Fixed-interval Gaussian smoothing and RTS recursion

Fix a history $h=y_{1:k}$. By [F3], the conditional law of $X_k$ is $N(m_{k\mathrm{\mid }k}(h),P_{k\mathrm{\mid }k})$. The next process noise is independent of the past by [F1], so [F4] extends this filtering kernel by $q_{k+1}$ and pushes it forward through $(x,q)\mapsto (x,F_{k+1}x+q)$. By [F5], $(X_k,q_{k+1})$ is jointly Gaussian, and the projection characterization in [F10] makes its linear image $(X_k,X_{k+1})$ jointly Gaussian. Its mean is $(m_{k\mathrm{\mid }k},F_{k+1}{m}_{k\mathrm{\mid }k})$, its state covariance is $P_{k\mathrm{\mid }k}$, its cross-covariance is $P_{k\mathrm{\mid }k}{F}_{k+1}^T$, and its next-state covariance is $P_{k+1\mathrm{\mid }k}$. The latter block is positive definite by hypothesis. Apply [F6] with that observed block: the conditional mean and covariance are exactly the displayed $Q_k$ parameters, so $V_k⪰0$. The history mean is affine by [F2], and representing each Gaussian section as a translate of fixed $N(0,V_k)$ plus [F8] gives joint measurability in $(h,x_{k+1})$. Thus $Q_k$ is an everywhere probability kernel.

Write $H=Y_{1:k}$, $V=X_{k+1}$, and $Z=Y_{k+1:T}$. By [F7], the full path law factors by taking the law of $(H,X_k)$, applying the transition kernel at $k+1$, and then a future observation kernel $R(V,dz)$ formed by the remaining transition and observation kernels. The first two factors disintegrate as ${\pi }_k(h,d{x}_k)K_{k+1}(x_k,dv)={\rho }_k(h,dv)Q_k(h,v,d{x}_k)$ by step 1.1, where ${\rho }_k$ is the one-step predictive law. Substitution gives the joint factorization as the law of $H$ followed by ${\rho }_k$, $Q_k$, and $R$. The future kernel depends on the past only through $V$, and its iterated integrals are measurable by [F8].

For any Borel set $A$ of $X_k$ and any Borel set $D$ in the product space of $(H,V,Z)$, integrate $1_A(x_k)1_D(h,v,z)$ in the factorization from step 2.1. This gives $\mathbb{P}(X_k\in A,(H,V,Z)\in D)={\int }_D{Q}_k(h,v,A){\mathbb{P}}_{(H,V,Z)}(dh,dv,dz)$. This is the RCD event identity for $Q_k$ given $(H,V,Z)$; its kernel properties were established in step 1.1. Since $(H,Z)=Y_{1:T}$, it is also a version of the conditional law given $(Y_{1:T},X_{k+1})$. The factorization uses model kernels rather than observation densities, and the only inverse is of $P_{k+1\mathrm{\mid }k}\succ 0$.

At time $T$, the conditioning data for the forward filter are exactly $y_{1:T}$. By [F3], the terminal filter kernel $N(m_{T\mathrm{\mid }T},P_{T\mathrm{\mid }T})$ is an everywhere conditional-law version of $X_T$ given these data, with $P_{T\mathrm{\mid }T}⪰0$. It is therefore the base case for the fixed-interval smoothing kernels in [F4].

Suppose the Gaussian kernel ${\psi }_{k+1\mathrm{\mid }T}(y,\cdot )$ is an RCD of $X_{k+1}$ given $Y_{1:T}=y$. Define ${\psi }_{k\mathrm{\mid }T}(y,A)=\int Q_k(y_{1:k},x,A){\psi }_{k+1\mathrm{\mid }T}(y,dx)$ for each Borel state set $A$. This is a probability kernel by [F5]–[F6]. To verify its RCD identity, define $\nu (D)=\int \text{\hspace{0.17em}⁣}\int 1_D(y,x){\psi }_{k+1\mathrm{\mid }T}(y,dx){\mathbb{P}}_Y(dy)$ for Borel $D$ in the product of data and state spaces. Measurability follows from [F6]; for pairwise disjoint $D_n$, monotone convergence gives countable additivity, and $\nu$ has total mass one. Its rectangle values equal those of the actual joint law of $(Y_{1:T},X_{k+1})$ by the RCD identity for ${\psi }_{k+1\mathrm{\mid }T}$. By [F8], the measures agree on every Borel set. Thus for any Borel data set $B$ and state set $A$, ${\int }_B{\psi }_{k\mathrm{\mid }T}(y,A){\mathbb{P}}_Y(dy)={\int }_{B\times {\mathbb{R}}^{d_{k+1}}}{Q}_k(y_{1:k},x,A){\mathbb{P}}_{(Y_{1:T},X_{k+1})}(dy,dx)=\mathbb{P}(X_k\in A,Y_{1:T}\in B)$, where the last equality is [F5] applied to the conditioning event $B\times {\mathbb{R}}^{d_{k+1}}$. Hence ${\psi }_{k\mathrm{\mid }T}$ is the required RCD.

Write $P^{-}=P_{k+1\mathrm{\mid }k}$ and $V_k=P_{k\mathrm{\mid }k}-C_k{P}^{-}{C}_k^T$ as in [F5]. By [F13], row substitution gives $L_k{U}_k=F_{k+1}{P}_{k\mathrm{\mid }k}$ and $L_k^T{C}_k^T=U_k$. Since $P^{-}=L_k{L}_k^T$, these equations yield $P^{-}{C}_k^T=F_{k+1}{P}_{k\mathrm{\mid }k}$ and hence $C_k=P_{k\mathrm{\mid }k}{F}_{k+1}^T(P^{-}{)}^{-1}$, exactly the gain in [F5]. On the product probability space carrying independent $U\sim N(0,V_k)$ and $W\sim N(m_{k+1\mathrm{\mid }T},P_{k+1\mathrm{\mid }T})$, the integral in step 2.1 is the law of $m_{k\mathrm{\mid }k}+C_k(W-m_{k+1\mathrm{\mid }k})+U$ by [F9]. The covariance matrices of $U$ and $W$ are positive semidefinite by [F5] and the induction hypothesis, so both Gaussian laws exist by [A1, F11]. The product coordinates are independent by [F12]; [F10] makes their stack jointly Gaussian, and its affine image is Gaussian by the projection characterization in [F11]. Independence makes the cross-covariance terms vanish, so its mean is $m_{k\mathrm{\mid }k}+C_k(m_{k+1\mathrm{\mid }T}-m_{k+1\mathrm{\mid }k})$ and its covariance is $V_k+C_k{P}_{k+1\mathrm{\mid }T}{C}_k^T=P_{k\mathrm{\mid }k}+C_k(P_{k+1\mathrm{\mid }T}-P^{-})C_k^T$. This is exactly the smoother update in the statement and algorithm. The first covariance expression is positive semidefinite, so the updated covariance is positive semidefinite as well.

Starting from step 1.1, apply steps 2.1–3.1 for $k=T-1,T-2,\dots ,0$. The induction constructs an everywhere smoothing kernel at each time and identifies its mean and covariance with the algorithm output. When $T=0$ only the base case is needed. Positive definiteness is used only for the forecast inverses in [F5] and the forward innovation solves in [F2]; no singular forecast is inverted.