更改

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>,

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>,

这蕴含着，对于任何<math>n \geq 1</math>，

这意味着，对于任何<math>n \geq 1</math>，

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>

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

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.

<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.

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

==Common heavy-tailed distributions==

==Common heavy-tailed distributions==