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From finite Gaussian comparisons to a sphere supremum

Statement

Assume the Axiom of Choice. Let $S = {t \in R^{d} : ∥ t ∥_{2} = 1}$ , where $d \geq 1$ . Let $X = (X_{t})_{t \in S}$ and $Y = (Y_{t})_{t \in S}$ be centered real Gaussian processes, possibly on different probability spaces. Suppose that each has continuous sample paths outside a measurable null set and that $\sup_{t \in S} X_{t}$ and $\sup_{t \in S} Y_{t}$ are measurable and integrable. If $E (X_{s} - X_{t})^{2} \geq E (Y_{s} - Y_{t})^{2}$ for every $s, t \in S$ , then $E \sup_{t \in S} X_{t} \geq E \sup_{t \in S} Y_{t}$ .

Here centered means that every coordinate has expectation zero; integrable means finite absolute expectation. A Gaussian process has a possibly singular multivariate normal evaluation vector at every finite list of indices. A finite $ε$ -net in $S$ is a finite subset whose points are within Euclidean distance $ε$ of every point of $S$ .

Statement references

Gaussian process

Stochastic processes and their finite-dimensional distributions

The Axiom of Choice

Continuity of a map between metric spaces, at a point and globally, in the ε-δ form

Expectation of a nonnegative or integrable random variable

Facts & Assumptions

Given: The sphere, processes, continuity and integrability assumptions, and increment-variance ordering in the Statement.

Goal: Compare the expectations of the two sphere suprema.

[F1]

Under AC, finite centered Gaussian vectors with finite covariance matrices and ordered increment variances have ordered expected maxima when the two vectors are independent and have at least two coordinates (Finite Sudakov–Fernique comparison for Gaussian maxima, Gaussian process, The Axiom of Choice).

[F2]

The product of two probability measures exists and has the rectangle formula; the rectangle criterion then gives independence of random vectors depending on separate coordinates (For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique, Independent random elements are characterized by finite rectangle probabilities).

[F3]

Finite maxima and increasing pointwise limits of measurable functions are measurable, and dominated convergence passes expectations through an almost-everywhere limit with an integrable majorant (Closure properties of measurable functions used by the integral, Dominated convergence).

[F4]

The rationals are countable and dense in the reals, and closed bounded subsets of $R^{d}$ are compact (ℚ is countably infinite, Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable, Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F5]

A one-dimensional normal law is an affine image of the standard normal density $e^{- x^{2} / 2} / \sqrt{2 π}$ , including the constant case (Standard normal and normal laws).

[F6]

A stochastic process is a family of random elements on one probability space; its finite evaluation vectors have the corresponding pushforward laws (Stochastic processes and their finite-dimensional distributions).

[F7]

Continuity means metric $ε$ - $δ$ continuity, using the Euclidean metrics for maps on the sphere and normalization away from zero (Continuity of a map between metric spaces, at a point and globally, in the ε-δ form).

[F8]

The expectation of a nonnegative or integrable random variable is its probability-space integral; for a real random variable, integrability means finite absolute expectation (Expectation of a nonnegative or integrable random variable).

Proof

1.1F4F7construct

Enumerate the distinct points of $D = {q / ∥ q ∥_{2} : q \in Q^{d} ∖ {0}}$ as $t_{1}, t_{2}, \dots$ if $d \geq 2$ ; for $d = 1$ , take $D = {- 1, 1}$ and repeat these points in the list. Rational coordinate approximation and continuity of normalization away from zero show that $D$ is dense in $S$ . Put $N_{k} = {t_{1}, \dots, t_{k + 1}}$ . These finite sets are nested and have dense union. For every $ε > 0$ , the open $ε$ -balls centered at points of $D$ cover compact $S$ ; a finite subcover is contained in some $N_{k}$ , so all later $N_{k}$ are $ε$ -nets.

2.1givenF3F6F7step 1.1

For either process $Z \in {X, Y}$ , let $M_{k}^{Z} = \max_{t \in N_{k}} Z_{t}$ and $M^{Z} = \sup_{t \in S} Z_{t}$ . On the measurable probability-one event of continuous sample paths, density gives $M_{k}^{Z} ↑ M^{Z}$ : for any $t \in S$ , values at a sequence from $D$ tending to $t$ converge to $Z_{t}$ . The finite maxima are measurable by [F3]. Since $t_{1} \in N_{k}$ , $Z_{t_{1}} \leq M_{k}^{Z} \leq M^{Z}$ , hence $∣ M_{k}^{Z} ∣ \leq ∣ Z_{t_{1}} ∣ + ∣ M^{Z} ∣$ on that event.

3.1F1F3F5F8step 2.1

The coordinate $Z_{t_{1}}$ has a normal law by the Gaussian-process definition. Its absolute expectation is finite: by [F5] it is bounded by $∣ m ∣ + σ E ∣ G ∣$ , where $G$ is standard normal and $E ∣ G ∣ = \sqrt{2 / π}$ by integrating its density on the two half-lines. Thus the majorant in step 2.1 is integrable. Dominated convergence gives $E M_{k}^{Z} \to E M^{Z}$ for each process.

4.1givenF1F2F6F8step 1.1step 3.1∎

For each $k$ , put independent copies of the two finite evaluation vectors on the product of their original probability spaces. The rectangle formula and criterion in [F2] give independence, and each copy has its original finite-dimensional law, means, and increment variances. If $N_{k}$ has fewer than two distinct points, repeat a point in its evaluation list; otherwise list its points once. The finite comparison [F1] gives $E M_{k}^{X} \geq E M_{k}^{Y}$ . Taking $k \to \infty$ using step 3.1 proves the desired comparison.

Remarks

To apply this proposition to a quadratic-form process, its increment ordering, almost-sure path continuity, and integrability of its supremum must be established separately.

Depends on

Used by · 1

Expected extreme eigenvalues from finite Gaussian comparison

Dependency tree · 2 levels

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