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189 views

Lower bounds for sub-Gaussians?

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_{\theta : …
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1 answer
71 views

Improved bounds on $\lVert XY\rVert_{\psi_2}$ via concentration data of the (bounded) random...

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 & x = 1\\ …
1 vote
1 answer
140 views

Is the space of bounded $\psi_\infty$ Orlicz norm random variables equal to $L^\infty$?

Let $\psi_\alpha(x) = \exp(x^\alpha)-1$ for $\alpha\geq 1$. Define $$ \psi_\infty(x) = \begin{cases}\infty & x>1\\1& x = 1\\ 0 & x <1 \end{cases} $$ to be such that for any $x>0$ $\psi_\infty(x) = \li …
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112 views

Is the product of sub-Gaussian polynomials in $\mathbb{R}[x]/(x^n-1)$ sub-Gaussian?

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 norm …