Long-run covariance of an L2-m-approximable shift is trace class

Put $\sigma =(\mathbb{E}\mathrm{\parallel }{X}_0{\mathrm{\parallel }}_H^2{)}^{1\mathrm{/}2}$. For each $j\ge 0$, the map $(x,y)\mapsto x\otimes y$ is continuous into $S_1(H)$, since [F2] and the triangle inequality give $\mathrm{\parallel }x\otimes y-x^{\mathrm{\prime }}\otimes y^{\mathrm{\prime }}{\mathrm{\parallel }}_1\le \mathrm{\parallel }x-x^{\mathrm{\prime }}{\mathrm{\parallel }}_H\mathrm{\parallel }y{\mathrm{\parallel }}_H+\mathrm{\parallel }{x}^{\mathrm{\prime }}{\mathrm{\parallel }}_H\mathrm{\parallel }y-y^{\mathrm{\prime }}{\mathrm{\parallel }}_H\mathrm{.}$ Likewise $(x,y)\mapsto x⊠y$ is continuous into $K$ by [F5]. Strong simple approximants to $X_0$ and $X_j$ therefore make both tensor-valued maps strongly measurable. Their norms have expectations at most $(\mathbb{E}\mathrm{\parallel }{X}_0{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}(\mathbb{E}\mathrm{\parallel }{X}_j{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}={\sigma }^2$ by Cauchy–Schwarz and stationarity.

Fix $j\ge 1$ and put $Z_j=X_{j,j-1}$. For any $h,g\in H$, the scalar maps $x\mapsto ⟨x,g{⟩}_H$ and $y\mapsto ⟨h,y{⟩}_H$ are continuous. By [F1] and [F4] their values at $X_0$ and $Z_j$ are independent, and the first has mean zero. The bounded linear functional $L_{h,g}(T)=⟨Th,g{⟩}_H$ on $S_1(H)$ commutes with Bochner integration by [F2]–[F3], so $L_{h,g}({\mathbb{E}}_B[X_0\otimes Z_j])=\mathbb{E}[⟨h,Z_j{⟩}_H⟨X_0,g{⟩}_H]=0.$ For each $h$, taking $g={\mathbb{E}}_B[X_0\otimes Z_j]h$ shows that this vector has zero squared norm, so ${\mathbb{E}}_B[X_0\otimes Z_j]=0$; reversing the tensor factors gives ${\mathbb{E}}_B[Z_j\otimes X_0]=0$ as well. For the kernel expectation, pair ${\mathbb{E}}_B[X_0⊠Z_j]$ with any product kernel $g⊠h$. By [F3] and [F5], the scalar pairing is $\mathbb{E}[⟨X_0,g{⟩}_H⟨Z_j,h{⟩}_H]$, which is zero by independence and centering. Such product kernels span a dense subspace of $K$ by [F5], so ${\mathbb{E}}_B[X_0⊠Z_j]=0$; similarly ${\mathbb{E}}_B[Z_j⊠X_0]=0$.

By step 1.2, stationarity, and [F1], for every $j\ge 1$, $\begin{array}{rl}{\displaystyle \mathrm{\parallel }{\mathbb{E}}_B[X_0\otimes X_j]{\mathrm{\parallel }}_1}& {\displaystyle =\mathrm{\parallel }{\mathbb{E}}_B[X_0\otimes (X_j-Z_j)]{\mathrm{\parallel }}_1}\\ {\displaystyle }& {\displaystyle \le \mathbb{E}[\mathrm{\parallel }{X}_0{\mathrm{\parallel }}_H\mathrm{\parallel }{X}_j-Z_j{\mathrm{\parallel }}_H]\le \sigma {\delta }_{j-1}\mathrm{.}}\end{array}$ The same bound holds for ${\mathbb{E}}_B[X_j\otimes X_0]$. It also holds for $\mathrm{\parallel }{\kappa }_j{\mathrm{\parallel }}_K$ and $\mathrm{\parallel }{\tilde{\kappa }}_j{\mathrm{\parallel }}_K$ by [F3] and [F5]. Since ${\delta }_0\le 2\sigma$ and ${\sum }_{m\ge 1}{\delta }_m<\mathrm{\infty }$, both series of trace-class terms and product-kernel terms are absolutely summable. The Banach property of $S_1(H)$ in [F2] and the Hilbert completeness of $K$ in [A1], [F5] give their norm limits.

For $n\ge 1$, set $S_n={\sum }_{t=1}^n{X}_t$ and ${\mathrm{\Gamma }}_n=n^{-1}{\mathbb{E}}_B[S_n\otimes S_n]$. Step 1.1 makes this Bochner expectation well-defined. It is positive and self-adjoint because for every $h\in H$, $⟨{\mathrm{\Gamma }}_{n}h,h{⟩}_H=n^{-1}\mathbb{E}⟨S_n,h{⟩}_H^2\ge 0.$ Expanding the finite sum and using stationarity gives ${\mathrm{\Gamma }}_n={\mathbb{E}}_B[X_0\otimes X_0]+{\sum }_{j=1}^{n-1}(1-\frac{j}{n})({\mathbb{E}}_B[X_0\otimes X_j]+{\mathbb{E}}_B[X_j\otimes X_0])\mathrm{.}$ Absolute summability from step 2.1 implies ${\mathrm{\Gamma }}_n\to \mathrm{\Gamma }$ in trace norm: for any fixed finite lag cutoff the coefficients tend to $1$, and the remaining trace-norm tail is uniformly bounded by the tail of the absolutely convergent series. Since $\mathrm{\parallel }T\mathrm{\parallel }\le \mathrm{\parallel }T{\mathrm{\parallel }}_1$, positivity and self-adjointness pass to the limit. Thus $\mathrm{\Gamma }$ is positive self-adjoint and trace class.

For $f,g\in H$, define the bounded linear functional ${\mathrm{\Lambda }}_{f,g}(k)=⟨T_{k}f,g{⟩}_H$ on $K$; [F5] gives $\mathrm{\mid }{\mathrm{\Lambda }}_{f,g}(k)\mathrm{\mid }\le \mathrm{\parallel }k{\mathrm{\parallel }}_K\mathrm{\parallel }f{\mathrm{\parallel }}_H\mathrm{\parallel }g{\mathrm{\parallel }}_H$. For each $j\ge 0$, [F5] and [F3] therefore give ${\mathrm{\Lambda }}_{f,g}({\kappa }_j)=\mathbb{E}⟨(X_0\otimes X_j)f,g{⟩}_H=⟨({\mathbb{E}}_B[X_0\otimes X_j])f,g{⟩}_H,$ and the same identity holds for the reversed terms. Passing to the absolutely convergent kernel and trace-norm series in step 2.1 yields $⟨T_{c}f,g{⟩}_H=⟨\mathrm{\Gamma }f,g{⟩}_H$ for every $f,g$. Hence $T_c=\mathrm{\Gamma }$, and $c$ is exactly the design's $L^2$ covariance-kernel equivalence class.