General finite-horizon state-space models by kernels

Assume AC (The Axiom of Choice) and fix a finite horizon $T\ge 0$. For $0\le t\le T$, let $(E_t,{\mathcal{E}}_t)$ be a standard-Borel state space; for $1\le t\le T$, let $(F_t,{\mathcal{F}}_t)$ be a standard-Borel observation space (Standard Borel spaces), meaning a measurable space bimeasurably isomorphic to a Polish space. A probability kernel has probability-measure sections and measurable evaluations as in Measure kernel and probability kernel. Specify an initial probability measure ${\mu }_0$ on $E_0$ (a measure of total mass one, as in Probability measures and probability spaces), transition probability kernels $K_t$ from $E_{t-1}$ to $E_t$, and observation probability kernels $G_t$ from $E_t$ to $F_t$, for $1\le t\le T$.

The trajectory space, with the finite product sigma-algebra generated by measurable coordinate rectangles (The product sigma-algebra and its finite iterates), is ${\mathcal{Z}}_T=E_0\times E_1\times F_1\times \cdots \times E_T\times F_T,$ with ${\mathcal{Z}}_0=E_0$. Its finite-horizon law $P_T$ is the unique probability law whose ordered iterated kernel integral is ${\mu }_0(d{x}_0){\prod }_{t=1}^T{K}_t(x_{t-1},d{x}_t)G_t(x_t,d{y}_t)\mathrm{.}$ The product notation means successive integration in the displayed order; it does not assert the existence of densities. Existence and uniqueness under AC are proved in Finite-horizon state-space kernel law exists and is unique ↗. Every finite product trajectory space is standard Borel by Finite products of standard Borel spaces are standard Borel.

The initial and transition laws specify the state evolution, while each $G_t$ specifies the observation law given the current state in this model. State evolution by itself therefore does not specify a joint state-observation law. No density or time-homogeneity assumption is imposed, and no strict stationarity assumption (shift invariance of all finite-dimensional laws) is required (Strict and weak stationarity for integer-indexed time series). This is a single law, not a parameter-indexed family. The model contract records a proposed law and makes no claim that it adequately describes an application (Scientific statistical model contracts).