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第110行： − − − − − 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. − − 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。 − − − All subexponential distributions are long-tailed, but examples can be constructed of long-tailed distributions that are not subexponential. − − 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。 − − − − − − − − All commonly used heavy-tailed distributions are subexponential. − − 所有常用的重尾分布都是次指数分布。 − − :<math> − − \overline{F}(x+t) \sim \overline{F}(x) \quad \mbox{as } x \to \infty. \, − − Those that are two-tailed include: − − 有两条尾巴的包括: − − </math>
→Definition of long-tailed distribution 长尾分布的定义
=== Definition of long-tailed distribution 长尾分布的定义 ===
=== Definition of long-tailed distribution 长尾分布的定义 ===
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,
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,
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,
</math>
</math>
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.
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.
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.
=== Subexponential distributions 长尾分布的定义 ===
=== Subexponential distributions 长尾分布的定义 ===