The perturbed-observation EnKF converges to the Kalman filter at fixed horizon

Couple the EnKF members with a reference ensemble by using the same iid initial vectors, process-error draws, and perturbed observations. At time $0$, $X_{N,i}^{a,0}=U_{N,i}^{a,0}$, so the paired row is exchangeable and the memberwise difference is zero in every finite $L^p$.

Fix $k\in \{1,\dots ,T\}$ and assume the induction hypothesis: the paired analysis row $(X_{N,i}^{a,k-1},U_{N,i}^{a,k-1}{)}_{i=1}^N$ is exchangeable and $\mathrm{\parallel }{X}_{N,1}^{a,k-1}-U_{N,1}^{a,k-1}{\mathrm{\parallel }}_p\to 0$ for every finite $p$. Couple the forecasts with the same $q_{N,i}^k$; then $X_{N,i}^{f,k}-U_{N,i}^{f,k}=F_k(X_{N,i}^{a,k-1}-U_{N,i}^{a,k-1})$. Thus forecast differences tend to zero in every finite $L^p$. The independent iid process-error and observation-perturbation rows have permutation-invariant laws and are independent of the paired analysis row; their product with that row is permutation-invariant, and the same forecast map is applied to each member. Hence the forecast pair row with its perturbations is exchangeable. By [F8] and [F1], each reference forecast is Gaussian, with mean and covariance $m^f=F_k{m}_{k-1\mathrm{\mid }k-1}$ and $P^f=F_k{P}_{k-1\mathrm{\mid }k-1}{F}_k^T+Q_k$ as in [F3]; independence across members follows from the independent per-member inputs.

Apply [L2] to the forecast pair row with $p=4$. The reference forecast covariance is $P_{k\mathrm{\mid }k-1}$ by [F1] and [F3], so $C_N^{f,k}\to P_{k\mathrm{\mid }k-1}$ in probability. Since $R_k\succ 0$, both $S_N=H_k{C}_N^{f,k}{H}_k^T+R_k$ and $S=H_k{P}_{k\mathrm{\mid }k-1}{H}_k^T+R_k$ dominate $R_k$ in quadratic form. [F5] bounds both inverse operator norms by $1\mathrm{/}{\lambda }_{\min }(R_k)$. The identity $S_N^{-1}-S^{-1}=S_N^{-1}(S-S_N)S^{-1}$ therefore gives $K_N^k=C_N^{f,k}{H}_k^T{S}_N^{-1}\to K_k=P_{k\mathrm{\mid }k-1}{H}_k^T{S}^{-1}$ in probability.

Let $D_{N,i}^k=y_k+{\eta }_{N,i}^k$. Subtracting the reference update from the EnKF update yields $X_{N,i}^{a,k}-U_{N,i}^{a,k}=(I-K_N^k{H}_k)(X_{N,i}^{f,k}-U_{N,i}^{f,k})+(K_N^k-K_k)(D_{N,i}^k-H_k{U}_{N,i}^{f,k})\mathrm{.}$ The first factor in the first product is bounded in probability because $K_N^k\to K_k$ in probability. The first factor in the second product tends to zero in probability, while $D_{N,i}^k-H_k{U}_{N,i}^{f,k}$ has a fixed finite-moment Gaussian law. Splitting on whether this latter norm exceeds an arbitrary $M$ shows the product tends to zero in probability. The same bounded-in-probability argument for the first product gives $X_{N,1}^{a,k}-U_{N,1}^{a,k}\to 0$ in probability. By [L1], the updated paired row is exchangeable.

The uniform moment bounds [L4] and the Gaussian reference law [L3] give a uniform bound on $\mathbb{E}\mathrm{\parallel }{X}_{N,1}^{a,k}-U_{N,1}^{a,k}{\mathrm{\parallel }}^s$ for every finite $s$. Apply [L5] with any $s>p$ to the probability convergence in step 4.1; the memberwise difference tends to zero in $L^p$ for every finite $p$. Together, the preceding steps establish the induction hypothesis at time $k$.

At any fixed $k$, the paired row is exchangeable, the reference members are iid with law ${\pi }_k$ by [L3], and [F4] identifies that Kalman law with the exact filtering law at the fixed data. At $k=0$, the memberwise difference is identically zero by step 1.1, hence tends to zero in $L^4$; for $k\ge 1$, it tends to zero in $L^4$ by step 5.1. The sample-moment result [L2] gives the claimed mean and covariance convergence. For bounded Lipschitz $\phi$, the empirical-average difference is at most $\mathrm{Lip}(\phi )N^{-1}{\sum }_i\mathrm{\parallel }{X}_{N,i}^{a,k}-U_{N,i}^{a,k}\mathrm{\parallel }$; its expectation tends to zero by exchangeability and the $L^1$ convergence in step 1.1 when $k=0$ and step 5.1 when $k\ge 1$, so the pointwise bound $\mathbb{P}(W\ge t)\le t^{-1}\mathbb{E}W$ from [F7] makes it vanish in probability. For the iid reference average, independence [F6] makes all distinct centered cross terms have expectation zero, so its variance is $\mathrm{Var}(\phi (U_{N,1}^{a,k}))\mathrm{/}N\le \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}^2\mathrm{/}N$. Apply [F7] to its squared centered deviation at threshold ${ϵ}^2$ to obtain convergence in probability to $\int \phi d{\pi }_k$. This proves the limits for each fixed $k$.

Step 1.1 is the base case, and for every $k\ge 1$ the four induction steps establish the induction hypothesis at time $k$ from time $k-1$. Thus the coupling bounds hold through $T$, and step 6.1 gives the three asserted limits for every fixed $k\le T$.