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Products of sub-Gaussian coordinates are sub-exponential

Statement

Let $U, V$ be centered real random variables, not necessarily independent, each sub-Gaussian with proxy $K^{2} > 0$ as in Sub-Gaussian variables and vectors, and sub-exponential variables. Then $W = U V - E [U V]$ is sub-exponential with parameters $(256 K^{4}, 16 K^{2})$ under the restricted-MGF convention of that definition.

Facts & Assumptions

Given: Centered real random variables $U, V$ with sub-Gaussian proxy $K^{2} > 0$ .

[F1]

A centered sub-Gaussian scalar with proxy $K^{2}$ has two-sided tail $P (∣ U ∣ \geq t) \leq 2 e^{- t^{2} / (2 K^{2})}$ for $t > 0$ (Sub-Gaussian variables and vectors, and sub-exponential variables, Finite sub-Gaussian maxima and epsilon-net reduction).

[F2]

For a nonnegative random variable $T$ , $E T = \int_{0}^{\infty} P (T > t) d t$ (Layer-cake formulas for random variables).

[F3]

Sub-exponential with parameters $(ν^{2}, b)$ means $E e^{λ W} \leq e^{λ^{2} ν^{2} / 2}$ for $∣ λ ∣ \leq 1 / b$ (Sub-Gaussian variables and vectors, and sub-exponential variables).

[F4]

Increasing nonnegative measurable partial sums pass to their pointwise limit under expectation. (Monotone convergence for the integral)

[F5]

Measurable partial sums dominated in absolute value by one integrable random variable converge in $L^{1}$ and in expectation to their pointwise limit. (Dominated convergence)

[F6]

Expectation is linear on finite sums of integrable random variables. (Linearity, monotonicity, and the modulus bound for expectation)

Proof

1.1F1F2algebra

For each integer $m \geq 1$ , [F1]–[F2] and the substitution $u = t^{2 m}$ give $E ∣ U ∣^{2 m} = 2 m \int_{0}^{\infty} t^{2 m - 1} P (∣ U ∣ > t) d t \leq 2 (2 K^{2})^{m} m! .$ The same bound holds for $V$ . The last integral is evaluated by $z = t^{2} / (2 K^{2})$ and $m - 1$ integrations by parts.

2.1step 1.1algebra

The pointwise inequality $2 ∣ U ∣^{m} ∣ V ∣^{m} \leq ∣ U ∣^{2 m} + ∣ V ∣^{2 m}$ and step 1.1 imply $E ∣ U V ∣^{m} \leq 2 (2 K^{2})^{m} m! .$ In particular $∣ E [U V] ∣ \leq E ∣ U V ∣ \leq 4 K^{2}$ .

3.1step 2.1algebra

For $m \geq 2$ , $∣ a - b ∣^{m} \leq 2^{m - 1} (∣ a ∣^{m} + ∣ b ∣^{m})$ and step 2.1 give $E ∣ W ∣^{m} \leq 2^{m - 1} (2 (2 K^{2})^{m} m! + (4 K^{2})^{m}) \leq (8 K^{2})^{m} m! .$ Also $E W = 0$ .

4.1F4F6step 2.1step 3.1algebra

Fix $∣ λ ∣ \leq 1 / (16 K^{2})$ and put $x = 8 K^{2} ∣ λ ∣ \leq 1 / 2$ . The variable $W$ is integrable by step 2.1. The nonnegative measurable partial sums $A_{N} = \sum_{m = 0}^{N} ∣ λ W ∣^{m} / m!$ increase pointwise to $e^{∣ λ W ∣}$ . Finite linearity [F6] and monotone convergence [F4], applied only to these nonnegative partial sums, give $E e^{∣ λ W ∣} = 1 + ∣ λ ∣ E ∣ W ∣ + \sum_{m \geq 2} \frac{∣ λ ∣^{m} E ∣ W ∣^{m}}{m!} \leq 1 + ∣ λ ∣ E ∣ W ∣ + \sum_{m \geq 2} x^{m} < \infty .$ Thus $e^{∣ λ W ∣}$ is an integrable dominating variable.

5.1F3F5F6step 3.1step 4.1algebra∎

The signed partial sums $S_{N} = \sum_{m = 0}^{N} (λ W)^{m} / m!$ converge pointwise to $e^{λ W}$ and satisfy $∣ S_{N} ∣ \leq A_{N} \leq e^{∣ λ W ∣}$ . Dominated convergence [F5] and finite linearity [F6] therefore justify taking expectations termwise; the resulting scalar series is absolutely convergent by step 4.1. Since $E W = 0$ , step 3.1 gives $E e^{λ W} = 1 + \sum_{m \geq 2} \frac{λ^{m} E W^{m}}{m!} \leq 1 + \sum_{m \geq 2} x^{m} \leq 1 + 128 K^{4} λ^{2} \leq e^{128 K^{4} λ^{2}} .$ This is the sub-exponential MGF bound in [F3] with $ν^{2} = 256 K^{4}$ and $b = 16 K^{2}$ .

Depends on

Used by

Dense sample covariance concentration for sub-Gaussian vectors Theorem

Dependency tree · two levels

43 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

Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Chapter 1, Lemma 1.12 (standard reference, not scraped)
Philippe Rigollet, 18.S997 High-Dimensional Statistics, Chapter 1, Section 1.3 (standard reference, not scraped)