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Sample covariance converges for exchangeable arrays close in L4 to iid rows

Statement

Let $d \geq 1$ . For each $N \geq 2$ , let $(X_{N, i}, U_{N, i})_{i = 1}^{N}$ be jointly exchangeable random pairs in $R^{d}$ . Assume that $U_{N, 1}, \dots, U_{N, N}$ are iid with the law of a fixed $U$ , $E ∥ U ∥^{4} < \infty$ , and $∥ X_{N, 1} - U_{N, 1} ∥_{4} \to 0$ . Define ${\overset{ˉ}{X}}_{N} = N^{- 1} \sum_{i} X_{N, i}, C_{N} = N^{- 1} \sum_{i} (X_{N, i} - {\overset{ˉ}{X}}_{N}) (X_{N, i} - {\overset{ˉ}{X}}_{N})^{T} .$ Here $Cov (U) : = E [(U - E U) (U - E U)^{T}]$ . Then ${\overset{ˉ}{X}}_{N} \to E U$ and $C_{N} \to Cov (U)$ in probability, entry by entry.

Facts & Assumptions

Given: The hypotheses in the Statement.

[A1]

Joint exchangeability means invariance of the paired row under a common permutation of its index.

[A2]

$U_{N, i}$ are iid with finite fourth moment, and $∥ X_{N, 1} - U_{N, 1} ∥_{4} \to 0$ .

[F1]

The preceding weak law states that if exchangeable rows are pairwise uncorrelated and are close in $L^{2}$ to a common row, their empirical means converge in $L^{2}$ . A weak law for exchangeable arrays close to uncorrelated reference rows

[F2]

Holder's inequality implies $∥ Z W ∥_{2} \leq ∥ Z ∥_{4} ∥ W ∥_{4}$ . Holder's inequality for random variables

Proof

Given: The hypotheses in the Statement.

1.1A1A2F1F2given

Holder's inequality [F2] on a probability space gives $∥ X_{N, 1} - U_{N, 1} ∥_{2} \leq ∥ X_{N, 1} - U_{N, 1} ∥_{4}$ . Apply [F1] to each coordinate to obtain ${\overset{ˉ}{X}}_{N} \to E U$ in probability.

2.1A1A2F1F2step 1.1algebra

Fix coordinates $r, s \in {1, \dots, d}$ and set $V_{N, i} = X_{N, i, r} X_{N, i, s}$ and $W_{N, i} = U_{N, i, r} U_{N, i, s}$ . The rows $(V_{N, i}, W_{N, i})$ are jointly exchangeable. By Cauchy–Schwarz, $∥ V_{N, 1} - W_{N, 1} ∥_{2} \leq ∥ X_{N, 1, r} - U_{N, 1, r} ∥_{4} ∥ X_{N, 1, s} ∥_{4} + ∥ U_{N, 1, r} ∥_{4} ∥ X_{N, 1, s} - U_{N, 1, s} ∥_{4} \to 0.$ The product bounds follow from [F2]; the factors tending to zero follow from [A2], and the remaining fourth moments are bounded by [A2]. Since the $W_{N, i}$ are iid and square-integrable, they are pairwise uncorrelated. Thus [F1] applied to these scalar rows gives $N^{- 1} \sum_{i} X_{N, i, r} X_{N, i, s} \to E [U_{r} U_{s}]$ in probability.

3.1step 1.1step 2.1algebra∎

Entrywise, $C_{N} = N^{- 1} \sum_{i} X_{N, i} X_{N, i}^{T} - {\overset{ˉ}{X}}_{N} {\overset{ˉ}{X}}_{N}^{T}$ . The first term converges to $E [U U^{T}]$ by step 2.1 and the second to $(E U) (E U)^{T}$ by step 1.1. Their difference is $Cov (U)$ , proving the claim.

Depends on

Used by

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

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jan Mandel, Loren Cobb, and Jonathan D. Beezley, On the Convergence of the Ensemble Kalman Filter, §2, Lemma 3 (standard reference, not scraped)