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Results tagged with orlicz-spaces
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user 101207
0
votes
0
answers
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
$$\lVert \vec X\rVert_{\psi_2} = \sup_{\theta : …
- 2,183
1
vote
1
answer
71
views
Improved bounds on $\lVert XY\rVert_{\psi_2}$ via concentration data of the (bounded) random...
Throughout I will use the language of Orlicz norms associated with the family of functions $\psi_a(x) = \exp(x^a)-1$ for $a\in[1,\infty)$, and
$$\psi_\infty(x) = \begin{cases}\infty & x>1\\1 & x = 1\\ …
- 2,183
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 …
- 2,183
0
votes
1
answer
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 …
- 2,183