Two-process coupling characterization of weak convergence

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 $E$ be a separable metric space with metric $\rho$ and its Borel sigma-field. On $E^{\mathbb N_0}$ use a compatible product metric, for example \[d(x,y)=\sum_{k=0}^\infty2^{-k-1}\min\{1,\rho(x_k,y_k)\}.\] Let $X,X'$ be processes in this path space and $\theta_n(x)_k=x_{n+k}$. Suppose $\theta_nX\Rightarrow X'$ as random path-space elements. A two-process coupling constructs only $X$ and $X'$ together, preserving their respective full laws; it does not introduce a separate copy with law $\mathcal L(\theta_nX)$ for each $n$.

Statement recorded in the supplied source.

Give a characterization of this weak convergence using a two-process coupling. In particular, can such a coupling be chosen so that $d(\theta_nX,\theta_nX')\to0$ almost surely as $n\to\infty$? [S2]

Short English statement

Can weak convergence of shifted processes be characterized by coupling just the original process and its limiting process?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9900007 (AMR-098-0007). 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, weak-convergence paragraph and Problem 3.3, p. 4. https://web.archive.org/web/20240327021031id_/http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf.