[proofplan] The proof is a conditional Gaussian regression argument. First we compute the conditional law of $X_t$ given $X_{t+1}$ and the observations $Y_{1:t}$, using the joint Gaussian law of $(X_t,X_{t+1})$ after the filtering pass. Then we condition this affine Gaussian representation on the full observation record $Y_{1:n}$ and use the Markov structure of the state-space model to replace future information by its effect on $X_{t+1}$. The resulting conditional mean and covariance are exactly the Rauch--Tung--Striebel backward recursions. [/proofplan] [step:Compute the backward conditional Gaussian regression] Fix $t \in \{1,\dots,n-1\}$. Let $\mathcal{F}_t := \sigma(Y_1,\dots,Y_t)$ and $\mathcal{F}_n := \sigma(Y_1,\dots,Y_n)$. By the Kalman filtering hypotheses, conditional on $\mathcal{F}_t$, the random vector \begin{align*} Z_t := \begin{pmatrix} X_t \\ X_{t+1} \end{pmatrix} \end{align*} is Gaussian with conditional mean \begin{align*} m_t := \begin{pmatrix} a_{t|t} \\ a_{t+1|t} \end{pmatrix} \end{align*} and conditional covariance matrix \begin{align*} \Sigma_t := \begin{pmatrix} P_{t|t} & P_{t|t}T_t^\top \\ T_tP_{t|t} & P_{t+1|t} \end{pmatrix}. \end{align*} Since $P_{t+1|t}$ is invertible, the conditional Gaussian regression formula gives \begin{align*} \mathbb{E}[X_t \mid X_{t+1},\mathcal{F}_t] &= a_{t|t}+P_{t|t}T_t^\top P_{t+1|t}^{-1}(X_{t+1}-a_{t+1|t}) \\ &= a_{t|t}+J_t(X_{t+1}-a_{t+1|t}), \end{align*} and \begin{align*} \operatorname{Cov}(X_t \mid X_{t+1},\mathcal{F}_t) &= P_{t|t}-P_{t|t}T_t^\top P_{t+1|t}^{-1}T_tP_{t|t} \\ &= P_{t|t}-J_tP_{t+1|t}J_t^\top. \end{align*} [guided] Fix $t \in \{1,\dots,n-1\}$ and define the observation sigma-algebras \begin{align*} \mathcal{F}_t &:= \sigma(Y_1,\dots,Y_t), \\ \mathcal{F}_n &:= \sigma(Y_1,\dots,Y_n). \end{align*} The forward Kalman filter gives a conditional Gaussian law for $X_t$ given $\mathcal{F}_t$ and for the one-step prediction $X_{t+1}$ given $\mathcal{F}_t$. Because the transition from $X_t$ to $X_{t+1}$ is linear Gaussian, the pair \begin{align*} Z_t := \begin{pmatrix} X_t \\ X_{t+1} \end{pmatrix} \end{align*} is also conditionally Gaussian given $\mathcal{F}_t$. Its conditional mean is \begin{align*} m_t := \begin{pmatrix} a_{t|t} \\ a_{t+1|t} \end{pmatrix}, \end{align*} and its conditional covariance is \begin{align*} \Sigma_t := \begin{pmatrix} P_{t|t} & \operatorname{Cov}(X_t,X_{t+1}\mid \mathcal{F}_t) \\ \operatorname{Cov}(X_{t+1},X_t\mid \mathcal{F}_t) & P_{t+1|t} \end{pmatrix}. \end{align*} By the hypothesis on the linear part of the state transition, \begin{align*} \operatorname{Cov}(X_t,X_{t+1}\mid \mathcal{F}_t)=P_{t|t}T_t^\top, \end{align*} and therefore \begin{align*} \Sigma_t = \begin{pmatrix} P_{t|t} & P_{t|t}T_t^\top \\ T_tP_{t|t} & P_{t+1|t} \end{pmatrix}. \end{align*} Now apply the conditional Gaussian regression formula to the block vector $(X_t,X_{t+1})$ conditional on $\mathcal{F}_t$. The required invertibility hypothesis is precisely the assumption that $P_{t+1|t}$ is invertible. Hence \begin{align*} \mathbb{E}[X_t \mid X_{t+1},\mathcal{F}_t] &= a_{t|t}+\operatorname{Cov}(X_t,X_{t+1}\mid \mathcal{F}_t)P_{t+1|t}^{-1}(X_{t+1}-a_{t+1|t}) \\ &= a_{t|t}+P_{t|t}T_t^\top P_{t+1|t}^{-1}(X_{t+1}-a_{t+1|t}). \end{align*} Since the smoothing gain is defined by \begin{align*} J_t := P_{t|t}T_t^\top P_{t+1|t}^{-1}, \end{align*} this becomes \begin{align*} \mathbb{E}[X_t \mid X_{t+1},\mathcal{F}_t] = a_{t|t}+J_t(X_{t+1}-a_{t+1|t}). \end{align*} The same Gaussian regression formula gives the conditional covariance \begin{align*} \operatorname{Cov}(X_t \mid X_{t+1},\mathcal{F}_t) &= P_{t|t} -\operatorname{Cov}(X_t,X_{t+1}\mid \mathcal{F}_t)P_{t+1|t}^{-1} \operatorname{Cov}(X_{t+1},X_t\mid \mathcal{F}_t) \\ &= P_{t|t}-P_{t|t}T_t^\top P_{t+1|t}^{-1}T_tP_{t|t}. \end{align*} Using $J_t=P_{t|t}T_t^\top P_{t+1|t}^{-1}$ and the symmetry of $P_{t+1|t}$, this is \begin{align*} \operatorname{Cov}(X_t \mid X_{t+1},\mathcal{F}_t) = P_{t|t}-J_tP_{t+1|t}J_t^\top. \end{align*} [/guided] [/step] [step:Condition the backward regression mean on the full observation record] The state-space Markov property gives \begin{align*} \mathbb{E}[X_t \mid X_{t+1},\mathcal{F}_n] = \mathbb{E}[X_t \mid X_{t+1},\mathcal{F}_t]. \end{align*} Taking conditional expectation with respect to $\mathcal{F}_n$ in the regression identity from the previous step yields \begin{align*} a_{t|n} &= \mathbb{E}[X_t \mid \mathcal{F}_n] \\ &= \mathbb{E}\!\left[\mathbb{E}[X_t \mid X_{t+1},\mathcal{F}_n]\mid \mathcal{F}_n\right] \\ &= \mathbb{E}\!\left[a_{t|t}+J_t(X_{t+1}-a_{t+1|t})\mid \mathcal{F}_n\right] \\ &= a_{t|t}+J_t(a_{t+1|n}-a_{t+1|t}). \end{align*} [guided] The future observations $Y_{t+1},\dots,Y_n$ affect $X_t$ only through their information about the later state $X_{t+1}$. Formally, the Markov property of the linear Gaussian state-space model implies \begin{align*} \mathbb{E}[X_t \mid X_{t+1},\mathcal{F}_n] = \mathbb{E}[X_t \mid X_{t+1},\mathcal{F}_t]. \end{align*} This is the point where the state-space structure is used: once $X_{t+1}$ and the past observations $\mathcal{F}_t$ are known, the future observations add no further direct information about $X_t$. Using the tower property of conditional expectation, we compute \begin{align*} a_{t|n} &= \mathbb{E}[X_t \mid \mathcal{F}_n] \\ &= \mathbb{E}\!\left[\mathbb{E}[X_t \mid X_{t+1},\mathcal{F}_n]\mid \mathcal{F}_n\right]. \end{align*} By the Markov property, the inner conditional expectation equals the backward regression mean already computed: \begin{align*} \mathbb{E}[X_t \mid X_{t+1},\mathcal{F}_n] = a_{t|t}+J_t(X_{t+1}-a_{t+1|t}). \end{align*} Therefore \begin{align*} a_{t|n} &= \mathbb{E}\!\left[a_{t|t}+J_t(X_{t+1}-a_{t+1|t})\mid \mathcal{F}_n\right] \\ &= a_{t|t}+J_t\left(\mathbb{E}[X_{t+1}\mid \mathcal{F}_n]-a_{t+1|t}\right). \end{align*} Since \begin{align*} a_{t+1|n}:=\mathbb{E}[X_{t+1}\mid \mathcal{F}_n], \end{align*} we obtain \begin{align*} a_{t|n}=a_{t|t}+J_t(a_{t+1|n}-a_{t+1|t}). \end{align*} [/guided] [/step] [step:Condition the backward regression covariance on the full observation record] Define the regression residual \begin{align*} \varepsilon_t := X_t-a_{t|t}-J_t(X_{t+1}-a_{t+1|t}). \end{align*} From the conditional Gaussian regression above, \begin{align*} \mathbb{E}[\varepsilon_t \mid X_{t+1},\mathcal{F}_t]=0, \qquad \operatorname{Cov}(\varepsilon_t \mid X_{t+1},\mathcal{F}_t) =P_{t|t}-J_tP_{t+1|t}J_t^\top. \end{align*} The same Markov conditional independence implies that $\varepsilon_t$ is conditionally independent of $\mathcal{F}_n$ given $(X_{t+1},\mathcal{F}_t)$. Hence \begin{align*} \operatorname{Cov}(\varepsilon_t \mid \mathcal{F}_n) = P_{t|t}-J_tP_{t+1|t}J_t^\top, \end{align*} and the cross-covariance between $\varepsilon_t$ and $X_{t+1}$ conditional on $\mathcal{F}_n$ is zero. Using the decomposition \begin{align*} X_t-a_{t|n} = \varepsilon_t+J_t(X_{t+1}-a_{t+1|n}), \end{align*} we obtain \begin{align*} P_{t|n} &= \operatorname{Cov}(X_t\mid \mathcal{F}_n) \\ &= \operatorname{Cov}(\varepsilon_t\mid \mathcal{F}_n) +J_t\operatorname{Cov}(X_{t+1}\mid \mathcal{F}_n)J_t^\top \\ &= P_{t|t}-J_tP_{t+1|t}J_t^\top+J_tP_{t+1|n}J_t^\top \\ &= P_{t|t}+J_t(P_{t+1|n}-P_{t+1|t})J_t^\top. \end{align*} [guided] Define the regression residual \begin{align*} \varepsilon_t := X_t-a_{t|t}-J_t(X_{t+1}-a_{t+1|t}). \end{align*} This residual is the part of $X_t$ that remains after the best affine prediction from $X_{t+1}$ and the information $\mathcal{F}_t$. From the conditional Gaussian regression computation, \begin{align*} \mathbb{E}[\varepsilon_t \mid X_{t+1},\mathcal{F}_t]=0 \end{align*} and \begin{align*} \operatorname{Cov}(\varepsilon_t \mid X_{t+1},\mathcal{F}_t) =P_{t|t}-J_tP_{t+1|t}J_t^\top. \end{align*} Because the joint law is Gaussian, this residual is conditionally independent of $X_{t+1}$ given $\mathcal{F}_t$. The state-space Markov property also says that the future observations give no additional direct information about this residual once $X_{t+1}$ and $\mathcal{F}_t$ are known. Thus conditioning on the full observation record preserves the residual covariance: \begin{align*} \operatorname{Cov}(\varepsilon_t \mid \mathcal{F}_n) = P_{t|t}-J_tP_{t+1|t}J_t^\top. \end{align*} Moreover, the conditional cross-covariance of $\varepsilon_t$ with $X_{t+1}$ given $\mathcal{F}_n$ is zero: \begin{align*} \operatorname{Cov}(\varepsilon_t,X_{t+1}\mid \mathcal{F}_n)=0. \end{align*} Now subtract the smoothing mean formula from the definition of $\varepsilon_t$. Since \begin{align*} a_{t|n}=a_{t|t}+J_t(a_{t+1|n}-a_{t+1|t}), \end{align*} we have \begin{align*} X_t-a_{t|n} &= X_t-a_{t|t}-J_t(a_{t+1|n}-a_{t+1|t}) \\ &= \varepsilon_t+J_t(X_{t+1}-a_{t+1|t})-J_t(a_{t+1|n}-a_{t+1|t}) \\ &= \varepsilon_t+J_t(X_{t+1}-a_{t+1|n}). \end{align*} Taking conditional covariance given $\mathcal{F}_n$ and using the vanishing cross-covariance gives \begin{align*} P_{t|n} &= \operatorname{Cov}(X_t\mid \mathcal{F}_n) \\ &= \operatorname{Cov}(\varepsilon_t+J_t(X_{t+1}-a_{t+1|n})\mid \mathcal{F}_n) \\ &= \operatorname{Cov}(\varepsilon_t\mid \mathcal{F}_n) +J_t\operatorname{Cov}(X_{t+1}\mid \mathcal{F}_n)J_t^\top. \end{align*} By definition, \begin{align*} P_{t+1|n}:=\operatorname{Cov}(X_{t+1}\mid \mathcal{F}_n), \end{align*} so \begin{align*} P_{t|n} &= P_{t|t}-J_tP_{t+1|t}J_t^\top+J_tP_{t+1|n}J_t^\top \\ &= P_{t|t}+J_t(P_{t+1|n}-P_{t+1|t})J_t^\top. \end{align*} [/guided] [/step] [step:Run the recursion backward from the terminal filtering distribution] At the terminal time $n$, the smoothing distribution equals the filtering distribution because no observations occur after time $n$: \begin{align*} a_{n|n}=\mathbb{E}[X_n\mid Y_{1:n}], \qquad P_{n|n}=\operatorname{Cov}(X_n\mid Y_{1:n}). \end{align*} Applying the two identities above successively for $t=n-1,n-2,\dots,1$ gives \begin{align*} a_{t|n} &= a_{t|t}+J_t(a_{t+1|n}-a_{t+1|t}), \\ P_{t|n} &= P_{t|t}+J_t(P_{t+1|n}-P_{t+1|t})J_t^\top, \end{align*} for every $t \in \{1,\dots,n-1\}$. This is the Rauch--Tung--Striebel smoothing recursion. [/step]