[Paper Review] On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables

Julyan Arbel, Olivier Marchal+1 more|arXiv (Cornell University)|Jan 26, 2019

Stochastic processes and financial applicationsEconomics, Econometrics and Finance23 references17 citations

TL;DR

This paper provides characterizations of the optimal proxy variance and strict sub-Gaussianity for bounded random variables using function variation analysis. It shows that symmetry is neither necessary nor sufficient for strict sub-Gaussianity, and establishes necessary and sufficient conditions for strict sub-Gaussianity through cumulant generating function analysis and critical point analysis of the deviation function Δ(σ²,λ).

ABSTRACT

We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize strict sub-Gaussianity, defined by a proxy variance equal to the (standard) variance? We address the questions in proposing conditions based on the study of functions variations. A particular focus is given to the relationship between strict sub-Gaussianity and symmetry of the distribution. In particular, we demonstrate that symmetry is neither sufficient nor necessary for strict sub-Gaussianity. In contrast, simple necessary conditions on the one hand, and simple sufficient conditions on the other hand, for strict sub-Gaussianity are provided. These results are illustrated via various applications to a number of bounded random variables, including Bernoulli, beta, binomial, uniform, Kumaraswamy, and triangular distributions.

Motivation & Objective

• To characterize the optimal proxy variance σ²_opt for almost surely bounded random variables using function variation analysis.
• To investigate the conditions under which a bounded random variable is strictly sub-Gaussian (i.e., σ²_opt = Var(X)).
• To clarify the relationship between symmetry (with respect to the mean) and strict sub-Gaussianity, challenging the assumption that symmetry implies strict sub-Gaussianity.
• To generalize results from previous work on beta and Dirichlet distributions to broader families including Bernoulli, binomial, Kumaraswamy, and triangular distributions.
• To provide a unified framework for analyzing sub-Gaussian properties via the cumulant generating function and deviation function Δ(σ²,λ).

Proposed method

• Uses the cumulant generating function K(λ) = ln 𝔼[exp(λ(X−μ)]) to define the deviation function Δ(σ²,λ) = σ²λ²/2 − K(λ), which quantifies the sub-Gaussian deviation.
• Analyzes the sign and critical points of Δ(σ²,λ) to determine the optimal proxy variance σ²_opt, leveraging the fact that σ²_opt is the infimum of σ² for which Δ(σ²,λ) ≥ 0 for all λ.
• Applies asymptotic expansion of Δ(σ²,λ) around λ=0 to derive conditions on σ² relative to Var(X), using the third and fourth cumulants.
• Employs continuity and monotonicity arguments in σ² and λ to prove that σ²_opt is the smallest σ² such that Δ(σ²,λ) ≥ 0 and vanishes at some λ₀ with zero first derivative.
• Uses the function Δ(σ²,λ) to derive necessary and sufficient conditions for strict sub-Gaussianity: Δ(σ²,λ) ≥ 0 and ∃λ₀ such that Δ(σ²,λ₀)=0 and ∂λΔ(σ²,λ₀)=0.
• Constructs counterexamples using mixtures of Dirac masses and beta distributions to demonstrate that symmetry does not imply strict sub-Gaussianity, and asymmetric distributions can still be strictly sub-Gaussian.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a bounded random variable to be strictly sub-Gaussian?
• RQ2 How can the optimal proxy variance σ²_opt be characterized for bounded random variables beyond the variance?
• RQ3 Is symmetry with respect to the mean a necessary or sufficient condition for strict sub-Gaussianity in bounded distributions?
• RQ4 Can the optimal proxy variance be computed for standard bounded distributions such as Bernoulli, beta, binomial, Kumaraswamy, and triangular?
• RQ5 Do the standard equivalences between symmetry and strict sub-Gaussianity (known for beta) extend to other bounded distributions?

Key findings

• For Bernoulli, beta, binomial, Kumaraswamy, and triangular distributions, symmetry with respect to the mean is equivalent to strict sub-Gaussianity.
• Symmetry is neither necessary nor sufficient for strict sub-Gaussianity in general: counterexamples exist with symmetric mixtures (e.g., two-component beta mixture) that are not strictly sub-Gaussian.
• An asymmetric three-component mixture of Dirac masses is strictly sub-Gaussian, demonstrating that asymmetry does not preclude strict sub-Gaussianity.
• The optimal proxy variance σ²_opt is characterized as the infimum of σ² such that the deviation function Δ(σ²,λ) ≥ 0 for all λ and vanishes at some λ₀ with zero derivative.
• The function Δ(σ²,λ) has a non-negative sign for σ² ≥ σ²_opt, is strictly positive for σ² > σ²_opt, and is negative in some interval for σ² < σ²_opt.
• The paper resolves an open problem by proving the uniqueness of the global maximum of a function related to the cumulant generating function for the Bernoulli distribution, confirming a conjecture by Berend and Kontorovich (2013).