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On the existence of a spectral gap for ergodic Markov chains

  • Xavier Mak
  • James P. Hobert
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Abstract

Characterizations are established regarding the existence of an L2-spectral gap for ergodic (that is, irreducible, aperiodic, and positive) Markov chains on infinite state spaces. In particular, it is shown that the existence of an L2-spectral gap is entirely determined by the behavior of the spectrum at 1. These results are applied in the analysis of a family of Markov chains constructed to show that the ergodic composition of two geometrically ergodic Markov kernels need not be geometrically ergodic.

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Cite

BibTeXRIS
BibTeX
@misc{mak2026existence,
  title = {{On the existence of a spectral gap for ergodic Markov chains}},
  author = {Xavier Mak and James P. Hobert},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24808},
  eprint = {2610.24808},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24808},
}
RIS
TY  - UNPB
TI  - On the existence of a spectral gap for ergodic Markov chains
AU  - Xavier Mak
AU  - James P. Hobert
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.24808
UR  - https://arxiv.org/abs/2610.24808
AB  - Characterizations are established regarding the existence of an L^2-spectral gap for ergodic (that is, irreducible, aperiodic, and positive) Markov chains on infinite state spaces. In particular, it is shown that the existence of an L^2-spectral gap is entirely determined by the behavior of the spectrum at 1. These results are applied in the analysis of a family of Markov chains constructed to show that the ergodic composition of two geometrically ergodic Markov kernels need not be geometrically ergodic.
ER  -