Localized Filtering Lower Bound from a 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 van Trees certificate for the data $(X,\mathcal{G},\mathcal{H},c)$ with tolerance $\epsilon$ and value $\varkappa$: there are a natural number $d\ge1$, 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 $\Theta=(\Theta_{1},\dots,\Theta_{d})$, and vectors $\alpha,z\in\mathbb{R}^{d}$ with $z$ nonzero, 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 random variables $\Theta_{1},\dots,\Theta_{d}$, the map $\mathsf{D}$ and the measure $\varrho_{0}$ satisfy hypotheses (i)--(iv) of the multivariate van Trees inequality, with $l$ there replaced by $d$ and with data space $(\mathsf{Y},\mathcal{Y})$ carrying $\varrho_{0}$; write $\mathcal{I}$ for the resulting van Trees information matrix, which is positive definite by hypothesis (iv) there;

(C3) $\bigl\lVert\alpha\cdot\Theta-\mathbf{1}_{\mathcal{H}}\,(c\cdot X)\bigr\rVert_{2}\le\epsilon$, where $\alpha\cdot\Theta=\sum_{j=1}^{d}\alpha_{j}\Theta_{j}$;

(C4) $(\alpha\cdot z)^{2}\ \ge\ \varkappa\,\bigl(z\cdot(\mathcal{I}z)\bigr)$, with the dot product and the matrix-vector product on the Euclidean space $\mathbb{R}^{d}$.

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

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

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