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PreprintCC BY-NC-ND 4.0arXiv

Conditional Stable Laws and Rare-Event Limits for Absorbing Markov Chains

  • Bernat Bassols Cornudella
  • Matheus M Castro
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Abstract

We establish conditional limit theorems, pointwise in the initial state, for absorbing Markov chains on a compact metric space M. We assume L1(M,ρ)-continuous transition densities, irreducibility and aperiodicity. For the observable fβ(x)=dM(x,x0)−β, with suitable x0 satisfying ρ(Br(x0))∼ Cd(x0)rd, we prove that the point-process of normalised observations converges to a Poisson random measure. This yields totally right-skewed α-stable laws for α=d/β∈(0,2) and, at the boundary value α=2, a Gaussian limit with the non-standard normalisation √(nlog n). We also establish a conditional central limit theorem for L2 observables, exponential deviation bounds for bounded observables and a conditional Poisson law for visits to shrinking targets. The results and remarks have been formalised in Lean~4.

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Cite

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BibTeX
@misc{cornudella2026conditional,
  title = {{Conditional Stable Laws and Rare-Event Limits for Absorbing Markov Chains}},
  author = {Bernat Bassols Cornudella and Matheus M Castro},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.22499},
  eprint = {2609.22499},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22499},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Conditional Stable Laws and Rare-Event Limits for Absorbing Markov Chains
AU  - Bernat Bassols Cornudella
AU  - Matheus M Castro
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.22499
UR  - https://arxiv.org/abs/2609.22499
AB  - We establish conditional limit theorems, pointwise in the initial state, for absorbing Markov chains on a compact metric space M. We assume L^1(M,ρ)-continuous transition densities, irreducibility and aperiodicity. For the observable f_β(x)=d_M(x,x_0)^−β, with suitable x_0 satisfying ρ(B_r(x_0))∼ C_d(x_0)r^d, we prove that the point-process of normalised observations converges to a Poisson random measure. This yields totally right-skewed α-stable laws for α=d/β∈(0,2) and, at the boundary value α=2, a Gaussian limit with the non-standard normalisation √(nlog n). We also establish a conditional central limit theorem for L^2 observables, exponential deviation bounds for bounded observables and a conditional Poisson law for visits to shrinking targets. The results and remarks have been formalised in Lean~4.
ER  -