Multi-time Markov renewal chains and stratified renewal theorems
Abstract
We develop stratified renewal theorems for multi-time observations of Markov-additive paths on standard Borel state spaces with nonnegative integer vector increments. Horizon directions with the same active constraints form strata. A compact-uniform inverse expansion yields a global limit in a topology combining hypi-convergence with exact restrictions to each stratum. At Brownian scale, the central theorem determines the joint terminal-edge law at fixed finite families of critical horizons about a common direction, under nondegenerate pairwise Gaussian contrasts. One Gaussian vector selects the exiting coordinates. Selections of the same coordinate at the same threshold share an edge; different selected pairs carry conditionally independent equilibrium marks. These marks retain both endpoint states, the full increment, age and overshoot. Under total-variation convergence of the state marginal to stationarity, pointwise scalar renewal convergence in total variation suffices for the marked argument. Regenerative conditions provide explicit criteria. Stationary scalar level processes retain dependence at fixed integer offsets. Under additional moment conditions, a second-order allocation theorem localizes optimal budgets and, with the marked hypotheses and nonsingular Gaussian contrasts, yields a cost-weighted equilibrium-edge mixture. An exact covariance criterion links uniform tightness of the unperturbed inverse field to the absence of a square-root allocation loss. Finite, denumerable and continuous modulating environments are included.
Related
- Global central limit theorems for Markov chains
- Global Central Limit Theorems for Markov chains with normal transition operator
- Central Limit Theorems for Sample Fréchet Means of Manifold-Valued Markov Chains
- Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains
- Occupation counts of rarely switching Markov chains: stratified local laws, support selection and the boundary layer
BibTeX
@misc{kordalis2026multitime,
title = {{Multi-time Markov renewal chains and stratified renewal theorems}},
author = {Leonidas Kordalis and Samis Trevezas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.07283},
eprint = {2607.07283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.07283},
}
RIS
TY - UNPB
TI - Multi-time Markov renewal chains and stratified renewal theorems
AU - Leonidas Kordalis
AU - Samis Trevezas
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2607.07283
UR - https://arxiv.org/abs/2607.07283
AB - We develop stratified renewal theorems for multi-time observations of Markov-additive paths on standard Borel state spaces with nonnegative integer vector increments. Horizon directions with the same active constraints form strata. A compact-uniform inverse expansion yields a global limit in a topology combining hypi-convergence with exact restrictions to each stratum. At Brownian scale, the central theorem determines the joint terminal-edge law at fixed finite families of critical horizons about a common direction, under nondegenerate pairwise Gaussian contrasts. One Gaussian vector selects the exiting coordinates. Selections of the same coordinate at the same threshold share an edge; different selected pairs carry conditionally independent equilibrium marks. These marks retain both endpoint states, the full increment, age and overshoot. Under total-variation convergence of the state marginal to stationarity, pointwise scalar renewal convergence in total variation suffices for the marked argument. Regenerative conditions provide explicit criteria. Stationary scalar level processes retain dependence at fixed integer offsets. Under additional moment conditions, a second-order allocation theorem localizes optimal budgets and, with the marked hypotheses and nonsingular Gaussian contrasts, yields a cost-weighted equilibrium-edge mixture. An exact covariance criterion links uniform tightness of the unperturbed inverse field to the absence of a square-root allocation loss. Finite, denumerable and continuous modulating environments are included.
ER -