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So suppose you have independent subgaussian random variables $X_i$. Is there any particular result that bounds the subgaussian norm of the sum $||\sum_{i=1}^n X_i||_{\psi_2}$ in terms of the subgaussian norms $||X_i||_{\psi_2}$ where $||\cdot||_{\psi_2}$ is the Orlicz norm with $\psi_2 = \exp(x^2)-1$. In other words

$$ ||X_i||_{\psi_2}=\inf\left\{ t: E\exp(\frac{X_i^2}{t^2}) \right\}\leq 2 $$

I know results when the $X_i$ are centered, in other words if $EX_i=0$ for all $i\in \{1,\cdots,n\}$ then we have

$$ \left|\left|\sum_{i=1}^n X_i \right|\right|_{\psi_2}^2 \leq C \sum_{i=1}^n ||X_i||^2_{\psi_2} $$

for some absolute constant $C$(see proposition 2.6.1 in Vershynin's High Dimensional Probability book). But I don't see any results for when the $X_i$ are not neccesarily centered.

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  • $\begingroup$ Keep reading the book and you'll see that centering a random variable only affects the subguassian norm by a constant. $\endgroup$ – user58955 Feb 14 at 7:03

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