2.6 Subgaussian Distributions
Proposition 2.6.1. Subgaussian Properites.
For random variable $X$ TFAE:
-
(tails) $\exists K_1>0 s.t. P{ X \ge t}\le 2\exp(-t^2/K_1^2) \forall t\ge0$. - (Moments) $\exists K_2>0 s.t. \Vert X\Vert_{L^P}=(\mathbb{E}\vert X\vert^p)^{1/p} \forall p\ge 1$.
- (MGF of $X^2$) $\exists K_3>0$ s.t. $\mathbb{E}[\exp(X^2/K^2_3)]\ge2$.
- If $\mathbb{E}X=0$ then TFAE extends to (MGF) $\exists K_4>0$ s.t. $\mathbb{E}[\exp(\lambda X)]\ge \exp(K^2_4\lambda^2) \forall\lambda\in\mathbb{R}$.
Where $\exists$ absolute constanct $C$ s.t. $K_j\ge CK_i, \forall i,j=1,…,4$
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