en>Michael Hardy: punctuation

2013-05-27T14:27:58Z

punctuation

← Older revision		Revision as of 16:27, 27 May 2013
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	The author's ~~title~~ is ~~Christy Brookins~~. ~~Invoicing~~ is ~~my profession~~. ~~Ohio~~ is ~~exactly where his house~~ is ~~and his family members loves it~~. ~~One~~ of the ~~extremely best things in~~ the ~~globe~~ for ~~him~~ is ~~doing ballet~~ and he'~~ll be beginning something else alongside~~ with it.<br><br>~~My blog post~~ - [~~http:~~//~~www~~.~~chk~~.~~woobi~~.co.kr/~~xe/?document_srl=346069 real psychic~~]		In [[probability theory]], a [[probability distribution]] is '''infinitely divisible''' if it can be expressed as the sum of the probability distributions of an arbitrary number of [[Independent and identically distributed random variables\|independent and identically distributed]] [[random variable]]s. The [[characteristic function (probability theory)\|characteristic function]] of any infinitely divisible distribution is then called an '''infinitely divisible characteristic function'''.<ref name="Lukacs">Lukacs, E. (1970) ''Characteristic Functions'', Griffin , London. p. 107</ref>

			More rigorously, the probability distribution ''F'' is infinitely divisible if, for every positive integer ''n'', there exist ''n'' independent identically distributed random variables ''X''<sub>''n''1</sub>, ..., ''X''<sub>''nn''</sub> whose sum ''S''<sub>''n''</sub> = ''X''<sub>''n''1</sub> + … + ''X''<sub>''nn''</sub> has the distribution ''F''.

			The concept of infinite divisibility of probability distributions was introduced in 1929 by [[Bruno de Finetti]]. This type of [[Decomposable distributions\|decomposition of a distribution]] is used in [[probability]] and [[statistics]] to find families of probability distributions that might be natural choices for certain models or applications. Infinitely divisible distributions play an important role in probability theory in the context of limit theorems.<ref name="Lukacs" />

			==Examples==

			The [[Poisson distribution]], the [[negative binomial distribution]], the [[Gamma distribution]] and the [[degenerate distribution]] are examples of infinitely divisible distributions; as are the [[normal distribution]], [[Cauchy distribution]] and all other members of the [[stable distribution]] family. The [[Uniform distribution (continuous)\|uniform distribution]] and the [[binomial distribution]] are not infinitely divisible.<ref>{{cite book\|title=Lévy Processes and Infinitely Divisible Distributions\|author=Sato, Ken-iti\|page=31\|year=1999\|publisher=Cambridge University Press\|isbn=978-0-521-55302-5}}</ref> The [[Student's t-distribution]] is infinitely divisible, while the distribution of the reciprocal of a random variable having a Student's t-distribution, is not.<ref>Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) ''Continuous Univariate Distributions, Volume 2,'' 2nd Edition. Wiley, ISBN 0-471-58494-0 (Chapter 28, page 368)</ref>

			==Limit theorem==

			Infinitely divisible distributions appear in a broad generalization of the [[central limit theorem]]: the limit as ''n'' → +∞ of the sum ''S''<sub>''n''</sub> = ''X''<sub>''n''1</sub> + … ''X''<sub>''nn''</sub> of [[statistical independence\|independent]] uniformly asymptotically negligible (u.a.n.) random variables within a triangular array
			:<math>
			\begin{array}{cccc}
			X_{11} \\
			X_{21} & X_{22} \\
			X_{31} & X_{32} & X_{33} \\
			\vdots & \vdots & \vdots & \ddots
			\end{array}
			</math>
			approaches — in the [[convergence in distribution\|weak sense]] — an infinitely divisible distribution. The '''uniformly asymptotically negligible (u.a.n.)''' condition is given by

			: <math>\lim_{n\to\infty} \, \max_{1 \le k \le n} \; P( \left\| X_{nk} \right\| > \varepsilon ) = 0 \text{ for every }\varepsilon > 0.</math>

			Thus, for example, if the uniform asymptotic negligibility (u.a.n.) condition is satisfied via an appropriate scaling of identically distributed random variables with finite [[variance]], the weak convergence is to the [[normal distribution]] in the classical version of the central limit theorem. More generally, if the u.a.n. condition is satisfied via a scaling of identically distributed random variables (with not necessarily finite second moment), then the weak convergence is to a [[stable distribution]]. On the other hand, for a [[triangular array]] of independent (unscaled) [[Bernoulli distribution\|Bernoulli random variables]] where the u.a.n. condition is satisfied through

			:<math>\lim_{n\rightarrow\infty} np_n = \lambda,</math>

			the weak convergence of the sum is to the Poisson distribution with mean ''λ'' as shown by the familiar proof of the [[Poisson_limit_theorem\|law of small numbers]].

			==Lévy process==
			{{main\|Lévy process}}

			Every infinitely divisible probability distribution corresponds in a natural way to a [[Lévy process]]. A Lévy process is a [[stochastic process]] { ''L<sub>t</sub>'' : ''t'' ≥ 0 } with stationary independent increments, where ''stationary'' means that for ''s'' < ''t'', the [[probability distribution]] of ''L''<sub>''t''</sub> − ''L''<sub>''s''</sub> depends only on ''t'' − ''s'' and where ''independent increments'' means that that difference ''L''<sub>''t''</sub> − ''L''<sub>''s''</sub> is [[statistical independence\|independent]] of the corresponding difference on any interval not overlapping with [''s'', ''t''], and similarly for any finite number of mutually non-overlapping intervals.

			If { ''L<sub>t</sub>'' : ''t'' ≥ 0 } is a Lévy process then, for any ''t'' ≥ 0, the random variable ''L''<sub>''t''</sub> will be infinitely divisible: for any ''n'', we can choose (''X''<sub>''n''0</sub>, ''X''<sub>''n''1</sub>, …, ''X''<sub>''nn''</sub>) = (''L''<sub>''t''/''n''</sub> − ''L''<sub>0</sub>, ''L''<sub>2''t''/''n''</sub> − ''L''<sub>''t''/''n''</sub>, …, ''L''<sub>''t''</sub> − ''L''<sub>(''n''-1)''t''/''n''</sub>). Similarly, ''L''<sub>''t''</sub> − ''L''<sub>''s''</sub> is infinitely divisible for any ''s'' < ''t''.

			On the other hand, if ''F'' is an infinitely divisible distribution, we can construct a Lévy process { ''L<sub>t</sub>'' : ''t'' ≥ 0 } from it. For any interval [''s'', ''t''] where ''t'' − ''s'' > 0 equals a [[rational number]] ''p''/''q'', we can define ''L''<sub>''t''</sub> − ''L''<sub>''s''</sub> to have the same distribution as ''X''<sub>''q''1</sub> + ''X''<sub>''q''2</sub> + … + ''X''<sub>''qp''</sub>. [[Irrational number\|Irrational]] values of ''t'' − ''s'' > 0 are handled via a continuity argument.

			==See also==
			*[[Cramér's theorem]]
			*[[Indecomposable distribution]]

			==Footnotes==
			{{reflist}}

			==References==
			* Domínguez-Molina, J.A.; Rocha-Arteaga, A. (2007) "On the Infinite Divisibility of some Skewed Symmetric Distributions". ''Statistics and Probability Letters'', 77 (6), 644–648 {{doi\|10.1016/j.spl.2006.09.014}}

			* Steutel, F. W. (1979), "Infinite Divisibility in Theory and Practice" (with discussion), ''Scandinavian Journal of Statistics.'' 6, 57–64.

			* Steutel, F. W. and Van Harn, K. (2003), ''Infinite Divisibility of Probability Distributions on the Real Line'' (Marcel Dekker).

			{{ProbDistributions\|Infinite divisibility}}

			[[Category:Theory of probability distributions]]
			[[Category:Types of probability distributions]]
			[[Category:Infinitely divisible probability distributions]]

en>The tree stump: Added more links to the original manuscript

2012-06-14T07:59:12Z

Added more links to the original manuscript

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The author's title is Christy Brookins. Invoicing is my profession. Ohio is exactly where his house is and his family members loves it. One of the extremely best things in the globe for him is doing ballet and he'll be beginning something else alongside with it.<br><br>My blog post - [http://www.chk.woobi.co.kr/xe/?document_srl=346069 real psychic]

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en>Michael Hardy: punctuation

en>The tree stump: Added more links to the original manuscript