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I'm interested in a class C of $R^1$-valued random variables $\xi$ which satisfy an inequality of the type $$ (1) \qquad E|\xi|^p \leq F(E|\xi|^2), $$ where $p>2$, $F$ is a certain non-decreasing on $(0; + \infty)$ function, $E\xi=0$ and random variables from class C must have tails which are heavier than Gaussian tails. One more requirement: inequality (1) must be homogenous, i.e. it must turn into an equivalent inequality if we replace $\xi$ by $k \xi$, where $k \in R^1$.

Do there exist such classes? Inequalities of type (1) are true for Gaussian and sub-Gaussian random variables, but the case when the tails are heavier seems to be non-trivial.

Thanks in advance.

Best wishes, Ievgen.

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  • $\begingroup$ What do you mean by a class of random variables? It seems easy to construct sets of examples. $\endgroup$
    – Douglas Zare
    Commented Aug 26, 2014 at 9:07
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    $\begingroup$ By a class of random variables I mean a more or less wide family, something more ample than a class consisting e.g. only of random variables with "centered" exponential distribution. It would be ideal to have a characterization of such class in terms of something like the order of decay of $P\{|\xi|>x\}$ as $x \to \infty$. $\endgroup$
    – Ievgen
    Commented Aug 26, 2014 at 10:52

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