Sum of subgaussian random variables Let $\{X_i\}_{i=1}^N$ be a set of $\nu$-subgaussian random variables, meaning
$$
\mathbb{E}(\exp(tX_i)) \leq e^{\nu t^2/2}.
$$
Suppose also that $X_i$ are normalized so that $\mathbb{E}(X_i) = 0$
and $\mathbb{E}(X_i^2) = 1$.
Let $a_i$ be a fixed (deterministic) sequence of coefficients, and define the random variables
$$
Y = \sum_{i=1}^N a_i X_i,
$$
and its normalization
$$
Z = \frac{Y}{\sqrt{\mathbb{E} Y^2}}. 
$$
If the $X_i$ are assumed to be independent, then it is not hard to check that $Z$ is also $\nu$-subgaussian, because one can compute its moment generating function.  
My question is whether this hypothesis is necessary.

Is $Z$ a $\nu$-subgaussian random variable, even if $X_i$ are not independent?

The reason one might hope that this is true is that the normalization of $Z$ takes care of any growth of $Z$ that might arise from the dependencies.
My best guess for how to prove this is to bound the joint distribution of $(X_i)$ by a multivariate Gaussian distribution, and then observe that $Y$ is some marginal and is therefore bounded by a Gaussian of the correct variance(?).  But I'm not sure how to fill in the details. 
 A: Not sure if this answers your question but I also stumbled upon this problem, hopefully my solution adds to the discussion for this question.
I am going to first assume $N = 2$, let $X_1$ and $X_2$ be two subgaussian random variables. By definition, we have,
$$ E(\exp{tX_i}) \leq \exp{(t^2\nu_i^2)/2}\ \ \text{for} \ i =1,2.$$
We have $Y = a_1 X_1 + a_2 X_2$. Consider,
$$ E(\exp(tY)) = E(\exp\{ta_1 X_1 + ta_2 X_2\}) = E(\exp{ta_1X_1}\cdot \exp{t a_2 X_2}).$$
Now by Cauchy-Schwarz inequality, we have,
\begin{align*}
E(\exp{ta_1X_1}\cdot \exp{t a_2 X_2}) &\leq (E(\exp{2ta_1 X_1}))^{1/2} (E(\exp{2ta_2 X_2}))^{-1/2}\\
&\leq \exp{t^2a_1^2 \nu_1^2} \cdot \exp{t^2 a_2^2 \nu_2^2}\\
&= \exp\{t^2 \sum_{i=1}^2 a_i^2 \nu_i^2\},
\end{align*}
where the last inequality was by using the definition of sub-gaussian r.v. $X_1$ and $X_2$. This gives us that $Y \sim SubG(\sum_{i=1}^2 a_i^2 \nu_i^2)$ is also a sub-gaussian random variable. I guess if you normalized $Y$ then the normalizing constant will appear in addition to the $a_1^2$. I think a similar logic could hold for $N$ random variables.
