Coupling characterization of setwise asymptotic stationarity

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(E,\mathcal E)$ be a measurable state space and let $X=(X_k)_{k\ge0}$ and $X'=(X'_k)_{k\ge0}$ be processes with values in the path space $(E^{\mathbb N_0},\mathcal E^{\otimes\mathbb N_0})$, where $\mathbb N_0=\{0,1,2,\ldots\}$. The shift is $\theta_n(x)_k=x_{n+k}$. Setwise convergence of laws means convergence of probabilities for every measurable path set. Total variation distance of two probability laws is $\sup_A|\mu(A)-\nu(A)|$. A coupling means a joint construction preserving the two complete marginal process laws. A distributional exact coupling is a coupling admitting almost surely finite random integers $T,T'$ with $(\theta_TX,T)\stackrel d=(\theta_{T'}X',T')$. The source notes that total-variation convergence would yield such a characterization when the limit $X'$ is stationary.

Statement recorded in the supplied source.

If setwise convergence of $\mathcal L(\theta_nX)$ to $\mathcal L(X')$ does not imply total-variation convergence, find a coupling characterization of \[\mathbb P(\theta_nX\in A)\to\mathbb P(X'\in A)\quad\text{for every measurable path set }A.\] [S2]

Short English statement

Find the right coupling criterion for setwise asymptotic stationarity if the usual total-variation coupling criterion is too strong.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9900006 (AMR-098-0006). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hermann Thorisson, Some Open Probability Problems, author preprint; original Universidad de Buenos Aires copy preserved in Internet Archive capture of 27 March 2024, Section 3, coupling definitions, p. 3, and Problem 3.2, p. 4. https://web.archive.org/web/20240327021031id_/http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf.