Interlaced maximal-correlation mixing for a uniformly mixing Markov chain
Let $(X_k)_{k \in \mathbb{Z}}$ be a strictly stationary, time-homogeneous, real-valued Markov chain; no further assumption is made on its state space or transition kernel. For $\sigma$-fields $\mathcal{A}, \mathcal{B}$ let $$\phi(\mathcal{A}, \mathcal{B}) = \sup\bigl\{|P(B \mid A) - P(B)| : A \in \mathcal{A},\ P(A) > 0,\ B \in \mathcal{B}\bigr\},$$ and let $\rho(\mathcal{A}, \mathcal{B})$ be the supremum of $|\operatorname{Corr}(f, g)|$ over square-integrable real $\mathcal{A}$-measurable $f$ and $\mathcal{B}$-measurable $g$ with positive variances (zero if there is no such pair). Put $$\phi(n) = \phi\bigl(\sigma(X_k : k \le 0), \sigma(X_k : k \ge n)\bigr), \qquad \rho^*(n) = \sup_{S, T} \rho\bigl(\sigma(X_k : k \in S), \sigma(X_k : k \in T)\bigr),$$ the second supremum over all nonempty disjoint sets $S, T \subset \mathbb{Z}$ with $\min_{s \in S, t \in T} |s - t| \ge n$, which may be interlaced and infinite. The chain is $\phi$-mixing if $\phi(n) \to 0$ and $\rho^*$-mixing (interlaced $\rho$-mixing) if $\rho^*(n) \to 0$ as $n \to \infty$. Every $\phi$-mixing sequence is $\rho$-mixing, since $\rho(\mathcal{A}, \mathcal{B}) \le 2\phi(\mathcal{A}, \mathcal{B})^{1/2}$. Bradley proved that a Markov chain is $\rho^*$-mixing as soon as $\inf P(A \cap B)/(P(A)P(B)) > 0$ for some $n$, the infimum over events $A \in \sigma(X_k : k \le 0)$ and $B \in \sigma(X_k : k \ge n)$ of positive probability; in particular strictly stationary, finite-state, irreducible, aperiodic chains are $\rho^*$-mixing. He also constructed a strictly stationary countable-state Markov chain that is $\rho$-mixing but not $\rho^*$-mixing.
Problem. Is every $\phi$-mixing such chain $\rho^*$-mixing?
Source: Bradley (2005), Question 5, for Markov chains not assumed stationary; its analogue for general random sequences, Question 2 there, was conjectured to have an affirmative answer by Bradley (1997, p. 226). Stated for strictly stationary chains by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Keywords
- AIM Open Applied Problems, Markov chains, maximal correlation, strong mixing conditions
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{bradley2005,
author = {Bradley, Richard C.},
title = {Basic properties of strong mixing conditions. {A} survey and some open questions},
journal = {Probab. Surv.},
volume = {2},
year = {2005},
pages = {107--144},
doi = {10.1214/154957805100000104},
eprint = {math/0511078},
archivePrefix = {arXiv},
}
@article{bradley1997,
author = {Bradley, Richard C.},
title = {Every ``lower psi-mixing'' {M}arkov chain is ``interlaced rho-mixing''},
journal = {Stochastic Process. Appl.},
volume = {72},
number = {2},
year = {1997},
pages = {221--239},
doi = {10.1016/S0304-4149(97)00090-2},
}
@article{bradley2001,
author = {Bradley, Richard C.},
title = {A stationary rho-mixing {M}arkov chain which is not ``interlaced'' rho-mixing},
journal = {J. Theoret. Probab.},
volume = {14},
number = {3},
year = {2001},
pages = {717--727},
doi = {10.1023/A:1017545123473},
}
@misc{aim-interlaced-maximal-correlation-mixing-for-a-uniformly-mixing-markov-chain,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Interlaced maximal-correlation mixing for a uniformly mixing Markov chain}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/994-interlaced-maximal-correlation-mixing-for-a-uniformly-mixing-markov-chain.md},
note = {Entry 994 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01157, "Interlaced maximal-correlation mixing for a uniformly mixing Markov chain", clopen.solutions.