Probabilistic Equality of Left and Right Tail Fields

Definitions and mathematical statement

Let $S$ be a finite nonempty set equipped with its discrete sigma-field, and let $(X_k)_{k\in{\mathbb{Z}}}$ be an $S$-valued process on a probability space $(\Omega,\mathcal F,\mathbb P)$. Assume strict stationarity: for every $r\geq1$, every $k_1,\ldots,k_r\in{\mathbb{Z}}$, and every $h\in{\mathbb{Z}}$, \[ (X_{k_1+h},\ldots,X_{k_r+h})\overset{\mathrm d}= (X_{k_1},\ldots,X_{k_r}). \] Define the left and right tail sigma-fields by \[ \mathcal T^-:=\bigcap_{n=0}^{\infty}\sigma(X_k:k\leq -n), \qquad \mathcal T^+:=\bigcap_{n=0}^{\infty}\sigma(X_k:k\geq n). \] Equality modulo null sets means that for every $A\in\mathcal T^-$ there exists $B\in\mathcal T^+$ with $\mathbb P(A\triangle B)=0$, and conversely for every $B\in\mathcal T^+$ there exists such an $A\in\mathcal T^-$.

Requested statement. For every process satisfying these assumptions, prove, using only probability theory and without entropy theory, that \[ \mathcal T^-=\mathcal T^+\qquad\text{modulo }\mathbb P\text{-null sets}. \] Unresolved qualifications. The supplied statement does not explicitly specify the time index set or the meaning of stationarity. The formulation above uses the conventional interpretation of a bi-infinite discrete-time strictly stationary process; this interpretation requires source confirmation. The source also does not formally delimit the permitted probabilistic methods or the excluded entropy-theoretic methods. Its target is a proof subject to this methodological restriction.

Short English statement

For a stationary process taking values in a finite set, show that the events determined by the arbitrarily remote past coincide, up to probability-zero differences, with those determined by the arbitrarily remote future. The requested proof must use only probability theory and exclude entropy theory. Bi-infinite discrete time and strict stationarity are the explicit interpretations used here; the source leaves the precise methodological boundary unspecified.

Sources

• Supplied local registry record: UnsolvedMath permanent ID $30004435$, title Probabilistic Equality of Left and Right Tail Fields, exact source problem code catalogue entry, in Oberwolfach Reports Open Problems. No individual proposer is identified in the supplied record.
• Supplied report citation: Mini-Workshop: One-sided and Two-sided Stochastic Descriptions (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/11, pp. 601--637. Supplied source URL: https://doi.org/10.4171/owr/2020/11. The record gives proposed year 2020 and an owr-year:2020 tag; their relationship to the citation's 2021 date is unresolved here.
• Supplied literature source, labelled OWR-2020-11: https://ems.press/content/serial-article-files/46847.
• Literature assessment dated exactly 2026-08-22: the supplied curation states that equality of the two tails is known, but that no proof using only probability theory was identified, so it labels the method-restricted problem open. This is supplied dated curation, not an independently verified literature assessment; the background expressly requests independent verification of time-sensitive status and cited claims.
• The registry supplies status open and statement status corrected_verified, with a statement-verification note reporting manual semantic, typographic, and LaTeX review. These are supplied registry labels and review claims, not independent mathematical verification of this draft.