Occupation counts of rarely switching Markov chains: stratified local laws, support selection and the boundary layer

Abstract

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 $1\le T\le \epsilon N$ with $\epsilon {\max }_i{q}_i<1$ 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 $\log (N/\sqrt{T})$ 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 $T\le N^{1/3}$, and a constant given by a finite-dimensional variational formula. For $T=\tau \log N$ the face of the mode minimizes $\tau {\lambda }_F+|F|-1$ at first order, with ${\lambda }_F$ the exit rate. If an admissible face has $\Delta \ge 1$ more states than another and an exit rate smaller by $\delta >0$, we locate every crossing of their largest probabilities for large $T\le N^{1/3}$ up to $O(1/\log N)$. For two irreducible faces the crossing is unique for large $N$ in $1\le T\le \epsilon N$, with coefficient $1/2$ of $\log \log N$. Near a facet, the continuous-time occupation density is the facet density times an explicit kernel, with relative error $O(1/T)$ at times $x\le cT$ off the facet, $c$ small. The counts obey a lattice form with extra error $T^2/N$ for $T\le {\epsilon }_1\sqrt{N}$. 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 $m\ge 3$, 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.

arXiv comment82 pages, 2 figures. v2 replaces 'Vertex crossings in a symmetric Markov multinomial model'. Most results of v1 are reproved here, mainly in Sections 7 and 8; Table 1 and the paragraph before it list which, and the others stay in v1. Code and Lean files at https://github.com/apemm/Kagey-Problems