# Sub-gaussian norm of sample mean?

Suppose $$X$$ is a mean-zero random variable with subgaussian norm $$k = \|X\|_{\Psi_2}:= \inf\{t>0: \mathbb{E} \exp(X^2/t^2)\leq 2\}.$$ What can we say about the sub-gaussian norm of $$m$$ iid samples $$X_1, \dots, X_m$$ from the same distribution as $$m$$? I believe that $$\bigg\| \frac{1}{m}\sum_{i=1}^{m}X_{i}\bigg\|_{\Psi_2}\leq c \frac{k}{\sqrt{m}}$$ for some absolute constant $$c>0$$.

It is easiest to use a different [but equivalent] definition of sub-Gaussianity. I use Proposition 2.5.2 of this book twice below in order to convert back and forth from your definition to the MGF definition.

If $$k := \|X\|_{\Psi_2}$$, then $$X$$ satisfies $$\mathbb{E} e^{\lambda X} \le e^{\lambda^2 (ck)^2}, \forall \lambda \in \mathbb{R}$$ for some universal constant $$c$$.

Then $$\mathbb{E} e^{\lambda \frac{1}{m} \sum_{i=1}^m X_i} \le e^{\frac{\lambda^2}{m} (c k)^2},$$ so the sample mean is sub-Gaussian with norm $$\left\|\frac{1}{m} \sum_{i=1}^m X_i \right\|_{\Psi_2} \le c' \frac{k}{\sqrt{m}}$$ for another universal constant $$c'$$.

• Great! Thanks for your response. – Alex Lapanowski May 13 '19 at 22:24

By Theorem 4 in

Talagrand, M. (1989). Isoperimetry and integrability of the sum of independentBanach-space valued random variables.Ann. Probab.17, 1546–1570,

the following inequality holds for each independent centered sequence $$\left(X_i\right)_{i\geqslant 1}$$: $$\left\lVert \sum_{i=1}^nX_i\right\rVert_{\Psi_2}\leqslant K \mathbb E\left\lvert\sum_{i=1}^nX_i\right\rvert+K \left(\sum_{i=1}^n\left\lVert X_i\right\rVert_{\Psi_2}^2\right)^{1/2}.$$ Here the constant $$K$$ is universal and is in particular independent on $$n$$ and the distribution of the $$X_i$$. When all the $$X_i$$ have the same distribution, it can be simplified (after having bounded the term $$\mathbb E\left\lvert\sum_{i=1}^nX_i\right\rvert$$ by the $$\mathbb L^2$$-norm, $$\left\lVert \sum_{i=1}^nX_i\right\rVert_{\Psi_2}\leqslant K\sqrt n\left\lVert X_1\right\rVert_2+K\sqrt n\left\lVert X_1\right\rVert_{\Psi_2}$$ and one can control the term $$\left\lVert X_1\right\rVert_2$$ by $$\left\lVert X_1\right\rVert_{\Psi_2}$$, up to maybe a universal constant.

• Thank you for your response. Both of you have been helpful. – Alex Lapanowski May 13 '19 at 22:24