propose constant 48 'convex sub-Gaussian constant'

constants / 48a.md

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+# Convex Sub-Gaussian Constant
+
+## Description of constant
+
+Let $X$ denote a univariate random variable. Say that $X$ is (-sub-Gaussian if for all $t \in \mathbf{R}$, there holds the exponential inequality $\log\left( \mathbf{E} \left[ \exp \left( t X \right) \right] \right) \leq \frac{1}{2} \cdot t^2$.
+
+Let $X$, $X^\prime$ denote two univariate random variables. Say that $X$ is majorised by $X^\prime$ in the convex ordering if for all convex functions $F$, it holds that $\mathbf{E} \left[ f \left( X \right) \right] \leq \mathbf{E} \left[ f \left( X^\prime \right) \right]$. Equivalently, by a result of Strassen, one can say that this holds iff one can couple $\left( X, X^\prime \right)$ so that $\mathbf{E} \left[ X^\prime \mid X \right] = X$ almost surely.
+
+It is possible to show (see e.g. [MO23]) that there is a universal constant $C \in \left( 0, \infty \right)$ such that for any 1-sub-Gaussian random variable $X$, the centered Gaussian random variable $X^\prime \sim \mathcal{N} \left( 0, C \right)$ dominates $X$ in the convex ordering. Write then $C_{CSG}$ for the infimum of all $C$ for which this comparison holds.
+
+The utility of this is that when one knows that a given random variable $X$ is sub-Gaussian (which is often), and one is interested in controlling the expectation of a convex function of $X$ (which is also rather often), then one can instead compute the same expectation for a Gaussian random variable (which is often much easier and more explicit) and thereby obtain a rigorous upper bound on the expectation of interest. In this regard, obtaining a smaller $C$ leads to sharper bounds.
+
+## Known upper bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| [$< \infty$] | [MO23] | From the solution provided, a constant could, in principle, be extracted. |
+| [$ < \infty$] | [vH25] | This work establishes that a similar comparison can even be made in the multivariate case, with dimension-free constant. |
+
+## Known lower bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| 1 | [N/A] | This can be obtained by taking $X$ to be a standard Gaussian random variable.
+| [$\sqrt{\pi / 2}$] | [N/A] | This can be obtained by taking $X$ to be a standard Rademacher random variable, and taking $f$ to be the absolute value function. |
+
+## Additional comments
+
+As mentioned in 'known upper bounds', the same comparison is known to hold in the case of multivariate sub-Gaussianity and multivariate normality (see [vH25] for details), and it is eventually of interest to compute a $C$ which works in all dimensions.
+
+## References
+
+[MO23] 'Sub-Gaussian random variables and convex ordering', asked at [MathOverflow](https://mathoverflow.net/questions/456236/sub-gaussian-random-variables-and-convex-ordering) \
+[vH25] van Handel, R. 'On the subgaussian comparison theorem'. [arXiv](http://arxiv.org/abs/2512.18588)
+

constants/48a.md

@@ -0,0 +1,35 @@
+# Convex Sub-Gaussian Constant
+
+## Description of constant
+
+Let $X$ denote a univariate random variable. Say that $X$ is (-sub-Gaussian if for all $t \in \mathbf{R}$, there holds the exponential inequality $\log\left( \mathbf{E} \left[ \exp \left( t X \right) \right] \right) \leq \frac{1}{2} \cdot t^2$.
+
+Let $X$, $X^\prime$ denote two univariate random variables. Say that $X$ is majorised by $X^\prime$ in the convex ordering if for all convex functions $F$, it holds that $\mathbf{E} \left[ f \left( X \right) \right] \leq \mathbf{E} \left[ f \left( X^\prime \right) \right]$. Equivalently, by a result of Strassen, one can say that this holds iff one can couple $\left( X, X^\prime \right)$ so that $\mathbf{E} \left[ X^\prime \mid X \right] = X$ almost surely.
+
+It is possible to show (see e.g. [MO23]) that there is a universal constant $C \in \left( 0, \infty \right)$ such that for any 1-sub-Gaussian random variable $X$, the centered Gaussian random variable $X^\prime \sim \mathcal{N} \left( 0, C \right)$ dominates $X$ in the convex ordering. Write then $C_{CSG}$ for the infimum of all $C$ for which this comparison holds.
+
+The utility of this is that when one knows that a given random variable $X$ is sub-Gaussian (which is often), and one is interested in controlling the expectation of a convex function of $X$ (which is also rather often), then one can instead compute the same expectation for a Gaussian random variable (which is often much easier and more explicit) and thereby obtain a rigorous upper bound on the expectation of interest. In this regard, obtaining a smaller $C$ leads to sharper bounds.
+
+## Known upper bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| [$< \infty$] | [MO23] | From the solution provided, a constant could, in principle, be extracted. |
+| [$ < \infty$] | [vH25] | This work establishes that a similar comparison can even be made in the multivariate case, with dimension-free constant. |
+
+## Known lower bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| 1 | [N/A] | This can be obtained by taking $X$ to be a standard Gaussian random variable.
+| [$\pi / 2$] | [N/A] | This can be obtained by taking $X$ to be a standard Rademacher random variable, and taking $f$ to be the absolute value function. |
+
+## Additional comments
+
+As mentioned in 'known upper bounds', the same comparison is known to hold in the case of multivariate sub-Gaussianity and multivariate normality (see [vH25] for details), and it is eventually of interest to compute a $C$ which works in all dimensions.
+
+## References
+
+[MO23] 'Sub-Gaussian random variables and convex ordering', asked at [MathOverflow](https://mathoverflow.net/questions/456236/sub-gaussian-random-variables-and-convex-ordering) \
+[vH25] van Handel, R. 'On the subgaussian comparison theorem'. [arXiv](http://arxiv.org/abs/2512.18588)
+