# Norm of subgaussian random variables which are not necessarily centered

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.

• Keep reading the book and you'll see that centering a random variable only affects the subguassian norm by a constant. – user58955 2 days ago