A geometrically summable Bernoulli shift

For every input sequence with $\mathrm{\mid }{ϵ}_t\mathrm{\mid }=1$, ${\sum }_{k\ge 0}\mathrm{\mid }{2}^{-k}{ϵ}_{t-k}\mathrm{\mid }\le 2$, so the series converges absolutely and uniformly over the product space. Its partial sums are product-measurable and bounded by $2$, so their pointwise limit is a Borel causal map. Finite partial-sum variances are ${\sum }_{k\le N}{4}^{-k}$ by [F2]; the uniform bound and dominated convergence give centering and $\mathbb{E}\mathrm{\parallel }{X}_0{\mathrm{\parallel }}_H^2={\sum }_{k\ge 0}{4}^{-k}=4\mathrm{/}3<\mathrm{\infty }$. For the replacement, the first $m$ terms cancel; finite tails are independent centered differences with variance $2{\sum }_{k=m}^N{4}^{-k}$ and are bounded in absolute value by $2{\sum }_{k=m}^{\mathrm{\infty }}{2}^{-k}$. Dominated convergence [F2] passes these variances to the infinite tail, giving ${\delta }_m^2=2{\sum }_{k=m}^{\mathrm{\infty }}{4}^{-k}=\frac{8}{3}{4}^{-m}\mathrm{.}$ Thus ${\sum }_{m\ge 1}{\delta }_m=\sqrt{8\mathrm{/}3}{\sum }_{m\ge 1}{2}^{-m}<\mathrm{\infty }$.

For $j\ge 0$, truncate both linear sums at index $N$. Their product is bounded by $4$ and converges pointwise to $X_0{X}_j$, so dominated convergence [F2] passes expectations to the limit. The finite sums share exactly innovations with indices $-k=j-(k+j)$; their coefficient products are $2^{-k}{2}^{-(k+j)}$, while all other products have zero expectation by [F2]. The finite shared-coefficient sums tend to ${\sum }_{k\ge 0}{2}^{-k}{2}^{-(k+j)}$, giving ${\gamma }_j=\mathbb{E}⟨X_0,X_j{⟩}_H=\frac{4}{3}{2}^{-j}\mathrm{.}$ Symmetry gives ${\gamma }_{-j}={\gamma }_j$. Therefore the scalar long-run variance is ${\gamma }_0+2{\sum }_{j\ge 1}{\gamma }_j=\frac{4}{3}+\frac{8}{3}{\sum }_{j\ge 1}{2}^{-j}=4,$ and the operator is $4(e\otimes e)$.

Step 1.1 gives centering, a finite second moment and summable physical-dependence errors; step 2.1 gives the long-run covariance. Applying [F3] yields convergence of the normalized $H$-valued sums to a centered Gaussian with covariance $4(e\otimes e)$.