3.9.1. Heavy-tailed distributions#

  • Does the CLT hold for heavy-tailed distributions?

3.9.1.1. Classification#

Heavy-tailed
\[\lim_{x\rightarrow+\infty} e^{tx}\overline{F}(x) = \infty \ , \quad \text{for all $t > 0$} \ ,\]

being \(\overline{F}(x) := \text{P}(X > x)\) the tail cumulative distribution function.

todo alternative definition in terms of moment generating function of \(X\)

Long-tailed distribution
\[\lim_{x \rightarrow + \infty} \text{P}(X > x + t | X > x) = 1 \ ,\]

for all \(t > 0\), or equivalently

\[\overline{F}(x+t) \sim \overline{F}(x) \ , \quad \text{for $x\rightarrow +\infty$} \ ,\]

being \(\overline{F}(x) := \text{P}(X > x)\) the tail cumulative distribution function.

Sub-exponential distribution

…

3.9.1.2. Examples#

  • Cauchy distribution

  • Power-law distribution