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Finite-fourth-moment root-n consistency of sample-centered covariance

Statement

Fix $p \geq 1$ and a Borel probability law $P$ on $R^{p}$ with $E_{P} ∥ X ∥_{2}^{4} < \infty$ , mean $μ$ , and covariance matrix $Σ$ . For each $n \geq 1$ , let $X_{n, 1}, \dots, X_{n, n}$ be iid with law $P$ ; the probability space may depend on $n$ . Put ${\overset{ˉ}{X}}_{n} : = \frac{1}{n} \sum_{i = 1}^{n} X_{n, i}, {\hat{Σ}}_{n}^{*} : = \frac{1}{n} \sum_{i = 1}^{n} (X_{n, i} - {\overset{ˉ}{X}}_{n}) (X_{n, i} - {\overset{ˉ}{X}}_{n})^{T},$ the sample-centered covariance with divisor $n$ from Sample covariance estimators and squared operator loss. For a matrix $B$ , define $∥ B ∥_{F} : = (\sum_{j, k = 1}^{p} B_{j k}^{2})^{1 / 2}$ . Then $∥ {\hat{Σ}}_{n}^{*} - Σ ∥_{F} = O_{P} (n^{- 1 / 2}),$ meaning $\sqrt{n} ∥ {\hat{Σ}}_{n}^{*} - Σ ∥_{F} = O_{P} (1)$ in the bounded-in-probability sense of Stochastic order and convergence notation for finite-dimensional random vectors. No Gaussian assumption is needed.

Facts & Assumptions

Given: A fixed finite $p \geq 1$ , a law $P$ on $R^{p}$ with $E_{P} ∥ X ∥_{2}^{4} < \infty$ , and for each $n$ an iid sample of size $n$ from $P$ .

[F1]

Identical distribution and IID families: iid means identically distributed and mutually independent, not only pairwise independent.

[F2]

Measurable coordinatewise functions preserve independence: measurable coordinatewise functions of an independent family remain independent.

[F3]

Covariance matrix of a finite-dimensional random vector and Sample covariance estimators and squared operator loss: $Σ_{j k} = E [(X_{j} - μ_{j}) (X_{k} - μ_{k})]$ and ${\hat{Σ}}_{n}^{*} = n^{- 1} \sum_{i} (X_{n, i} - {\overset{ˉ}{X}}_{n}) (X_{n, i} - {\overset{ˉ}{X}}_{n})^{T}$ .

[F4]

Variance and covariance identities for random variables and Independence forces covariance to vanish: covariance is bilinear on finite sums, and independent square-integrable real random variables have zero covariance.

[F5]

Linearity, monotonicity, and the modulus bound for expectation: expectation is linear for integrable variables.

[F6]

Markov's inequality for random variables: for nonnegative $Z$ and $a > 0$ , $P (Z \geq a) \leq E [Z] / a$ .

[F7]

Stochastic order and convergence notation for finite-dimensional random vectors: $Z_{n} = O_{P} (1)$ means, using the Euclidean norm of its finite vectorization, that for every $η > 0$ some $M > 0$ satisfies ${lim sup}_{n} P (∣ Z_{n} ∣ > M) \leq η$ .

[F8]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: the Euclidean norm satisfies the triangle inequality in every finite dimension.

Proof

technique · direct

1.1F1F2F3F8givenalgebra

Since $t^{2} \leq 1 + t^{4}$ for $t \geq 0$ , the fourth-moment assumption gives $E ∥ X ∥_{2}^{2} < \infty$ . Then $∥ X ∥_{2} \leq 1 + ∥ X ∥_{2}^{2}$ gives $E ∥ X ∥_{2} < \infty$ , so the coordinatewise mean $μ$ is finite. Put $Y_{n, i} = X_{n, i} - μ$ and $M_{n} = n^{- 1} \sum_{i} Y_{n, i} Y_{n, i}^{T}$ . By [F8], $∥ Y_{n, 1} ∥_{2} \leq ∥ X_{n, 1} ∥_{2} + ∥ μ ∥_{2}$ . For $a, b \geq 0$ , $2 a b \leq a^{2} + b^{2}$ gives $(a + b)^{2} \leq 2 (a^{2} + b^{2})$ and then $(a + b)^{4} \leq 4 (a^{2} + b^{2})^{2} \leq 8 (a^{4} + b^{4})$ . Hence $E ∥ Y_{n, 1} ∥_{2}^{4} \leq 8 (E ∥ X ∥_{2}^{4} + ∥ μ ∥_{2}^{4}) < \infty$ . For each $j, k$ , the variables $W_{n, i}^{j k} : = Y_{n, i, j} Y_{n, i, k} - Σ_{j k}$ are centered and square-integrable because $(Y_{n, i, j} Y_{n, i, k})^{2} \leq ∥ Y_{n, i} ∥_{2}^{4}$ . The map $y \mapsto y_{j} y_{k} - Σ_{j k}$ is Borel, so [F1] and [F2] make $(W_{n, i}^{j k})_{i = 1}^{n}$ independent.

2.1F1F2F3F4F5F6step 1.1algebra

Let ${\overset{ˉ}{Y}}_{n} = n^{- 1} \sum_{i} Y_{n, i}$ . By [F1] and [F2], the coordinate variables $Y_{n, i, j}$ are independent across $i$ . Thus [F4] makes their cross-covariances zero, and [F4] and [F5] give $E ∥ {\overset{ˉ}{Y}}_{n} ∥_{2}^{2} = tr (Σ) / n$ . Also $∥ {\overset{ˉ}{Y}}_{n} {\overset{ˉ}{Y}}_{n}^{T} ∥_{F} = ∥ {\overset{ˉ}{Y}}_{n} ∥_{2}^{2}$ by expanding both squares. For every $C > 0$ , [F6] therefore gives $P (\sqrt{n} ∥ {\overset{ˉ}{Y}}_{n} {\overset{ˉ}{Y}}_{n}^{T} ∥_{F} > C) \leq tr (Σ) / (C \sqrt{n}) \to 0$ . Thus this centered-mean term converges to zero in probability.

2.2F1F2F3F4F6F7step 1.1algebra

For fixed $j, k$ , all $W_{n, i}^{j k}$ are the same Borel function of the iid vectors $Y_{n, i}$ , so they are iid: their independence follows from [F1] and [F2], and equal laws follow because a measurable map sends equal input laws to equal output laws. Thus [F4] gives $E [(M_{n, j k} - Σ_{j k})^{2}] = n^{- 2} \sum_{i} Var (W_{n, i}^{j k}) = n^{- 1} Var (W_{n, 1}^{j k})$ . Since $E [Y_{n, 1, j} Y_{n, 1, k}] = Σ_{j k}$ , $Var (W_{n, 1}^{j k}) = E [Y_{n, 1, j}^{2} Y_{n, 1, k}^{2}] - Σ_{j k}^{2} \leq E [Y_{n, 1, j}^{2} Y_{n, 1, k}^{2}]$ . Summing over $j, k$ and using $\sum_{j, k} Y_{n, 1, j}^{2} Y_{n, 1, k}^{2} = ∥ Y_{n, 1} ∥_{2}^{4}$ yields $E ∥ M_{n} - Σ ∥_{F}^{2} \leq n^{- 1} E ∥ Y_{n, 1} ∥_{2}^{4}$ . Consequently [F6] gives, for every $C > 0$ , $P (\sqrt{n} ∥ M_{n} - Σ ∥_{F} > C) \leq E ∥ Y_{n, 1} ∥_{2}^{4} / C^{2}$ , so $\sqrt{n} ∥ M_{n} - Σ ∥_{F} = O_{P} (1)$ by [F7].

3.1F3F7F8step 2.2step 2.1algebra∎

Expanding the centered sum in [F3] gives ${\hat{Σ}}_{n}^{*} = M_{n} - {\overset{ˉ}{Y}}_{n} {\overset{ˉ}{Y}}_{n}^{T}$ . For a matrix $B$ , let $vec (B) \in R^{p^{2}}$ list its entries; by the definition of the Frobenius norm, $∥ vec (B) ∥_{2} = ∥ B ∥_{F}$ . Applying [F8] to these vectors gives $∥ B + C ∥_{F} \leq ∥ B ∥_{F} + ∥ C ∥_{F}$ , including the triangle inequality used below. For any $η > 0$ , choose $C_{1}$ so the limsup tail of $\sqrt{n} ∥ M_{n} - Σ ∥_{F}$ above $C_{1}$ is at most $η / 2$ by step 2.2. For all sufficiently large $n$ , the probability that $\sqrt{n} ∥ {\overset{ˉ}{Y}}_{n} {\overset{ˉ}{Y}}_{n}^{T} ∥_{F} > 1$ is at most $η / 2$ by step 2.1. The displayed Frobenius triangle inequality then bounds the limsup probability that $\sqrt{n} ∥ {\hat{Σ}}_{n}^{*} - Σ ∥_{F} > C_{1} + 1$ by $η$ . This is $\sqrt{n} ∥ {\hat{Σ}}_{n}^{*} - Σ ∥_{F} = O_{P} (1)$ by [F7], which is the claimed root-n rate.

Depends on

Used by

A simple population eigengap gives root-n sample PCA eigenspace consistency Corollary

Dependency tree · two levels

44 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

Online Bayesian Data Analysis, Lecture Notes 8: Learning Signal Representations (11 April 2016) (standard reference, not scraped)
Soumendu Sundar Mukherjee, Multivariate Analysis, Lecture 8: Principal Component Analysis (standard reference, not scraped)