Convex sub-Gaussian comparison constant
Description of constant
Let $X$ be a real random variable. Following [vH25], in dimension $n$ we call $X$ $1$-subgaussian if \(\mathbf{E}[X]=0 \quad\text{and}\quad \mathbf{P}\!\left[\lvert\langle v,X\rangle\rvert>x\right]\le 2e^{-x^2/2}\) for all $x\ge 0$ and $v\in S^{n-1}$. [vH25-def-subg]
For real random variables $X,Y$, write $X\preceq_{cx}Y$ if \(\mathbf{E}[f(X)]\le \mathbf{E}[f(Y)]\) for every convex $f:\mathbf{R}\to\mathbf{R}$ for which both expectations are finite.
Define \(C_{48}:=\inf\Bigl\{C>0:\ \forall\ \text{$1$-sub-Gaussian }X,\ \exists\ Z\sim\mathcal{N}(0,C)\ \text{with }X\preceq_{cx}Z\Bigr\}.\)
Theorem 1.1 in [vH25] implies $C_{48}<\infty$ (take $n=1$). [vH25-thm11]
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $<\infty$ | [vH25] | A universal Gaussian comparator exists in every dimension; in particular, in dimension $1$, which gives finiteness of $C_{48}$. [vH25-thm11] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Elementary | Taking $X\sim\mathcal{N}(0,1)$ and testing with $f(x)=x^2$ gives $1=\mathbf{E}[X^2]\le \mathbf{E}[Z^2]=C$, so any admissible $C$ must satisfy $C\ge 1$. |
Additional comments and links
- The comparison theorem is dimension-free: the same universal constant works for all $n\ge 1$. [vH25-thm11]
- A Strassen-type reformulation is given as Corollary 1.2: one can construct $X$ and a standard Gaussian $G$ on a common space so that $X=c\,\mathbf{E}[G\mid X]$. [vH25-cor12]
- Historical discussion: MathOverflow question on sub-Gaussian variables and convex ordering.
References
- [vH25] van Handel, Ramon. On the subgaussian comparison theorem. arXiv:2512.18588 (2025). DOI: 10.48550/arXiv.2512.18588. Google Scholar. arXiv PDF. Author PDF.
- [vH25-def-subg]
loc: Author PDF p.1, Introduction (definition paragraph).
quote: “A random vector $X$ in $\mathbf{R}^n$ is said to be $1$-subgaussian if $\mathbf{E}[X]=0$ and $\mathbf{P}[|\langle v, X\rangle| > x] \leq 2e^{-x^2/2}$ for all $x \geq 0$ and $v \in S^{n-1}$” - [vH25-thm11]
loc: Author PDF p.1, Theorem 1.1.
quote: “Let $X$ be any $1$-subgaussian random vector in $\mathbf{R}^n$ and $G \sim N(0, I_n)$ be a standard Gaussian vector in $\mathbf{R}^n$. Then $\mathbf{E}[f(X)] \leq \mathbf{E}[f(cG)]$ for every convex function $f : \mathbf{R}^n \to \mathbf{R}$, where $c$ is a universal constant.” - [vH25-cor12]
loc: Author PDF p.1, Corollary 1.2.
quote: “There is a universal constant $c$ such that for every $1$-subgaussian vector $X$ in $\mathbf{R}^n$, we can construct $X$ and a standard Gaussian vector $G \sim N(0, I_n)$ on a common probability space such that $X = c\mathbf{E}[G|X]$.”
- [vH25-def-subg]