<p>I am reading a <a href="http://arxiv.org/pdf/1405.3629v1.pdf" rel="nofollow">paper</a> which is mainly about Dobrushin's contraction coefficient and its generalization. In page 27, the following is defined:
Consider an arbitrary, non-negative, convex function $\psi:\mathbb{R}\mapsto \mathbb{R}$ with $\psi(0)&lt;1$. Then for an integrable random variable $X$ 
$$||X||_{\psi}:=\inf \{c&gt;0:~\mathbb{E}[\psi(\frac{X}{c})]\leq 1\}.$$</p>

<p>Then letting $\psi^*$ be the convex conjugate of $\psi$, we can write the following for arbitrary random variables $X$ and $Y$ by Young's inequality: 
$$XY\leq \psi(X)+\psi^*(Y)$$ and hence, 
$$\mathbb{E}[XY]\leq 2||X||_{\psi}||Y||_{\psi^*}.$$</p>

<p>First of all, in the typical definition of Orlicz norm, it is assumed that $\psi(0)=0$ and also $\psi$ is non-decreasing. Why are those properties removed here? </p>

<p>Secondly, the last inequality is not clear to me. Can anybody proivde a hint on how to show that?</p>

CC BY-SA 3.0
2015-02-13T05:10:33.293Z
What do you know about $\frac{X}{\|X\|_{\phi}}\frac{Y}{\|Y\|_{\phi^*}}$ by Young's inequality and by the definition of the norm?
Sasho Nikolov
CC BY-SA 3.0
2015-02-13T07:18:42.760Z
Please don't cross-post so soon together, (http://mathoverflow.net/questions/196356/orlicz-norm-and-a-result-on-expectation), or at least link to the previous question. Thanks.
usul
<pr.probability><st.statistics><convex-geometry>
Orlicz Norm and a result on expectation
SAmath
# Orlicz Norm and a result on expectation

(Asked by: username_0 on 2015-02-13)

I am reading a [paper](http://arxiv.org/pdf/1405.3629v1.pdf) which is mainly about Dobrushin's contraction coefficient and its generalization. In page 27, the following is defined:
Consider an arbitrary, non-negative, convex function $\psi:\mathbb{R}\mapsto \mathbb{R}$ with $\psi(0)<1$. Then for an integrable random variable $X$
$$||X||\_{\psi}:=\inf \{c>0:~\mathbb{E}[\psi(\frac{X}{c})]\leq 1\}.$$

Then letting $\psi^\*$ be the convex conjugate of $\psi$, we can write the following for arbitrary random variables $X$ and $Y$ by Young's inequality:
$$XY\leq \psi(X)+\psi^\*(Y)$$ and hence,
$$\mathbb{E}[XY]\leq 2||X||\_{\psi}||Y||\_{\psi^\*}.$$

First of all, in the typical definition of Orlicz norm, it is assumed that $\psi(0)=0$ and also $\psi$ is non-decreasing. Why are those properties removed here?

Secondly, the last inequality is not clear to me. Can anybody proivde a hint on how to show that?

**Tags:** pr.probability, st.statistics, convex-geometry

> **Comments:**
> **username_1**: What do you know about $\frac{X}{\|X\|_{\phi}}\frac{Y}{\|Y\|_{\phi^*}}$ by Young's inequality and by the definition of the norm?  
> **username_2**: Please don't cross-post so soon together, (http://mathoverflow.net/questions/196356/orlicz-norm-and-a-result-on-expectation), or at least link to the previous question. Thanks.  

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## 0 answers

