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第19行：
第19行： − The distribution of a [[random variable]] ''X'' with [[cumulative distribution function|distribution function]] ''F'' is said to have a heavy (right) tail if the [[moment generating function]] of ''X'', ''M<sub>X</sub>''(''t''), is infinite for all ''t'' > 0.<ref name="ReferenceA">Rolski, Schmidli, Scmidt, Teugels, ''Stochastic Processes for Insurance and Finance'', 1999</ref> − The distribution of a random variable X with distribution function F is said to have a heavy (right) tail if the moment generating function of X, MX(t), is infinite for all t > 0.+ − 如果''<math>X</math>''的矩生成函数, ''M<sub>X</sub>''(''t'')对于所有''t'' > 0都是无限的，则具有分布函数''F''的随机变量''X''的分布被称为重尾（右）。<ref name="ReferenceA">Rolski, Schmidli, Scmidt, Teugels, ''Stochastic Processes for Insurance and Finance'', 1999</ref> − + − − That means − − − − 第38行：
第30行： − + − An implication of this is that − − − − 第49行：
第36行： − − − This is also written in terms of the tail distribution function 第71行：
第55行： − + − 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, 第86行：
第67行： − − or equivalently 第96行：
第75行： − 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. 第112行：
第83行： − + − Subexponentiality is defined in terms of [[Convolution of probability distributions|convolutions of probability distributions]]. For two independent, identically distributed [[random variables]] <math> X_1,X_2</math> with common distribution function <math>F</math> the convolution of <math>F</math> with itself, <math>F^{*2}</math> is convolution square, using [[Lebesgue–Stieltjes integration]], by: − − Subexponentiality is defined in terms of [[Convolution of probability distributions|convolutions of probability distributions]]. For two independent, identically distributed [[random variables]] <math> X_1,X_2</math> with common distribution function <math>F</math> the convolution of <math>F</math> with itself, <math>F^{*2}</math> is convolution square, using [[Lebesgue–Stieltjes integration]], by: 第125行：
第93行： − − and the ''n''-fold convolution <math>F^{*n}</math> is defined inductively by the rule: 第135行：
第101行： + − The tail distribution function <math>\overline{F}</math> is defined as <math>\overline{F}(x) = 1-F(x)</math>. − 尾分布函数<math>\overline{F}</math>定义为<math>\overline{F}(x) = 1-F(x)</math>。 − − A distribution <math>F</math> on the positive half-line is subexponential<ref name="Asmussen"/><ref>{{Cite web|url=https://www.researchgate.net/publication/242637603_A_Theorem_on_Sums_of_Independent_Positive_Random_Variables_and_Its_Applications_to_Branching_Random_Processes|title=A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes|last=Chistyakov|first=V. P.|date=1964|website=ResearchGate|language=en|archive-url=|archive-date=|access-date=April 7, 2019}}</ref><ref>{{Cite web|url=https://projecteuclid.org/download/pdf_1/euclid.aop/1176996225|title=The Class of Subexponential Distributions|last=Teugels|first=Jozef L.|authorlink=|date=1975|website=|publisher=Annals of Probability|publication-place=[[KU Leuven|University of Louvain]]|archive-url=|archive-date=|access-date=April 7, 2019}}</ref> if 第150行：
第113行： − − This implies<ref name="Embrechts">{{cite book |author1=Embrechts P. |author2=Klueppelberg C. |author3=Mikosch T. |title=Modelling extremal events for insurance and finance |publisher=Springer | series = Stochastic Modelling and Applied Probability|location=Berlin |year=1997 | volume=33| doi = 10.1007/978-3-642-33483-2|isbn=978-3-642-08242-9 }}</ref> that, for any <math>n \geq 1</math>, 第160行：
第121行： − The probabilistic interpretation<ref name="Embrechts"/> of this is that, for a sum of <math>n</math> [[statistical independence|independent]] [[random variables]] <math>X_1,\ldots,X_n</math> with common distribution <math>F</math>, 第169行：
第129行： − This is often known as the principle of the single big jump<ref>{{Cite journal | last1 = Foss | first1 = S. | last2 = Konstantopoulos | first2 = T. | last3 = Zachary | first3 = S. | doi = 10.1007/s10959-007-0081-2 | title = Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments | journal = Journal of Theoretical Probability| volume = 20 | issue = 3 | pages = 581 | year = 2007 | arxiv = math/0509605| pmid = | url = http://www.math.nsc.ru/LBRT/v1/foss/fkz_revised.pdf| pmc = | citeseerx = 10.1.1.210.1699 }}</ref> or catastrophe principle.<ref>{{cite web| url = http://rigorandrelevance.wordpress.com/2014/01/09/catastrophes-conspiracies-and-subexponential-distributions-part-iii/ | title = Catastrophes, Conspiracies, and Subexponential Distributions (Part III) | first = Adam | last = Wierman | authorlink = Adam Wierman | date = January 9, 2014 | accessdate = January 9, 2014 | website = Rigor + Relevance blog | publisher = RSRG, Caltech}}</ref> − − − A distribution <math>F</math> on the whole real line is subexponential if the distribution − <math>F I([0,\infty))</math> is.<ref>{{cite journal | last = Willekens | first = E. | title = Subexponentiality on the real line | journal = Technical Report | publisher = K.U. Leuven | year = 1986}}</ref> Here <math>I([0,\infty))</math> is the [[indicator function]] of the positive half-line. Alternatively, a random variable <math>X</math> supported on the real line is subexponential if and only if <math>X^+ = \max(0,X)</math> is subexponential. − − − All subexponential distributions are long-tailed, but examples can be constructed of long-tailed distributions that are not subexponential.
→定义
=== 重尾分布的定义 ===
=== 重尾分布的定义 ===
如果''<math>X</math>''的矩生成函数, ''<math>M<sub>X</sub></math>''(''<math>t</math>'')对于所有''<math>t</math>'' > 0都是无限的，则具有分布函数''<math>F</math>''的随机变量''<math>X</math>''的分布被称为重尾（右）。<ref name="ReferenceA">Rolski, Schmidli, Scmidt, Teugels, ''Stochastic Processes for Insurance and Finance'', 1999</ref>
也就是说:<math>
也就是说
:<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.
这意味着:<math>
这意味着
:<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>
也可以写成<font color="#ff8000">尾分布函数 the tail distribution function </font>：
也可以写成<font color="#ff8000">尾分布函数 the tail distribution function </font>：
</math>
</math>
=== Definition of long-tailed distribution 长尾分布的定义 ===
=== 长尾分布的定义 ===
如果对于所有''t''>0，则称具有分布函数''F''的随机变量''X''的分布为有较长的右尾，
如果对于所有''t''>0，则称具有分布函数''F''的随机变量''X''的分布为有较长的右尾，
或等同于
或等同于
</math>
</math>
对于右尾长尾分布量具有直观的解释，即如果长尾量超过某个高水平，则概率将接近1，它将超过其他更高的水平。
对于右尾长尾分布量具有直观的解释，即如果长尾量超过某个高水平，则概率将接近1，它将超过其他更高的水平。
所有长尾分布都是重尾分布，但反过来不一定成立，且可以构造出非长尾分布的重尾分布。
所有长尾分布都是重尾分布，但反过来不一定成立，且可以构造出非长尾分布的重尾分布。
=== Subexponential distributions 次指数分布 ===
===次指数分布 ===
次指数性是根据概率分布的<font color="#ff8000">卷积 Convolution </font>定义的。对于具有共同分布函数<math>F</math>的两个独立且分布均匀的随机变量<math> X_1,X_2</math>，<math>F</math>与自身的卷积，<math>F^{*2}</math>是卷积的平方，使用Lebesgue–Stieltjes积分，方法如下：
次指数性是根据概率分布的<font color="#ff8000">卷积 Convolution </font>定义的。对于具有共同分布函数<math>F</math>的两个独立且分布均匀的随机变量<math> X_1,X_2</math>，<math>F</math>与自身的卷积，<math>F^{*2}</math>是卷积的平方，使用Lebesgue–Stieltjes积分，方法如下：
</math>
</math>
''n''倍卷积<math>F^{*n}</math>定义如下：
''n''倍卷积<math>F^{*n}</math>定义如下：
</math>
</math>
尾分布函数<math>\overline{F}</math>定义为<math>\overline{F}(x) = 1-F(x)</math>。
如果满足以下条件，则正半线上的分布<math>F</math>为次指数<ref name="Asmussen"/><ref>{{Cite web|url=https://www.researchgate.net/publication/242637603_A_Theorem_on_Sums_of_Independent_Positive_Random_Variables_and_Its_Applications_to_Branching_Random_Processes|title=A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes|last=Chistyakov|first=V. P.|date=1964|website=ResearchGate|language=en|archive-url=|archive-date=|access-date=April 7, 2019}}</ref><ref>{{Cite web|url=https://projecteuclid.org/download/pdf_1/euclid.aop/1176996225|title=The Class of Subexponential Distributions|last=Teugels|first=Jozef L.|authorlink=|date=1975|website=|publisher=Annals of Probability|publication-place=[[KU Leuven|University of Louvain]]|archive-url=|archive-date=|access-date=April 7, 2019}}</ref>
如果满足以下条件，则正半线上的分布<math>F</math>为次指数<ref name="Asmussen"/><ref>{{Cite web|url=https://www.researchgate.net/publication/242637603_A_Theorem_on_Sums_of_Independent_Positive_Random_Variables_and_Its_Applications_to_Branching_Random_Processes|title=A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes|last=Chistyakov|first=V. P.|date=1964|website=ResearchGate|language=en|archive-url=|archive-date=|access-date=April 7, 2019}}</ref><ref>{{Cite web|url=https://projecteuclid.org/download/pdf_1/euclid.aop/1176996225|title=The Class of Subexponential Distributions|last=Teugels|first=Jozef L.|authorlink=|date=1975|website=|publisher=Annals of Probability|publication-place=[[KU Leuven|University of Louvain]]|archive-url=|archive-date=|access-date=April 7, 2019}}</ref>
</math>
</math>
这意味着<ref name="Embrechts">{{cite book |author1=Embrechts P. |author2=Klueppelberg C. |author3=Mikosch T. |title=Modelling extremal events for insurance and finance |publisher=Springer | series = Stochastic Modelling and Applied Probability|location=Berlin |year=1997 | volume=33| doi = 10.1007/978-3-642-33483-2|isbn=978-3-642-08242-9 }}</ref>，对于任何<math>n \geq 1</math>，
这意味着<ref name="Embrechts">{{cite book |author1=Embrechts P. |author2=Klueppelberg C. |author3=Mikosch T. |title=Modelling extremal events for insurance and finance |publisher=Springer | series = Stochastic Modelling and Applied Probability|location=Berlin |year=1997 | volume=33| doi = 10.1007/978-3-642-33483-2|isbn=978-3-642-08242-9 }}</ref>，对于任何<math>n \geq 1</math>，
</math>
</math>
对此的概率解释<ref name="Embrechts"/>是，对于具有共同分布<math>F</math>的<math>n</math>个独立随机变量<math>X_1,\ldots,X_n</math>的总和
对此的概率解释<ref name="Embrechts"/>是，对于具有共同分布<math>F</math>的<math>n</math>个独立随机变量<math>X_1,\ldots,X_n</math>的总和
</math>
</math>
这通常被称为<font color="#ff8000">单跳 single big jump</font><ref>{{Cite journal | last1 = Foss | first1 = S. | last2 = Konstantopoulos | first2 = T. | last3 = Zachary | first3 = S. | doi = 10.1007/s10959-007-0081-2 | title = Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments | journal = Journal of Theoretical Probability| volume = 20 | issue = 3 | pages = 581 | year = 2007 | arxiv = math/0509605| pmid = | url = http://www.math.nsc.ru/LBRT/v1/foss/fkz_revised.pdf| pmc = | citeseerx = 10.1.1.210.1699 }}</ref>或<font color="#ff8000">突变理论 catastrophe principle</font> <ref>{{cite web| url = http://rigorandrelevance.wordpress.com/2014/01/09/catastrophes-conspiracies-and-subexponential-distributions-part-iii/ | title = Catastrophes, Conspiracies, and Subexponential Distributions (Part III) | first = Adam | last = Wierman | authorlink = Adam Wierman | date = January 9, 2014 | accessdate = January 9, 2014 | website = Rigor + Relevance blog | publisher = RSRG, Caltech}}</ref>。
这通常被称为<font color="#ff8000">单跳 single big jump</font><ref>{{Cite journal | last1 = Foss | first1 = S. | last2 = Konstantopoulos | first2 = T. | last3 = Zachary | first3 = S. | doi = 10.1007/s10959-007-0081-2 | title = Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments | journal = Journal of Theoretical Probability| volume = 20 | issue = 3 | pages = 581 | year = 2007 | arxiv = math/0509605| pmid = | url = http://www.math.nsc.ru/LBRT/v1/foss/fkz_revised.pdf| pmc = | citeseerx = 10.1.1.210.1699 }}</ref>或<font color="#ff8000">突变理论 catastrophe principle</font> <ref>{{cite web| url = http://rigorandrelevance.wordpress.com/2014/01/09/catastrophes-conspiracies-and-subexponential-distributions-part-iii/ | title = Catastrophes, Conspiracies, and Subexponential Distributions (Part III) | first = Adam | last = Wierman | authorlink = Adam Wierman | date = January 9, 2014 | accessdate = January 9, 2014 | website = Rigor + Relevance blog | publisher = RSRG, Caltech}}</ref>。
如果分布<math>F I([0,\infty))</m4ath>为实数，则<math>F</math>为整个实数上的次指数分布。<ref>{{cite journal | last = Willekens | first = E. | title = Subexponentiality on the real line | journal = Technical Report | publisher = K.U. Leuven | year = 1986}}</ref>此时<math>I([0,\infty))</math>是正半轴的指标函数。或者，当且仅当<math>X^+ = \max(0,X)</math>是次指数时，实数上支持的随机变量<math>X</math>才是次指数。
如果分布<math>F I([0,\infty))</m4ath>为实数，则<math>F</math>为整个实数上的次指数分布。<ref>{{cite journal | last = Willekens | first = E. | title = Subexponentiality on the real line | journal = Technical Report | publisher = K.U. Leuven | year = 1986}}</ref>此时<math>I([0,\infty))</math>是正半轴的指标函数。或者，当且仅当<math>X^+ = \max(0,X)</math>是次指数时，实数上支持的随机变量<math>X</math>才是次指数。
所有次指数分布都是长尾分布，但可以构造出非次指数分布的长尾分布的示例。
所有次指数分布都是长尾分布，但可以构造出非次指数分布的长尾分布的示例。