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Extreme-value Lipschitz bounds and centered norm reduction

Statement

Assume the Axiom of Choice. For a real symmetric $r \times r$ matrix $A$ , $r \geq 1$ , define $λ_{+} (A) = \sup_{∥ v ∥_{2} = 1} v^{T} A v$ and $λ_{-} (A) = \sup_{∥ v ∥_{2} = 1} (- v^{T} A v)$ over real vectors. Use the operator norm on $C^{{1, \dots, r}}$ with its Euclidean norm. Put $N = r (r + 1) / 2$ and define $T : R^{N} \to R^{r \times r}$ by $T (z)_{i j} = T (z)_{j i} = z_{i j}$ for $1 \leq i \leq j \leq r$ . Both functions $z \mapsto λ_{\pm} (T (z))$ are $\sqrt{2}$ -Lipschitz, meaning that their absolute difference at $z, w$ is at most $\sqrt{2} ∥ z - w ∥_{2}$ .

Let $Z = (Z_{i j})_{i \leq j}$ have mutually independent standard normal coordinates on a common probability space $(Ω, F, P)$ , and put $Y = T (Z)$ . Set $m = E λ_{+} (Y) = E λ_{-} (Y)$ and $W = \max (∣ λ_{+} (Y) - m ∣, ∣ λ_{-} (Y) - m ∣)$ . Then $m$ is finite and nonnegative, $0 \leq ∥ Y ∥ \leq m + W$ , and $W$ has every finite positive even moment.

Statement references

The Axiom of Choice

Triangular Gaussian coordinates: metric and moment control

Expected extreme eigenvalues from finite Gaussian comparison

Probability measures and probability spaces

The operator norm as the least bound and as the unit-sphere or unit-ball supremum

Independent random elements

Standard real and normalized complex Gaussian matrix laws

Expectation of a nonnegative or integrable random variable

Moments, variance, and covariance on a probability space

Facts & Assumptions

[A1]

The Axiom of Choice is assumed, as required by both Gaussian suppliers in [F1] and [F2]. The Axiom of Choice.

[F1]

Under Choice, the triangular map satisfies $∥ T (z) - T (w) ∥ \leq \sqrt{2} ∥ z - w ∥_{2}$ , and $∥ Y ∥$ is measurable with $E ∥ Y ∥^{2 k} < \infty$ for every integer $k \geq 1$ . Triangular Gaussian coordinates: metric and moment control.

[F2]

Under Choice, $λ_{\pm} (Y)$ are measurable integrable real random variables, their expectations have the common finite value $m \geq 0$ , and $∥ Y ∥ = \max (λ_{+} (Y), λ_{-} (Y))$ . Expected extreme eigenvalues from finite Gaussian comparison.

[F3]

Euclidean Cauchy–Schwarz gives $∣ v^{T} B v ∣ \leq ∥ v ∥_{2} ∥ B v ∥_{2}$ for real $v, B$ . The operator norm bounds $∥ B u ∥_{2} \leq ∥ B ∥ ∥ u ∥_{2}$ , including real vectors viewed in the complex space. Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation, The operator norm as the least bound and as the unit-sphere or unit-ball supremum.

[F4]

Finite sums, real scalar multiples and absolute values of measurable real-valued functions are measurable. Closure properties of measurable functions used by the integral.

[F5]

If $h$ is a nonnegative finite-valued measurable function and $q > 0$ , then $h^{q}$ is measurable. Positive real powers preserve measurability.

[F6]

The nonnegative integral is monotone, homogeneous for positive scalars, and additive for two nonnegative measurable functions, even before their finiteness is known. Monotonicity and nonnegative homogeneity of the nonnegative integral, Additivity of the nonnegative Lebesgue integral.

[F7]

A probability measure satisfies $P (Ω) = 1$ . A constant $c \geq 0$ is the simple measurable function $c χ_{Ω}$ ; its simple integral is $c P (Ω)$ , and its nonnegative Lebesgue integral agrees with that simple integral. Probability measures and probability spaces, The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions.

[F8]

Independence of the real coordinates means independence of their generated sigma-algebras. A standard real Gaussian scalar has Lebesgue density $(2 π)^{- 1 / 2} e^{- x^{2} / 2}$ . The Countable Choice hypothesis of the Gaussian-law definition follows from [A1] by restriction to countable families. Independent random elements, Standard real and normalized complex Gaussian matrix laws, The Axiom of Choice.

[F9]

For a nonnegative measurable random variable $X$ , $E X = \int_{Ω} X d P$ , with an extended value allowed; for an integrable real random variable the same integral is its finite expectation and mean. For $p > 0$ , its $p$ th absolute moment is $E ∣ X ∣^{p}$ when $∣ X ∣^{p}$ is integrable. Expectation of a nonnegative or integrable random variable, Moments, variance, and covariance on a probability space.

Proof

Given: The objects and hypotheses in the Statement.

Goal: Establish all the stated estimates.

1.1A1F1F3algebra

For every real unit $v$ and real symmetric $B$ , [F3] gives $∣ v^{T} B v ∣ \leq ∥ B ∥$ . Thus the two signed suprema defining $λ_{\pm} (B)$ are finite and satisfy $∣ λ_{\pm} (B) ∣ \leq ∥ B ∥$ , because the real unit sphere is nonempty for $r \geq 1$ . For symmetric $A, B$ , each signed quadratic value for $A$ is at most its value for $B$ plus $∥ A - B ∥$ ; taking suprema gives $λ_{\pm} (A) \leq λ_{\pm} (B) + ∥ A - B ∥$ . Interchanging $A, B$ gives $∣ λ_{\pm} (A) - λ_{\pm} (B) ∣ \leq ∥ A - B ∥$ . Apply this with $A = T (z)$ and $B = T (w)$ and then use [F1], under [A1], to obtain the stated $\sqrt{2}$ -Lipschitz bounds.

2.1A1F2step 1.1algebra

Under [A1], [F2] gives the common finite mean $m \geq 0$ and $∥ Y ∥ = m + \max (λ_{+} (Y) - m, λ_{-} (Y) - m) \leq m + W$ . Also $∥ Y ∥ \geq 0$ . By step 1.1 and the scalar triangle inequality, $∣ λ_{\pm} (Y) - m ∣ \leq ∥ Y ∥ + m$ , hence $0 \leq W \leq ∥ Y ∥ + m$ .

3.1F1F2F4F5F7step 2.1algebra

By [F7], constants are simple measurable functions. With the measurability of $λ_{\pm} (Y)$ in [F2], [F4] therefore makes $a = ∣ λ_{+} (Y) - m ∣$ and $b = ∣ λ_{-} (Y) - m ∣$ measurable. The identity $W = (a + b + ∣ a - b ∣) / 2$ and [F4] prove that $W$ is measurable. It is finite-valued by step 2.1. For each integer $k \geq 1$ , [F5] applied with $q = 2 k > 0$ to $W$ and to the nonnegative measurable $∥ Y ∥$ from [F1] proves measurability of $W^{2 k}$ and $∥ Y ∥^{2 k}$ .

4.1F1F4F6F7F9step 1.1step 2.1step 3.1algebra∎

Fix an integer $k \geq 1$ . By [F9], expectations of the nonnegative measurable functions below are their nonnegative Lebesgue integrals. The nonnegative constant $C = m^{2 k}$ is finite; by [F7], $E C = C P (Ω) = C$ . Step 2.1 gives $W^{2 k} \leq (∥ Y ∥ + m)^{2 k} \leq 2^{2 k} (∥ Y ∥^{2 k} + C)$ : bound a sum of two nonnegative numbers by twice their maximum, raise to the positive integer power, and bound the maximum of the two powers by their sum. All functions here are measurable by step 3.1, [F4] and [F7]. Monotonicity, positive homogeneity and additivity in [F6] give $E W^{2 k} \leq 2^{2 k} (E ∥ Y ∥^{2 k} + C) < \infty$ by [F1]. Since $W \geq 0$ , this finite integral gives its finite $2 k$ th absolute moment in [F9]. Together with steps 1.1 and 2.1 this proves every assertion.

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