locate every crossing of their largest probabilities for large T≤ N1/3 up to O(1/log N). For two irreducible faces the crossing is unique for large N in 1≤ T≤ε N, with coefficient 1/2 of loglog N. Near a facet, the continuous-time occupation density is the facet density times an explicit kernel, with relative error O(1/T) at times x≤ cT off the facet, c small. The counts obey a lattice form with extra error T2/N for T≤ε1√N. The kernel is Bessel for x=o(T) at facet profiles where the removed state has equal entry and exit laws, and at a profile where the rate has a first correction, of order m≥3, if and only if x=o(T1−2/m). On the complete graph with the uniform start, each face of at least 2 states crosses a vertex exactly once.
Related
- Finite-sample certification of coverage regret for budgeted document selection under adaptive stratified auditing
- Continuous-Time Mean-Variance Portfolio Selection Reinforcement Learning Problems for a Markov Regime-Switching System with No-Shorting Constraints
- A support preserving homotopy for the de Rham complex with boundary decay estimates
- Selection of Marginally Stable Patterns by Boundary Conditions
- Zero Forcing, Support Varieties, and Boundary Complexes
BibTeX
@misc{pemmasani2026occupation,
title = {{Occupation counts of rarely switching Markov chains: stratified local laws, support selection and the boundary layer}},
author = {Arjun Pemmasani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09258},
eprint = {2610.09258},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09258},
}
RIS
TY - UNPB
TI - Occupation counts of rarely switching Markov chains: stratified local laws, support selection and the boundary layer
AU - Arjun Pemmasani
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.09258
UR - https://arxiv.org/abs/2610.09258
AB - A Markov chain with exit rates q_i runs for N steps with transition matrix I+(T/N)Q. On each irreducible face that carries start mass, we prove a local limit theorem for the occupation counts with an explicit constant and relative error O(1/T), uniformly for 1≤ T≤ε N with εmax_iq_i<1 and on compact subsets of the open face. Up to O(1), minus the log-probability of counts n with support F is |F|−1 times log(N/√T) plus N times the Donsker and Varadhan functional of I+(T/N)Q at n/N. Every admissible face has an explicit power of T in its largest probability for large T≤ N^1/3, and a constant given by a finite-dimensional variational formula. For T=τlog N the face of the mode minimizes τλ_F+|F|−1 at first order, with λ_F the exit rate. If an admissible face has Δ≥1 more states than another and an exit rate smaller by δ>0, we locate every crossing of their largest probabilities for large T≤ N^1/3 up to O(1/log N). For two irreducible faces the crossing is unique for large N in 1≤ T≤ε N, with coefficient 1/2 of loglog N. Near a facet, the continuous-time occupation density is the facet density times an explicit kernel, with relative error O(1/T) at times x≤ cT off the facet, c small. The counts obey a lattice form with extra error T^2/N for T≤ε_1√N. The kernel is Bessel for x=o(T) at facet profiles where the removed state has equal entry and exit laws, and at a profile where the rate has a first correction, of order m≥3, if and only if x=o(T^1−2/m). On the complete graph with the uniform start, each face of at least 2 states crosses a vertex exactly once.
ER -