LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Quadratic Gaussian process and its increment metric

Statement

Assume Choice and let $Y = T (Z)$ be the triangular Gaussian matrix of Triangular Gaussian coordinates: metric and moment control. On the real unit sphere $S = {v \in R^{r} : ∥ v ∥_{2} = 1}$ set $Q_{v} = v^{T} Y v$ . This is a centered Gaussian process with continuous sample paths. For $v, w \in S$ , writing $D_{i j} = v_{i} v_{j} - w_{i} w_{j}$ , one has $E (Q_{v} - Q_{w})^{2} = \sum_{i} D_{i i}^{2} + 4 \sum_{i < j} D_{i j}^{2} \leq 4 ∥ v - w ∥_{2}^{2}$ . The process $H_{v} = 2 v^{T} g$ , for an independent-standard-normal vector $g \in R^{r}$ , is centered Gaussian with increment variance exactly $4 ∥ v - w ∥_{2}^{2}$ .

Facts & Assumptions

[F1]

Finite independent standard normal scalars stack into a centered multivariate normal vector with identity covariance; every scalar projection is normal. A finite stack of independent Gaussian vectors is jointly Gaussian, Multivariate normal law, including singular covariance.

[F2]

Gaussian processes are defined by normal finite linear combinations, allowing singular laws. Gaussian coordinates have mean zero and second moment one, and distinct independent coordinates have zero cross moment. Gaussian process, Gaussian entry moments and triangular independence.

[F3]

Finite integrable sums have linear expectation; real vectors satisfy Cauchy–Schwarz, and their Euclidean norm induces the metric $d (v, w) = ∥ v - w ∥_{2}$ (restricted to $S$ when appropriate). The Lebesgue integral is linear on L¹(μ), Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation.

[F4]

A real stochastic process is a family of real random variables indexed by a set and defined on a common probability space. Stochastic processes and their finite-dimensional distributions.

[F5]

A map between metric spaces is continuous if at every point, for each $ε > 0$ , some $δ > 0$ makes input distance less than $δ$ imply output distance less than $ε$ . Here sample paths are maps from $S$ with its restricted Euclidean metric to $R$ with distance $∣ a - b ∣$ . Continuity of a map between metric spaces, at a point and globally, in the ε-δ form.

Proof

Given: The objects and hypotheses in the Statement.

Goal: Establish all the stated estimates.

1.1F1F2F3F4algebra

Expand $Q_{v} = \sum_{i} v_{i}^{2} Z_{i i} + 2 \sum_{i < j} v_{i} v_{j} Z_{i j}$ . By [F1], for each fixed $v$ this scalar projection is a measurable real random variable on the common probability space carrying $Z$ . Thus $Q = (Q_{v})_{v \in S}$ is a stochastic process by [F4]. Every finite linear combination of the $Q_{v}$ is a scalar projection of the jointly Gaussian triangular coordinates, so [F1] and [F2] give the Gaussian-process assertion; [F3] gives zero mean. Expanding the square of $Q_{v} - Q_{w}$ , the zero cross moments in [F2] give exactly $\sum_{i} D_{i i}^{2} + 4 \sum_{i < j} D_{i j}^{2}$ .

2.1F1F2F3F4step 1.1algebra

The preceding variance is at most $2 \sum_{i, j} D_{i j}^{2} = 4 (1 - (v^{T} w)^{2})$ . Put $t = v^{T} w \in [- 1, 1]$ by [F3]. Since $1 - t^{2} \leq 2 (1 - t)$ , this is at most $8 (1 - t) = 4 ∥ v - w ∥_{2}^{2}$ . By [F1], each $H_{v} = 2 v^{T} g$ is a measurable scalar projection on the common probability space carrying $g$ , so [F4] makes $H = (H_{v})_{v \in S}$ a stochastic process. Every finite linear combination of the $H_{v}$ is also a scalar projection of $g$ , so [F1] and [F2] prove that $H$ is centered Gaussian; direct expansion using its identity covariance gives $E (H_{v} - H_{w})^{2} = 4 \sum_{i} (v_{i} - w_{i})^{2}$ .

3.1F3F5step 1.1step 2.1algebra∎

Fix any outcome $ω$ for the quadratic process and put $A = Y (ω)$ , a finite real matrix. For $v, w \in S$ , the Euclidean norm gives $∣ v_{i} ∣, ∣ w_{i} ∣ \leq 1$ and $∣ v_{i} - w_{i} ∣ \leq ∥ v - w ∥_{2}$ . Consequently $∣ v_{i} v_{j} - w_{i} w_{j} ∣ = ∣ v_{i} (v_{j} - w_{j}) + w_{j} (v_{i} - w_{i}) ∣ \leq 2 ∥ v - w ∥_{2}$ , and finite summation yields $∣ Q_{v} (ω) - Q_{w} (ω) ∣ \leq L_{Q} (ω) ∥ v - w ∥_{2}$ , where $L_{Q} (ω) = 2 \sum_{i, j} ∣ A_{i j} ∣ < \infty$ . For any outcome of the linear process, Cauchy–Schwarz in [F3] gives $∣ H_{v} (ω) - H_{w} (ω) ∣ \leq L_{H} (ω) ∥ v - w ∥_{2}$ , where $L_{H} (ω) = 2 ∥ g (ω) ∥_{2} < \infty$ . For either path, write $L$ for its nonnegative finite bound. Given $ε > 0$ and any $w \in S$ , take $δ = ε / (1 + L) > 0$ ; then $∥ v - w ∥_{2} < δ$ implies that the output distance is at most $L ∥ v - w ∥_{2} < ε$ . By [F5] every path of each process is continuous.

Depends on

Used by · 1

Expected extreme eigenvalues from finite Gaussian comparison

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.