Strict stationarity; the $\alpha$-, $\rho$-, $\varphi$-mixing coefficients; asymptotic variance

processSigma — For arbitrary types $\Omega$ and $E$ where only $E$ carries a σ-algebra, a family of maps $Y_i : \Omega \to E$ indexed by $i \in \mathbb{N}$, and a set $s \subseteq \mathbb{N}$: the σ-algebra on $\Omega$ defined as the supremum over $i \in s$ of the pullback σ-algebras $\{\,Y_i^{-1}(B) : B \subseteq E \text{ measurable}\,\}$ — that is, the smallest σ-algebra on $\Omega$ making every $Y_i$ with $i \in s$ measurable. No measurability of the $Y_i$ relative to any ambient structure on $\Omega$ is assumed (none exists in this definition). For $s = \varnothing$ the supremum is the trivial σ-algebra $\{\varnothing, \Omega\}$. The definition is marked reducible.

IsStrictlyStationary — For types $\Omega$ and $E$ both with σ-algebras, a measure $P$ on $\Omega$ (not assumed to be a probability measure), and $Y : \mathbb{N} \to \Omega \to E$: the proposition that for every $k \in \mathbb{N}$, the pushforward of $P$ under the map $\omega \mapsto (n \mapsto Y_{n+k}(\omega))$, a map into the product space of sequences $\mathbb{N} \to E$, equals the pushforward of $P$ under $\omega \mapsto (n \mapsto Y_n(\omega))$. No measurability hypothesis on $Y$ appears anywhere; the pushforward of a measure under a map that is not almost-everywhere measurable is by convention the zero measure, so for non-measurable $Y$ the equality can hold degenerately with both sides equal to the zero measure.

alphaMixingCoef — For a measure $P$ on $\Omega$, maps $Y : \mathbb{N} \to \Omega \to E$, and $n \in \mathbb{N}$: the real supremum of the set of all reals $r$ such that there exist $k \in \mathbb{N}$ and sets $A, B \subseteq \Omega$ with $A$ measurable with respect to the σ-algebra generated by $Y_0,\dots,Y_k$ (the supremum of the pullback σ-algebras of $Y_i$, $i \le k$), $B$ measurable with respect to the σ-algebra generated by $\{Y_i : i \ge k+n\}$, and $r = \bigl|\mathrm{real}(P(A \cap B)) - \mathrm{real}(P(A))\cdot \mathrm{real}(P(B))\bigr|$, where $\mathrm{real}$ converts $[0,\infty]$ to $\mathbb{R}$ with $\infty \mapsto 0$. The supremum ranges over all splitting times $k$ jointly with $A$ and $B$; $n = 0$ is allowed, in which case the two generating index ranges overlap at $k$. The candidate set contains $0$ (take $A = B = \varnothing$). Junk conventions: sets of infinite measure enter through the value $0$; the real supremum of a set unbounded above is $0$ (possible when $P$ is not finite).

rhoMixingCoef — For $P$, $Y$, $n$ as above: the real supremum of the set of all reals $r$ such that there exist $k \in \mathbb{N}$ and functions $U, V : \Omega \to \mathbb{R}$ with $U$ measurable as a map from $\Omega$ equipped with the σ-algebra generated by $Y_0,\dots,Y_k$, $V$ measurable from $\Omega$ equipped with the σ-algebra generated by $\{Y_i : i \ge k+n\}$, both $U$ and $V$ in $L^2(P)$ (this membership is with respect to the ambient σ-algebra on $\Omega$), and $r = |\mathrm{cov}(U,V;P)| \,/\, \bigl(\sqrt{\mathrm{Var}(U;P)}\cdot\sqrt{\mathrm{Var}(V;P)}\bigr)$. Here $\mathrm{cov}(U,V;P)$ is the Bochner integral $\int \bigl(U(\omega) - \int U\,dP\bigr)\bigl(V(\omega) - \int V\,dP\bigr)\,dP(\omega)$ (equal to $0$ by convention if the integrand is not integrable), and $\mathrm{Var}(U;P)$ is the variance $\int (U - \int U\,dP)^2\,dP$ computed in the extended nonnegative reals and converted to $\mathbb{R}$ with $\infty \mapsto 0$. Junk conventions: when either variance is $0$ (e.g. $U$ constant almost everywhere) the denominator is $0$ and real division by $0$ yields $0$, so such pairs contribute the value $0$; if no admissible pair exists at all (for instance when constants fail to be in $L^2$, as for an infinite measure $P$, and nothing else qualifies) the candidate set is empty and the real supremum of the empty set is $0$; a candidate set unbounded above likewise yields $0$.

phiMixingCoef — For $P$, $Y$, $n$ as above: the real supremum of the set of all reals $r$ such that there exist $k \in \mathbb{N}$ and sets $A, B \subseteq \Omega$ with $A$ measurable with respect to the σ-algebra generated by $Y_0,\dots,Y_k$ and $P(A) \ne 0$ (as an extended-nonnegative value), $B$ measurable with respect to the σ-algebra generated by $\{Y_i : i \ge k+n\}$, and $r = \bigl|\mathrm{real}(P(A \cap B))/\mathrm{real}(P(A)) - \mathrm{real}(P(B))\bigr|$, with $\mathrm{real}$ the $[0,\infty] \to \mathbb{R}$ conversion sending $\infty$ to $0$. Junk conventions: a set $A$ with $P(A) = \infty$ satisfies $P(A) \ne 0$ yet has $\mathrm{real}(P(A)) = 0$, and real division by $0$ yields $0$, so such an $A$ contributes $r = \mathrm{real}(P(B))$; if no admissible $A$ with $P(A) \ne 0$ exists (e.g. $P$ the zero measure) the candidate set is empty and its real supremum is $0$; an unbounded candidate set also yields $0$.

seqAsymptoticVariance — For a measure $P$ on $\Omega$ and real-valued maps $Y : \mathbb{N} \to \Omega \to \mathbb{R}$: the real number $\int Y_0(\omega)^2\,dP(\omega) \;+\; 2\sum_{k=0}^{\infty} \int Y_0(\omega)\,Y_{k+1}(\omega)\,dP(\omega)$. All integrals are Bochner integrals, each equal to $0$ by convention when its integrand is not integrable; the infinite series is a topological sum, equal to $0$ by convention when the family of integrals is not summable. The cross terms always pair $Y_0$ with $Y_{k+1}$ for $k = 0, 1, 2, \dots$ (lags $1, 2, \dots$); no means are subtracted anywhere, and no stationarity, measurability, or probability-measure hypotheses appear.

BoundedInProbability — For maps $Z : \mathbb{N} \to \Omega \to \mathbb{R}$ and a measure $P$ on $\Omega$ (not assumed to be a probability measure): the proposition that for every real $\varepsilon > 0$ there exists a real $K$ (no sign constraint — $K$ may be negative) such that for every $n \in \mathbb{N}$, $\mathrm{real}\bigl(P\{\omega : K < |Z_n(\omega)|\}\bigr) \le \varepsilon$, where $\mathrm{real}$ converts $[0,\infty]$ to $\mathbb{R}$ with $\infty \mapsto 0$. The set $\{\omega : K < |Z_n(\omega)|\}$ is not required to be measurable ($P$ is applied to it as an outer measure), and no measurability of $Z_n$ is assumed. Junk convention: if that set has $P$-measure $\infty$, the conversion yields $0$ and the inequality holds trivially.