L2-m-approximable causal Bernoulli shifts

Let $\mathcal{B}$ be the Borel sigma-algebra of $[0,1]$, generated by intervals with rational endpoints; let $dt$ be Lebesgue measure restricted to $\mathcal{B}$, and set $H=L^2([0,1],\mathcal{B},dt;\mathbb{R})$, with inner product $⟨x,y{⟩}_H={\int }_0^{1}x(u)y(u)du$. We use the quotient-by-null-functions convention for $L^2$ (The space $L^p(\mu )$ as the quotient by null functions). The measure space is finite and countably generated, so under Countable Choice $H$ is separable (If $\mu$ is sigma-finite and $\mathcal{A}$ is countably generated, then $L^p(\mu )$ is separable for $1\le p<\mathrm{\infty }$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), Separability: the existence of an at most countable dense subset); $L^2$ with the integral pairing is a Hilbert space verifies that this real pairing makes the quotient a Hilbert space.

Let $(S,\mathcal{S},\nu )$ be a probability space and $({\delta }_t{)}_{t\in \mathbb{Z}}$ an independent identically distributed family of $S$-valued random elements with law $\nu$ (Random elements and real random variables, Law or distribution of a random element, Independent random elements). A causal Bernoulli shift with innovation law $\nu$ is a process $X_t=f({\delta }_t,{\delta }_{t-1},{\delta }_{t-2},\dots ),t\in \mathbb{Z},$ where $f:S^{{\mathbb{N}}_0}\to H$ is measurable from the product sigma-algebra ${\mathcal{S}}^{\otimes {\mathbb{N}}_0}$ to the Borel sigma-algebra of $H$. This product-measurability formulation is meaningful for a merely measurable innovation space $S$.

On an independent product extension, take replacement innovations ${\delta }_{t,m,k}^{\mathrm{\prime }}$, indexed by $t\in \mathbb{Z}$, $m\in {\mathbb{N}}_0$, and integers $k\le t-m$, all with law $\nu$, mutually independent, and independent of the original innovations. Such a countable family and its product with the original probability space exist under Countable Choice and Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain, Countably many independent copies of a prescribed law exist, For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique). For $m\ge 1$ define $X_{t,m}=f({\delta }_t,\dots ,{\delta }_{t-m+1},{\delta }_{t,m,t-m}^{\mathrm{\prime }},{\delta }_{t,m,t-m-1}^{\mathrm{\prime }},\dots ),$ and define the full replacement $X_{t,0}=f({\delta }_{t,0,t}^{\mathrm{\prime }},{\delta }_{t,0,t-1}^{\mathrm{\prime }},\dots )$. Each $X_{t,m}$ has the same marginal law as $X_t$; for fixed $m\ge 1$ the sequence $(X_{t,m}{)}_{t\in \mathbb{Z}}$ is strictly stationary and $m$-dependent because its original-innovation windows have length $m$ and its replacement arrays are independent across $t$.

For every fixed $m\ge 0$, the joint process $(X_t,X_{t,m}{)}_{t\in \mathbb{Z}}$ is strictly stationary: shifting all indices relabels a family of mutually independent innovations with the same law and commutes with the defining measurable maps. Consequently $D_{t,m}:=X_t-X_{t,m}$ is stationary and $\mathrm{\parallel }{D}_{t,m}{\mathrm{\parallel }}_{L^2(H)}={\delta }_m$ for every $t$.

Write $\mathrm{\parallel }X{\mathrm{\parallel }}_{L^2(H)}=(\mathbb{E}\mathrm{\parallel }X{\mathrm{\parallel }}_H^2{)}^{1\mathrm{/}2}$ and ${\delta }_m=\mathrm{\parallel }{X}_0-X_{0,m}{\mathrm{\parallel }}_{L^2(H)},m\ge 0.$ The process is $L^2$-$m$-approximable when $\mathbb{E}\mathrm{\parallel }{X}_0{\mathrm{\parallel }}_H^2<\mathrm{\infty }\text{and}{\sum }_{m\ge 1}{\delta }_m<\mathrm{\infty }\mathrm{.}$ The same marginal law gives ${\delta }_0\le 2\mathrm{\parallel }{X}_0{\mathrm{\parallel }}_{L^2(H)}<\mathrm{\infty }$. For a Bochner-integrable $H$-valued random element $Y$, write ${\mathbb{E}}_{B}Y=\int Yd\mathbb{P}$; the moment condition makes $X_t$ and every $X_{t,m}$ Bochner integrable (Bochner-integrable function, Bochner integrability criterion).

Here strict stationarity means invariance of every finite-dimensional law under integer shifts. A process is $m$-dependent if, for every $a\in \mathbb{Z}$, the sigma-algebras generated by $(X_s{)}_{s\le a}$ and $(X_s{)}_{s\ge a+m+1}$ are independent (Independent sigma-algebras and independent events). The stated input windows for $X_{t,m}$ are disjoint across these two index ranges, so they satisfy this definition.

Every Borel $H$-valued random element here is strongly measurable. Indeed, fix a dense sequence $(h_j)$ in the separable space $H$. For each $N$, send $x$ to the first point among $h_0,\dots ,h_N$ minimizing $\mathrm{\parallel }x-h_j{\mathrm{\parallel }}_H$. Its finitely many tie-broken cells are Borel because the distance functions are continuous, so this is a finite-valued measurable map; its distance from $x$ tends to zero since the sequence is dense. Composition with any Borel $H$-valued random element gives the simple approximants in Strongly measurable Banach-valued function. The phrase “Borel measurable” in the design is thus applied to the target $H$; the innovation domain uses its stated product sigma-algebra.