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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 $c>0$.

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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 $c'$.

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

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

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