Uniform finite-horizon moment bounds for perturbed-observation EnKF ensembles

By [A1] and [F2]–[F3], each Gaussian vector is an affine image of finitely many standard normals. For every finite $p$, the integral of $\mathrm{\mid }x{\mathrm{\mid }}^p{e}^{-x^2\mathrm{/}2}$ is finite because for sufficiently large $\mathrm{\mid }x\mathrm{\mid }$, $\mathrm{\mid }x{\mathrm{\mid }}^p\le e^{x^2\mathrm{/}4}$ and $e^{-x^2\mathrm{/}4}$ is integrable; bounded intervals contribute a finite integral. Finite-dimensional triangle inequalities then give finite moments for each vector input, with laws independent of $N$ and $i$. This proves the asserted state bound at $k=0$.

Assume the induction hypothesis that at time $k-1$ the state moments are bounded uniformly in $N,i$ at every finite order. The forecast formula in [F1] is a fixed linear map plus a Gaussian process-error draw. The triangle inequality and $(a+b{)}^p\le 2^{p-1}(a^p+b^p)$ for $a,b\ge 0$ therefore give uniform bounds at every order for the forecast members $X_{N,i}^{f,k}$.

Write the forecast sample covariance as $C_N=N^{-1}{\sum }_i{X}_i{X}_i^T-\bar{X}{\bar{X}}^T$. Since $\mathrm{\parallel }x{x}^T{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=\mathrm{\parallel }x{\mathrm{\parallel }}^2$, Jensen's inequality for finite averages bounds the $p$th moment of the first term by $N^{-1}{\sum }_i\mathbb{E}\mathrm{\parallel }{X}_i{\mathrm{\parallel }}^{2p}$ and bounds $\mathbb{E}\mathrm{\parallel }\bar{X}{\mathrm{\parallel }}^{2p}$ by the same average. Thus [F1] and the forecast bounds in step 2.1 give ${\sup }_N\mathbb{E}\mathrm{\parallel }{C}_N{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^p<\mathrm{\infty }$ for every finite $p$.

The matrix $H_k{C}_N{H}_k^T$ is positive semidefinite, so $S_N=H_k{C}_N{H}_k^T+R_k$ has quadratic form at least that of $R_k$. Let $r_k>0$ be the least eigenvalue of $R_k$, which exists by [F4]. Every eigenvalue of $S_N$ is at least $r_k$, hence $\mathrm{\parallel }{S}_N^{-1}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le r_k^{-1}$. It follows from $K_N=C_N{H}_k^T{S}_N^{-1}$ and step 3.1 that ${\sup }_N\mathbb{E}\mathrm{\parallel }{K}_N{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^p<\mathrm{\infty }$ at every finite order.

Let $D_i=y_k+{\eta }_{N,i}^k$. The update in [F1] gives $\mathrm{\parallel }{X}_{N,i}^{a,k}\mathrm{\parallel }\le \mathrm{\parallel }{X}_{N,i}^{f,k}\mathrm{\parallel }+\mathrm{\parallel }{K}_N{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\mathrm{\parallel }{H}_k{X}_{N,i}^{f,k}\mathrm{\parallel }+\mathrm{\parallel }{K}_N{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\mathrm{\parallel }{D}_i\mathrm{\parallel }$. Apply [F5] with exponent pair $(2,2)$ to the middle product; for the last product use that $D_i$ is independent of $K_N$ and has finite Gaussian moments. The forecast, gain, and perturbation bounds in steps 2.1 and 4.1 therefore bound the $p$th moment of the updated state uniformly in $N,i$. This proves the induction step at time $k$.

Step 1.1 proves the state bound at time zero. For each $k\in \{1,\dots ,T\}$, step 5.1 derives both the state and gain bounds at time $k$ from the state bound at time $k-1$. Finite induction therefore proves the state bounds for every $0\le k\le T$ and the gain bounds for every $1\le k\le T$, for each finite $p$.