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I am wondering if anybody knows some application for the random variable $x$ satisfying the following condition:

for $t\ge 1, a>0, c>0$

$$ P(|x|\ge t)\geq \frac{c}{t^a} $$

It looks like thhis is a $p$-stable random variable, but it is not.

Any source would be very helpful.

Thank you.

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up vote 1 down vote accepted

Assuming you choose $c$ such that $P(|x|\ge 1)=1$, this is the Pareto distribution with $x_m=1$. Wikipedia lists a number of empirical examples where the law is approximately satisfied. I don't know if the law ever arises naturally in a non-approximate way.

The law cannot be stable: it has a discontinuous PDF (at $t=1$) but the convolution $aX+bY$ (for $a,b\ne 0$) will always have a continuous PDF.

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I did not understand what kind of application did you propose. Did you wanted to say that its natural that the similar distributions will arise, or you had in mind some particular applicaions? – Michael Jul 12 '12 at 4:46
@Michael: The real-world applications that could be modeled by this distribution are given in the Wikipedia article I linked to. But, these models are just rough approximations: all of these examples could be more accurately described by more sophisticated distributions, at the cost of additional complexity. – Generic Human Jul 23 '12 at 11:32

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