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Finite-second-moment IID sample covariance converges

Statement

Let $d \geq 1$ be finite and let $X_{1}, X_{2}, \dots$ be IID $R^{d}$ -valued random vectors with $E ∥ X_{1} ∥^{2} < \infty$ . Put $μ_{j} = E X_{1 j}$ and $Σ_{j k} = E [(X_{1 j} - μ_{j}) (X_{1 k} - μ_{k})]$ , the coordinate covariance entries from Moments, variance, and covariance on a probability space. Define ${\overset{ˉ}{X}}_{n} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}, {\hat{Σ}}_{n} = \frac{1}{n} \sum_{i = 1}^{n} (X_{i} - {\overset{ˉ}{X}}_{n}) (X_{i} - {\overset{ˉ}{X}}_{n})^{T} .$ Then ${\overset{ˉ}{X}}_{n} \to μ$ in probability and every entry of ${\hat{Σ}}_{n}$ converges in probability to the corresponding entry of $Σ$ .

Facts & Assumptions

Given: The IID vectors, finite second moment, and definitions in the statement.

[F1]

IID means independent random elements with a common law; it has no moment assumption. Identical distribution and IID families

[F2]

The covariance entry is $Σ_{j k} = E [(X_{1 j} - μ_{j}) (X_{1 k} - μ_{k})]$ . Moments, variance, and covariance on a probability space

[F3]

For an integrable IID real sequence, its sample mean converges in $L^{1}$ and hence in probability to its expectation. Khinchin weak law for integrable IID variables

[F4]

Measurable coordinatewise maps of an independent family remain independent. Measurable coordinatewise functions preserve independence

Proof

1.1F1F3F4given

For each coordinate $j$ , the sequence $(X_{i j})_{i}$ is IID by [F1] and [F4], and is integrable since $E ∣ X_{1 j} ∣ \leq 1 + E ∣ X_{1 j} ∣^{2} < \infty$ . For each pair $j, k$ , $(X_{i j} X_{i k})_{i}$ is also IID by [F1] and [F4], and is integrable because $2 ∣ X_{1 j} X_{1 k} ∣ \leq X_{1 j}^{2} + X_{1 k}^{2}$ . Apply [F3] to each of these finitely many sequences. This gives ${\overset{ˉ}{X}}_{n, j} \to μ_{j}$ and $n^{- 1} \sum_{i} X_{i j} X_{i k} \to E [X_{1 j} X_{1 k}]$ in probability.

2.1F2step 1.1algebra∎

For each $j, k$ , expand ${\hat{Σ}}_{n, j k} = \frac{1}{n} \sum_{i} X_{i j} X_{i k} - {\overset{ˉ}{X}}_{n, j} {\overset{ˉ}{X}}_{n, k} .$ Step 1.1 makes the right side converge in probability to $E [X_{1 j} X_{1 k}] - μ_{j} μ_{k} = Σ_{j k}$ ; the same step gives the claimed mean convergence. Since $d$ is finite, entrywise convergence holds simultaneously in probability.

Depends on

Used by

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Sources

University of Washington STAT 581, Chapter 8: Bootstrap and Jackknife Estimation of Sampling Distributions, §4 (standard reference, not scraped)
Bradley Efron and Robert Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Behaviormetrika 12 (1985), §1 (standard reference, not scraped)