Sub-Gaussian Variables: Theory & Applications

• Sub-Gaussian variables are defined by Gaussian-like tail probabilities and moment growth, ensuring tight concentration bounds.
• They are characterized via moment generating function bounds, Orlicz norms, and moment inequalities that extend Gaussian insights to broader distributions.
• Their robust concentration properties underpin applications in high-dimensional statistics, random matrix theory, and compressed sensing.

A sub-Gaussian random variable is a real-valued random variable whose tail behavior, moment growth, and moment-generating function mirror that of a centered Gaussian variable up to a scale factor. Sub-Gaussianity is of central importance in modern probability theory, high-dimensional statistics, @@@@1@@@@, statistical learning theory, and signal processing, due to its robust control of concentration phenomena and its closure properties under summation and linear transformations. The structure of sub-Gaussian random variables and their generalizations, such as φ–sub-Gaussian laws, enables the transfer of Gaussian-type tail inequalities and norm controls to a wide class of random objects well beyond the strictly Gaussian setting, including bounded, discrete, and complex-valued variables.

1. Definitions and Characterizations

Sub-Gaussianity admits several equivalent formulations, all expressing a uniform Gaussian-type upper bound on tails, moments, or the moment-generating function:

• @@@@2@@@@ Bound: $X$ is sub-Gaussian if there exists $\sigma>0$ such that

$$\mathbb{E}\left[\exp\left(\lambda(X-\mathbb{E}X)\right)\right] \leq \exp\left(\frac{\lambda^2 \sigma^2}{2}\right),\quad \forall \lambda \in \mathbb{R}$$

The minimal such $\sigma^2$ is called the (optimal) sub-Gaussian variance proxy or sub-Gaussian parameter (Rudelson et al., 2013).

• Tail Bound: There exists $C>0$ such that for all $t\geq0$,

$$\mathbb{P}\left(|X-\mathbb{E}X| > t\right) \leq 2\exp\left(-t^2/C^2\right)$$

The minimal $C$ is, up to constants, the same as above (Ostrovsky et al., 2014).

• Orlicz-$\psi_2$ (Luxemburg) Norm:

$$\|X - \mathbb{E}X\|_{\psi_2} := \inf\left\{K > 0 : \mathbb{E}\exp\left(\frac{(X-\mathbb{E}X)^2}{K^2}\right) \leq 2\right\} < \infty$$

• Moment Growth: For all $p \geq 1$,

$$\left(\mathbb{E}|X-\mathbb{E}X|^p\right)^{1/p} \leq CK \sqrt{p}$$

for some absolute constant $C$.

These norms are equivalent up to universal constants (Garivier et al., 2024, Ostrovsky et al., 2014), and any bounded random variable is sub-Gaussian (Steinke et al., 2017, Ostrovsky et al., 2014). The concept generalizes to $(\sigma, \rho)$-sub-Gaussianity in which an additional multiplicative constant appears in the MGF bound (Li, 2024).

2. Variance Proxy and Strict Sub-Gaussianity

The variance proxy (or optimal variance parameter) $\sigma^2_{\text{opt}}$ of $X$ is the infimum of $\sigma^2>0$ such that $X$ is $\sigma^2$-sub-Gaussian (Arbel et al., 2019, Atouani et al., 7 Oct 2025). It is characterized explicitly by: $$\sigma^2_{\text{opt}} = \max_{\lambda\in\mathbb{R}} \frac{2}{\lambda^2} \ln \mathbb{E}[e^{\lambda(X - \mathbb{E}X)}] = \sup_{\lambda\neq0} \frac{\mathbb{E}[e^{\lambda(X - \mathbb{E}X)}]}{e^{\lambda^2 \sigma^2_{\text{opt}}/2}}$$ Strict sub-Gaussianity occurs when $\sigma^2_{\text{opt}} = \operatorname{Var}(X)$. Cumulant-based necessary conditions for strict sub-Gaussianity require vanishing third central moment and nonpositive excess kurtosis: $$\mathbb{E}[(X-\mu)^3]=0,\quad \mathbb{E}[(X-\mu)^4] - 3(\operatorname{Var}(X))^2 \leq 0$$ Sufficient conditions involve the comparison of even centered moments to those of the Gaussian (Arbel et al., 2019). For symmetric bounded variables, strictness is characterized by additional moment inequalities but symmetry is neither necessary nor sufficient in general—this is clarified for families such as Bernoulli, binomial, symmetric beta, and uniform distributions (Atouani et al., 7 Oct 2025, Ostrovsky et al., 2014).

For truncated Gaussians, strictness holds if and only if the truncation is symmetric; for truncated exponentials, strict sub-Gaussianity never occurs (Barreto et al., 2024).

3. Concentration Inequalities for Linear and Quadratic Forms

Sub-Gaussian variables are fundamental in concentration-of-measure results. Salient inequalities include:

• Hoeffding-Type Bounds for Sums: For independent, centered, strictly sub-Gaussian $X_i$ with variances $\sigma_i^2$, the sum $S_n=\sum_{i=1}^n X_i$ satisfies

$$\mathbb{P}(|S_n|>t) \leq 2\exp\left(-\frac{t^2}{2\sum_{i=1}^n \sigma_i^2}\right)$$

(Ostrovsky et al., 2014, Steinke et al., 2017).

• Maximal Partial Sum Bound: For independent sub-Gaussian $X_i$ (zero mean, parameter $\sigma_i^2$), letting $S_k = \sum_{i=1}^k X_i$,

$$\mathbb{P} \left(\max_{1\leq k\leq N} S_k > \epsilon \right) \leq \exp\left(- \frac{1}{N^2} \sum_{i=1}^N \frac{\epsilon^2}{2 \sigma_i^2}\right)$$

(Castro et al., 2011).

• Hanson–Wright Inequality: For $X = (X_1,\dots,X_n)$ with independent, mean-zero, $\|X_i\|_{\psi_2} \leq K$, and any $A\in\mathbb{R}^{n\times n}$,

$$\mathbb{P}(|X^\top A X - \mathbb{E} X^\top A X| > t) \leq 2\exp \left[ - c \min \left(\frac{t^2}{K^4\|A\|_{\mathrm{HS}}^2}, \frac{t}{K^2 \|A\|_{\mathrm{op}}} \right) \right]$$

This governs tails of quadratic forms and extends, with modifications, to settings with dependent coordinates (Rudelson et al., 2013, Zajkowski, 2018).

• Generalization via Orlicz-Norms: The Luxemburg Orlicz $\psi_2$-norm controls moments and allows for an alternative route to sub-Gaussian (and sub-exponential) tail bounds, especially in settings where variables are dependent (Zajkowski, 2018).

4. Canonical and Structured Examples

Sub-Gaussianity encompasses a broad range of distributions:

• Gaussian and Rademacher variables: Exactly sub-Gaussian; the variance proxy coincides with the variance (Ostrovsky et al., 2014, Li, 2024).
• Bernoulli and Discrete Laws: The optimal sub-Gaussian parameter for a centered indicator (Bernoulli) variable with parameter $p$ is $Q(p) = \sqrt{\frac{1-2p}{4\ln((1-p)/p)}}$ (Ostrovsky et al., 2014, Atouani et al., 7 Oct 2025). Symmetry is necessary and sufficient for strict sub-Gaussianity in the Bernoulli case ($p=1/2$) (Arbel et al., 2019).
• Uniform Distributions: The discrete uniform distribution over $N$ points is strictly sub-Gaussian, with variance proxy matching the variance (Atouani et al., 7 Oct 2025).
• Complex random sums: Linear combinations of Bernoulli or bounded variables with deterministic (e.g., Fourier) coefficients are sub-Gaussian with explicit norm scaling as the square root of the variance, underlying RIP properties in compressive sensing (Meštrović, 2018, Meštrović, 2018).
• φ–Sub-Gaussian Generalization: Variables satisfying $\mathbb{E}\exp(tX) \leq \exp(\varphi(at))$ for all $t$ and some Orlicz $N$-function $\varphi$ form a large class for which concentration and RIP properties generalize with only minor modifications (Chen et al., 2024).

5. Variance Proxy Computation: Methods and Algorithmic Aspects

Precise determination of the optimal sub-Gaussian parameter is tractable for several explicit laws, and a general theory is available for bounded-support variables:

• Variational Characterization: For a random variable $X$, the optimal variance proxy $\sigma^2_{\mathrm{opt}}$ is determined as the maximum over $\lambda$ of $$h(\lambda) = (2/\lambda^2) \ln \mathbb{E}\exp(\lambda(X-\mu))$$ (Arbel et al., 2019, Atouani et al., 7 Oct 2025). For truncated or non-symmetric laws, the equality $\sigma^2_{\mathrm{opt}} = \operatorname{Var}(X)$ may fail, and the proxy is characterized by solving a nonlinear system tied to the log-MGF and its derivatives (Barreto et al., 2024).
• Algorithmic Routine: Root-finding procedures based on the system $\lambda M'(\lambda) - 2 M(\lambda) = 0$ (where $M(\lambda)$ is the log-MGF) allow explicit computation of the optimal proxy for discrete and continuous distributions admitting an explicit MGF. This methodology is implemented in open-source tools for high-precision variance proxy computation (Atouani et al., 7 Oct 2025).
• Explicit Formulas: For the truncated Gaussian $Y_T\sim N(\mu,\sigma^2)$ conditioned to $(a,b)$, the variance proxy is

$$\sigma^2 \left[ 1 - \frac{2(\phi(\alpha)-\phi(\beta))}{(\alpha+\beta)(\Phi(\beta)-\Phi(\alpha))}\right]$$

with $\alpha=(a-\mu)/\sigma$, $\beta=(b-\mu)/\sigma$ (Barreto et al., 2024).

6. Applications and Structural Closure

Sub-Gaussian variables are pivotal in several high-dimensional and algorithmic domains:

• Johnson–Lindenstrauss Embedding: Sub-Gaussian variables, and their sparse analogues, are uniquely suited for fast random projection and dimensionality-reduction schemes, with concentration in output norm controlled by the sub-Gaussian parameter (Garivier et al., 2024).
• Random Matrix Theory & Compressed Sensing: Matrix ensembles with entries consisting of i.i.d. sub-Gaussian variables exhibit Restricted Isometry Property (RIP) with high probability, and the operational probability bounds reduce to sub-Gaussian concentrations via norm equivalences (Rudelson et al., 2013, Chen et al., 2024, Meštrović, 2018, Meštrović, 2018).
• Sums, Linear, and Quadratic Forms: Sub-Gaussianity is preserved under summation and linear transformations, with the sub-Gaussian norm of the sum controlled by the $\ell_2$ sum of the individual norms (Li, 2024, Steinke et al., 2017, Ostrovsky et al., 2014).
• Mixtures and Dependence: Mixtures (i.e., convex combinations) of (strictly) sub-Gaussian variables with common or bounded proxy remain (strictly) sub-Gaussian (Ostrovsky et al., 2014, Atouani et al., 7 Oct 2025). Extensions to dependent observations admit tail bounds for quadratic forms via Orlicz norm control, albeit with some loss in norm tightness for the non-Gaussian component (e.g., use of Hilbert–Schmidt norm instead of operator norm in the Hanson–Wright type bounds) (Zajkowski, 2018).
• Closure under Affine Transformation: The sub-Gaussian property is preserved under scaling and centering, with explicit rules for the adjustment of variance proxy and multiplicative constant (Li, 2024). For φ–sub-Gaussian variables, analogous closure properties hold for sums and tensor products (Chen et al., 2024).

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The theory of sub-Gaussian variables underpins much of modern high-dimensional analysis by enabling Gaussian-type tail estimates and exponential moment controls in non-Gaussian and even non-independent settings, with rigorous avenues for the optimized calibration of variance proxies and the extension to general convex Orlicz norms. The ongoing delineation of strict sub-Gaussianity, variance proxy computation, and their interplay with symmetry continues to refine the range and precision of concentration tools in statistics, geometry, and learning theory (Arbel et al., 2019, Atouani et al., 7 Oct 2025, Barreto et al., 2024).