TITLE: What is a long-tailed distribution for physicists?
QUESTION [0 upvotes]: What is the most common definition of long tailed distribution for physicists? I am looking for definition and examples. Examples should have arguments why the distribution is or is not long tailed.
I know that there are a few definitions.

REPLY [1 votes]: Ok, I'll start. Definition and list (filled with mistakes). Please, comment/add/edit/etc.
Definition
${lim}_{x\to\infty} P(x) = P(x+t),$ for $\forall\; t > 0$ 
$P(x)$ is finite i.e. $P(x) = \int p(x) dx = $ const.
Clarifying definitions
$p(x) :=$ "probability density function of measurable $x \in \Re_+$"
$P(x) :=$ "cumulative distribution function" = $\int_{0}^x p(x)\;dx$
$a > 0$
$0 < f(x) < const $
$g(x) \ne 0$
Long tailed

all exponential distributions: $p = f(x) exp(-ax)$
all power law distributions: $p = x^{-\alpha}$, where $\alpha < -1$ 
all "stretched exponentials": $p = f(x) exp(-ax^{g(x)})$

Non Long tailed

Uniform distribution: p(x) = const
All "increasing or equal" distributions: $|p(x)| \le |p(x+a)|  $ for $\forall a$ 
power law distributions with $p(x) = x^\alpha$, where $-1 \le \alpha \le 0$ (integral not finite)
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