ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A sub-Gaussian vector norm bound

Example

Let $d \geq 1$ and let $Z \in R^{d}$ be a centered sub-Gaussian vector with proxy $σ^{2} > 0$ . Choosing $ϵ = 1 / 2$ in the net reduction gives, for every $t > 0$ , $P (∥ Z ∥_{2} \geq t) \leq 2 \cdot 5^{d} e^{- t^{2} / (8 σ^{2})} .$ If $σ^{2} = 0$ , the same probability is $0$ .

Facts & Assumptions

Given: A centered sub-Gaussian vector $Z \in R^{d}$ , with $d \geq 1$ and proxy $σ^{2} \geq 0$ .

[F1]

A centered vector with proxy $σ^{2}$ has each projection $⟨ u, Z ⟩$ sub-Gaussian with proxy $σ^{2} ∥ u ∥_{2}^{2}$ (Sub-Gaussian variables and vectors, and sub-exponential variables).

[F2]

For $0 < ϵ < 1$ , the net reduction gives $P (∥ Z ∥_{2} \geq t) \leq 2 (1 + 2 / ϵ)^{d} \exp (- (1 - ϵ)^{2} t^{2} / (2 σ^{2}))$ when $σ^{2} > 0$ , and zero tail when $σ^{2} = 0$ (Finite sub-Gaussian maxima and epsilon-net reduction).

Verification

1.1givenalgebra

The choice $ϵ = 1 / 2$ is in $(0, 1)$ and gives the finite-net factor $(1 + 2 / ϵ)^{d} = 5^{d}$ .

2.1F1F2step 1.1algebra∎

By [F1], $Z$ satisfies the centered projection condition required by [F2]. Apply [F2] with the value from step 1.1. Since $(1 - 1 / 2)^{2} / (2 σ^{2}) = 1 / (8 σ^{2})$ , the stated positive-proxy bound follows; the zero-proxy clause of [F2] gives the boundary case.

Depends on

Used by

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Dependency tree · two levels

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Sources

Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Chapter 1, Lemma 1.18 and Theorem 1.19 (standard reference, not scraped)
John Duchi, Lecture Notes on Statistics and Information Theory, Chapter 4, sub-Gaussian vector convention (standard reference, not scraped)