961
个编辑
更改
→Definition of heavy-tailed distribution
:<math>
:<math>
\int_{-\infty}^\infty e^{t x} \,dF(x) = \infty \quad \mbox{for all } t>0.
\int_{-\infty}^\infty e^{t x} \,dF(x) = \infty \quad \mbox{for all } t>0.
:<math>
:<math>
\lim_{x \to \infty} e^{t x}\Pr[X>x] = \infty \quad \mbox{for all } t>0.\,
\lim_{x \to \infty} e^{t x}\Pr[X>x] = \infty \quad \mbox{for all } t>0.\,
<math>
<math>
\overline{F^{*2}}(x) \sim 2\overline{F}(x) \quad \mbox{as } x \to \infty.
\overline{F^{*2}}(x) \sim 2\overline{F}(x) \quad \mbox{as } x \to \infty.
: <math>\overline{F}(x) \equiv \Pr[X>x] \, </math>
: <math>\overline{F}(x) \equiv \Pr[X>x] \, </math>
</math>
</math>
This implies that, for any n \geq 1,
This implies that, for any n \geq 1,
:<math>
:<math>
\lim_{x \to \infty} e^{t x}\overline{F}(x) = \infty \quad \mbox{for all } t >0.\,
\lim_{x \to \infty} e^{t x}\overline{F}(x) = \infty \quad \mbox{for all } t >0.\,
\overline{F^{*n}}(x) \sim n\overline{F}(x) \quad \mbox{as } x \to \infty.
\overline{F^{*n}}(x) \sim n\overline{F}(x) \quad \mbox{as } x \to \infty.
</math>
</math>
===Definition of long-tailed distribution===
===Definition of long-tailed distribution===