All Questions

For a random variable $X$, define $$\lVert X\rVert_{\psi_2} =\inf \{k>0\mid \mathbb{E}[\exp((X/k)^2)]\leq 2\}$$ and for a random vector $\vec X$, define $$\lVert \vec X\rVert_{\psi_2} = \sup_{\...

Throughout I will use the language of Orlicz norms associated with the family of functions $\psi_a(x) = \exp(x^a)-1$ for $a\in[1,\infty)$, and $$\psi_\infty(x) = \begin{cases}\infty & x>1\\1 &...

Let $\psi_\alpha(x) := \exp(x^\alpha)-1$. It is well-known that for $\alpha\geq 1$ that $$\lVert X\rVert_{\psi_\alpha} = \inf\{k>0\mid \mathbb{E}[\psi_\alpha(|X|/k)] \leq 1\}$$ defines an Orlicz ...

Let $X_{i}$ be a collection of iid random variables of cardinality $n$, and let $S_{n}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i}.$ Let $|| X||:=\inf_{B}\{E[\exp(X/B)-1]\leq 1\}$. This is the so-called sub-...

Let $\phi$ be an $N$-function, (i.e. $\phi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ is convex and satisfies $\lim_{t \to 0} \frac{\phi(t)}{t} = 0, \lim_{t\to \infty} \frac{\phi(t)}{t} = \infty$)....