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PreprintCC BY 4.0arXiv

Polynomial Decay in Absorbing Continuous-Time Markov Chains

  • Phil Pollett
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Abstract

Let P(t) be the transition function of an absorbing continuous-time Markov chain on a countable state space. We study asymptotic relations of the form pij(t)∼ aij L(t), t→∞, where L is independent of i and j. We obtain general conditions under which the coefficient matrix A=(aij) has rank one and describe the resulting consequences for survival probabilities and conditional distributions. The analysis applies to both reducible and irreducible chains. We also construct an irreducible counterexample, based on a killed random walk on a homogeneous tree, for which a common asymptotic scale exists but the coefficient matrix has rank greater than one. This shows that irreducibility alone does not imply rank-one asymptotics. The results identify conditions under which fixed-state transition asymptotics determine the asymptotic behaviour of the entire transition function and clarify the limitations of such conclusions in the absence of additional structure.

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Cite

BibTeXRIS
BibTeX
@misc{pollett2026polynomial,
  title = {{Polynomial Decay in Absorbing Continuous-Time Markov Chains}},
  author = {Phil Pollett},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27335},
  eprint = {2609.27335},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27335},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Polynomial Decay in Absorbing Continuous-Time Markov Chains
AU  - Phil Pollett
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27335
UR  - https://arxiv.org/abs/2609.27335
AB  - Let P(t) be the transition function of an absorbing continuous-time Markov chain on a countable state space. We study asymptotic relations of the form p_ij(t)∼ a_ij L(t), t→∞, where L is independent of i and j. We obtain general conditions under which the coefficient matrix A=(a_ij) has rank one and describe the resulting consequences for survival probabilities and conditional distributions. The analysis applies to both reducible and irreducible chains. We also construct an irreducible counterexample, based on a killed random walk on a homogeneous tree, for which a common asymptotic scale exists but the coefficient matrix has rank greater than one. This shows that irreducibility alone does not imply rank-one asymptotics. The results identify conditions under which fixed-state transition asymptotics determine the asymptotic behaviour of the entire transition function and clarify the limitations of such conclusions in the absence of additional structure.
ER  -