Localized Filtering Lower Bound from a Mixture-Weight van Trees Certificate

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$, let $l\ge1$ be a natural number, and let $X=(X^{1},\dots,X^{l})$ be a tuple of square-integrable random variables on it. For each $\gamma\in\{1,\dots,l\}$ fix a conditional expectation $M^{\gamma}$ of $X^{\gamma}$ given $\mathcal{G}$, which exists by the existence and uniqueness theorem, and set $\varepsilon^{\gamma}=X^{\gamma}-M^{\gamma}$, which is square-integrable by the closure properties of square-integrability, each $M^{\gamma}$ being square-integrable by condition (ii) of the conditional-expectation definition. Let $c$ be a vector of the Euclidean space $\mathbb{R}^{l}$, let $\mathcal{H}\in\mathcal{G}$ be an event, and write $\mathbf{1}_{\mathcal{H}}$ for its indicator, $c\cdot X=\sum_{\gamma=1}^{l}c^{\gamma}X^{\gamma}$, $c\cdot M=\sum_{\gamma=1}^{l}c^{\gamma}M^{\gamma}$ and $c\cdot\varepsilon=\sum_{\gamma=1}^{l}c^{\gamma}\varepsilon^{\gamma}$, all square-integrable by the closure properties of square-integrability. Write $\mathbb{E}$ for the expectation, $\lVert U\rVert_{2}$ for the mean-square norm, and $\sqrt{\cdot}$ for the nonnegative square root. Let $\epsilon\ge0$ and $\varkappa\ge0$ be real numbers.

Assume the following mixture-weight van Trees certificate for the data $(X,\mathcal{G},\mathcal{H},c)$ with tolerance $\epsilon$ and value $\varkappa$: there are natural numbers $d\ge1$ and $n\ge1$, a measurable space $(\mathsf{Y},\mathcal{Y})$, a $\sigma$-finite measure $\varrho_{0}$ on it, a map $\mathsf{D}:\Omega\to\mathsf{Y}$ measurable with respect to $\mathcal{F}$ and $\mathcal{Y}$, square-integrable random variables $\Theta_{1},\dots,\Theta_{d}$ with $\Theta=(\Theta_{1},\dots,\Theta_{d})$, a function $p:\mathbb{R}^{d}\times\mathsf{Y}\to[0,\infty)$, vectors $\alpha$, $z$, $a_{1},\dots,a_{n}$ in $\mathbb{R}^{d}$, and a map $G:\mathsf{Y}\to\mathbb{R}$, such that:

(C1) every $\mathcal{G}$-measurable square-integrable random variable is almost surely equal to $g(\mathsf{D})=g\circ\mathsf{D}$ for some $g:\mathsf{Y}\to\mathbb{R}$ measurable with respect to $\mathcal{Y}$ and the Borel $\sigma$-algebra for which $g(\mathsf{D})$ is square-integrable;

(C2$'$) the probability space $(\Omega,\mathcal{F},\mathbb{P})$, the natural number $d$ (in the role of $l$), the measurable space $(\mathsf{Y},\mathcal{Y})$ carrying $\varrho_{0}$ (in the roles of the data space $(Y,\mathcal{G})$ and its $\sigma$-finite measure $\mu$ of the two items cited next; the letter $\mathcal{G}$ of those items denotes their data-space $\sigma$-algebra, not the sub-$\sigma$-algebra $\mathcal{G}$ of this lemma), the random variables $\Theta_{1},\dots,\Theta_{d}$ and the map $\mathsf{D}$ satisfy hypothesis (i) of the multivariate van Trees inequality with the function $p$ as the density there (that is, $p$ is measurable with respect to the product $\sigma$-algebra of hypothesis (i) and the Borel $\sigma$-algebra, and the joint law of $(\Theta,\mathsf{D})$ is the measure with density $p$ with respect to the product of Lebesgue measure on $\mathbb{R}^{d}$ and $\varrho_{0}$), and hypotheses (ii) and (iii) of that theorem and hypothesis (iv$'$) of the mixture-weight van Trees lemma, all formed with this $p$; write $\mathcal{I}_{z}$ for the mixture-weight information of that lemma in the direction $z$ for the shifts $a_{1},\dots,a_{n}$, a real number in $[0,\infty)$ (finite by claim 2 of that lemma, and nonnegative as the integral of a nonnegative function);

(C3) the map $G$ is measurable with respect to $\mathcal{Y}$ and the Borel $\sigma$-algebra, $G(\mathsf{D})=G\circ\mathsf{D}$ is square-integrable, and

$$\bigl\lVert\alpha\cdot\Theta+G(\mathsf{D})-\mathbf{1}_{\mathcal{H}}\,(c\cdot X)\bigr\rVert_{2}\le\epsilon,$$

where $\alpha\cdot\Theta=\sum_{j=1}^{d}\alpha_{j}\Theta_{j}$;

(C4$'$) $\alpha\cdot z>0$ and

$$\alpha\cdot z\ \ge\ \sqrt{\mathcal{I}_{z}}\,\Bigl(\max_{1\le j\le n}|\alpha\cdot a_{j}|+\sqrt{\varkappa}\Bigr),$$

with the dot product on the Euclidean space $\mathbb{R}^{d}$ and the absolute value.

(The condition $\lVert\alpha\cdot\Theta-\mathbf{1}_{\mathcal{H}}(c\cdot X)\rVert_{2}\le\epsilon$ is the special case of (C3) in which $G$ is the zero map, which is trivially measurable with $G(\mathsf D)=0$ square-integrable.)

Then the following hold.

1. (Localization of the filtering error.) The random variables $\mathbf{1}_{\mathcal{H}}(c\cdot X)$ and $\mathbf{1}_{\mathcal{H}}(c\cdot M)$ are square-integrable, $\mathbf{1}_{\mathcal{H}}(c\cdot M)$ is a conditional expectation of $\mathbf{1}_{\mathcal{H}}(c\cdot X)$ given $\mathcal{G}$, and

$$\mathbb{E}\Bigl[\mathbf{1}_{\mathcal{H}}\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}c^{\gamma}c^{\delta}\,\varepsilon^{\gamma}\varepsilon^{\delta}\Bigr]\ =\ \mathbb{E}\bigl[\mathbf{1}_{\mathcal{H}}\,(c\cdot\varepsilon)^{2}\bigr].$$

2. (Certified lower bound.) If $\sqrt{\varkappa}\ge\epsilon$ then

$$\mathbb{E}\bigl[\mathbf{1}_{\mathcal{H}}\,(c\cdot\varepsilon)^{2}\bigr]\ \ge\ \bigl(\sqrt{\varkappa}-\epsilon\bigr)^{2}.$$

Both quantities are unchanged if the conditional expectations $M^{\gamma}$ are replaced by any other conditional expectations of the $X^{\gamma}$ given $\mathcal{G}$.