Finite-dependent functional moving average

The vector $e$ has norm one, so $X_t$ has mean zero and $\mathbb{E}\mathrm{\parallel }{X}_t{\mathrm{\parallel }}_H^2=\mathbb{E}({ϵ}_t+2{ϵ}_{t-1}{)}^2=5$ by [F2]. For $m=1$, the replacement changes only ${ϵ}_{-1}$ in $X_0$, so $X_0-X_{0,1}=2({ϵ}_{-1}-{ϵ}_{0,1,-1}^{\mathrm{\prime }})e$ and [F2] gives ${\delta }_1^2=4(1+1)=8$. For $m\ge 2$, both innovations on which $X_0$ depends are retained, so $X_{0,m}=X_0$ and ${\delta }_m=0$. Hence ${\sum }_{m\ge 1}{\delta }_m=2\sqrt{2}<\mathrm{\infty }$.

The lag-zero covariance is $5(e\otimes e)$. At lag one, the only nonzero product in $({ϵ}_0+2{ϵ}_{-1})({ϵ}_1+2{ϵ}_0)$ is $2{ϵ}_0^2$, so ${\mathbb{E}}_B[X_0\otimes X_1]=2(e\otimes e)$. Reversing the factors gives the same operator. At every lag $\mathrm{\mid }j\mathrm{\mid }\ge 2$ the innovation sets are disjoint, so independence and centering make the covariance zero. Therefore the long-run covariance sum is $(5+2+2)(e\otimes e)=9(e\otimes e)$.

Fix a cut $a\in \mathbb{Z}$. The variables $({ϵ}_j{)}_{j\in \mathbb{Z}}$ are independent by the iid assumption, so their generated sigma-algebras form an independent family by Independent random elements and Independent sigma-algebras and independent events. Each $X_s$ is measurable with respect to $\sigma ({ϵ}_s,{ϵ}_{s-1})$. Thus $\sigma (X_s:s\le a)\subseteq \sigma ({ϵ}_j:j\le a)$, while $\sigma (X_s:s\ge a+2)\subseteq \sigma ({ϵ}_j:j\ge a+1)$. By [F4] the two source sigma-algebras are independent; their sub-sigma-algebras generated by the process groups inherit independence. The process is therefore $1$-dependent under the definition in L2-m-approximable causal Bernoulli shifts.

The moment, centering and summability conditions were verified in step 1.1, and step 1.2 gives the covariance required by [F3]. Applying [F3] proves the asserted sample-sum convergence; the Gaussian limit is allowed to have finite rank, as it does here. This conclusion concerns the $H$-valued sample sums.