A Gaussian AR(1) sensor and its long-run variance

For each scalar $x$, $S_{\psi }(V_x)={\int }_{D}x{e}_1\cdot (e_1\mathrm{/}\lambda (D))d\lambda =x\lambda (D)\mathrm{/}\lambda (D)=x$. Thus the random-field sensor output $Y_n$ equals $X_n$.

Independence of ${ϵ}_{n+1}$ from the natural past and its stated law imply $\mathbb{P}(X_{n+1}\in A\mid \sigma (X_0,\dots ,X_n))=P(X_n,A)$ for every Borel $A$, with $P(x,\cdot )=N(\rho x,{\eta }^2)$. The stipulated stationary law $\nu$ is invariant in the sense of [F4].

Stationarity and the recursion give $\mathbb{E}[X_{n+1}^2]={\rho }^2\mathbb{E}[X_n^2]+{\eta }^2$, because ${ϵ}_{n+1}$ is centered and independent of $X_n$. Thus ${\gamma }_0=\mathbb{E}[X_n^2]={\eta }^2\mathrm{/}(1-{\rho }^2)$. Iterating the recursion gives $X_k={\rho }^k{X}_0+{\sum }_{j=1}^k{\rho }^{k-j}{ϵ}_j$; the innovation sum is centered and independent of $X_0$, so [F5] yields ${\gamma }_k={\rho }^k{\gamma }_0$ for every $k\ge 0$.

Since $\mathrm{\mid }\rho \mathrm{\mid }<1$, [F6] gives ${\sum }_{k\ge 1}\mathrm{\mid }{\gamma }_k\mathrm{\mid }={\gamma }_0{\sum }_{k\ge 1}\mathrm{\mid }\rho {\mathrm{\mid }}^k<\mathrm{\infty }$ and ${\gamma }_0+2{\sum }_{k\ge 1}{\gamma }_k=\frac{{\eta }^2}{1-{\rho }^2}(1+\frac{2\rho }{1-\rho })=\frac{{\eta }^2}{(1-\rho {)}^2}\mathrm{.}$ The last equality uses $1-{\rho }^2=(1-\rho )(1+\rho )$ and $\mathrm{\mid }\rho \mathrm{\mid }<1$, so $1+\rho >0$.

The centered sum $S_n:=n^{-1\mathrm{/}2}{\sum }_{j=0}^{n-1}{Y}_j$ is normally distributed because the finite vector $(X_0,\dots ,X_{n-1})$ is jointly Gaussian. Expanding its variance and counting the $n-k$ pairs at lag $k$ gives $v_n:=\mathrm{Var}(S_n)={\gamma }_0+2{\sum }_{k=1}^{n-1}(1-\frac{k}{n}){\gamma }_k\mathrm{.}$

On the counting-measure space $\mathbb{N}$, the summands ${\mathbf{1}}_{\{k<n\}}(1-k\mathrm{/}n){\gamma }_k$ converge pointwise to ${\gamma }_k$ and are dominated in absolute value by the summable sequence $\mathrm{\mid }{\gamma }_k\mathrm{\mid }$. By [F8], $v_n\to {\gamma }_0+2{\sum }_{k\ge 1}{\gamma }_k={\eta }^2\mathrm{/}(1-\rho {)}^2=:v>0$.

Let $G$ have law $N(0,1)$. Since $S_n$ is centered normal with variance $v_n$, its law equals that of $\sqrt{v_n}G$. For every bounded continuous $h:\mathbb{R}\to \mathbb{R}$, $h(\sqrt{v_n}G)\to h(\sqrt{v}G)$ pointwise and is bounded by $\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}$; [F8] gives convergence of expectations. By [F7], $S_n\Rightarrow N(0,v)$. The chain is stationary Markov, the covariance series is absolutely summable by step 3.1, and the CLT has now been proved, so [F9] identifies $v$ as its covariance long-run variance.