Linear-Gaussian state-space models with independent noises

Assume AC (The Axiom of Choice) and fix a finite horizon $T\ge 0$. Let the state dimension at time $t$ be a positive integer $d_t$ for $0\le t\le T$, and the observation dimension at time $t$ be a positive integer $p_t$ for $1\le t\le T$. Choose deterministic matrices $F_t\in {\mathbb{R}}^{d_t\times d_{t-1}}$ and $H_t\in {\mathbb{R}}^{p_t\times d_t}$, a mean $m_0\in {\mathbb{R}}^{d_0}$, and symmetric positive-semidefinite matrices, meaning $v^{T}Pv\ge 0$ for every real vector $v$ of matching dimension, $P_0\in {\mathbb{R}}^{d_0\times d_0}$, $Q_t\in {\mathbb{R}}^{d_t\times d_t}$, and $R_t\in {\mathbb{R}}^{p_t\times p_t}$. A linear-Gaussian state-space model with independent noises is the general finite-horizon kernel model (General finite-horizon state-space models by kernels) with initial law ${\mu }_0=N_{d_0}(m_0,P_0)$ and transition and observation kernels $K_t(x,\cdot )=N_{d_t}(F_{t}x,Q_t),G_t(x,\cdot )=N_{p_t}(H_{t}x,R_t)\mathrm{.}$ All Gaussian laws use the convention of Multivariate normal law, including singular covariance, which allows singular covariance matrices.

Its independent-noise representation consists of random vectors $x_0,q_1,r_1,\dots ,q_T,r_T$ on a common probability space, with $x_0\sim N_{d_0}(m_0,P_0)$, $q_t\sim N_{d_t}(0,Q_t)$, $r_t\sim N_{p_t}(0,R_t)$, all mutually independent in the sense of Independent random elements (their generated sigma-algebras are independent), and equations $x_t=F_t{x}_{t-1}+q_t,y_t=H_t{x}_t+r_t(1\le t\le T)\mathrm{.}$ The local path-law theorem proves that every allowed parameter tuple has such a finite representation, that its induced kernels are the displayed $K_t,G_t$, and that its path law is the general kernel-product law.

This definition imposes the stated noise law and independence assumptions; it does not establish their empirical adequacy. It supplies no filtering algorithm, posterior recursion, or convergence claim. Any separate formula that inverts an observed innovation covariance must assume that covariance is positive definite at the point of inversion.