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 $c>0$.
 A: 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 $c'$.
A: 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.
