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→‎Definition of long-tailed distribution 长尾分布的定义

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=== Definition of long-tailed distribution 长尾分布的定义 ===

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The distribution of a [[random variable]] ''X'' with [[cumulative distribution function|distribution function]] ''F'' is said to have a long right tail~~<ref name="Asmussen"/>~~ if for all ''t'' > 0,

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The distribution of a [[random variable]] ''X'' with [[cumulative distribution function|distribution function]] ''F'' is said to have a long right tail if for all ''t'' > 0,

The distribution of a random variable X with distribution function F is said to have a long right tail[1] if for all t > 0,

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</math>

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F I([0,\infty)) is. Here I([0,\infty)) is the indicator function of the positive half-line. Alternatively, a random variable X supported on the real line is subexponential if and only if X^+ = \max(0,X) is subexponential.

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~~F i ([0，infty))是。这里 i ([0，infty))是正半直线的指示函数。或者，实数行上支持的随机变量 x 是子指数当且仅当 x ^ + = max (0，x)是子指数。~~

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~~All subexponential distributions are long-tailed, but examples can be constructed of long-tailed distributions that are not subexponential.~~

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~~所有的次指数分布都是长尾分布，但是例子可以由非次指数的长尾分布构造。~~

This has the intuitive interpretation for a right-tailed long-tailed distributed quantity that if the long-tailed quantity exceeds some high level, the probability approaches 1 that it will exceed any other higher level.

All long-tailed distributions are heavy-tailed, but the converse is false, and it is possible to construct heavy-tailed distributions that are not long-tailed.

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~~All commonly used heavy-tailed distributions are subexponential.~~

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~~所有常用的重尾分布都是次指数分布。~~

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~~:<math>~~

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~~\overline{F}(x+t) \sim \overline{F}(x) \quad \mbox{as } x \to \infty. \,~~

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~~Those that are two-tailed include:~~

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~~有两条尾巴的包括:~~

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~~</math>~~

=== Subexponential distributions 长尾分布的定义 ===

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