en>Cedar101: /* Details */ fix for mathjax

2013-12-17T00:39:28Z

Details: fix for mathjax

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	~~Hello and welcome~~. ~~My title is Figures Wunder. My day occupation~~ is a ~~meter reader. One~~ of ~~the extremely best things~~ in ~~the world for me~~ is ~~to do aerobics and now I'm attempting to earn money~~ with it. ~~For many years he~~'~~s been residing~~ in ~~North Dakota and his family enjoys it~~.<br><br>~~my page - home std test~~ ([~~http:~~//~~www~~.~~lankaclipstv~~.~~com~~/~~user~~/~~SFergerso Click At this website~~])		Let <math>(E, \mathcal A, \mu)</math> be some [[measure space]] with <math>\sigma</math>-[[sigma finite measure\|finite measure]] <math>\mu</math>. The '''Poisson random measure''' with intensity [[Measure (mathematics)\|measure]] <math>\mu</math> is a family of [[random variables]] <math>\{N_A\}_{A\in\mathcal{A}}</math> defined on some [[probability space]] <math>(\Omega, \mathcal F, \mathrm{P})</math> such that

			i) <math>\forall A\in\mathcal{A},\quad N_A</math> is a [[Poisson distribution\|Poisson random variable]] with rate <math>\mu(A)</math>.

			ii) If sets <math>A_1,A_2,\ldots,A_n\in\mathcal{A}</math> don't intersect then the corresponding [[random variables]] from i) are mutually [[Statistical independence\|independent]].

			iii) <math>\forall\omega\in\Omega\;N_{\bullet}(\omega)</math> is a measure on <math>(E, \mathcal A)</math>

			==Existence==
			If <math>\mu\equiv 0</math> then <math>N\equiv 0</math> satisfies the conditions i)–iii). Otherwise, in the case of [[finite measure]] <math>\mu</math>, given <math>Z</math>, a [[Poisson distribution\|Poisson random variable]] with rate <math>\mu(E)</math>, and <math>X_1, X_2,\ldots</math>, mutually [[Statistical independence\|independent]] [[random variable]]s with [[Probability distribution\|distribution]] <math>\frac{\mu}{\mu(E)}</math>, define <math>N_{\cdot}(\omega) = \sum\limits_{i=1}^{Z(\omega)} \delta_{X_i(\omega)}(\cdot)</math> where <math>\delta_c(A)</math> is a [[degenerate distribution\|degenerate measure]] located in <math>c</math>. Then <math>N</math> will be a Poisson random measure. In the case <math>\mu</math> is not finite the [[Measure (mathematics)\|measure]] <math>N</math> can be obtained from the measures constructed above on parts of <math>E</math> where <math>\mu</math> is finite.

			==Applications==
			This kind of [[random measure]] is often used when describing jumps of [[stochastic process]]es, in particular in [[Lévy–Itō decomposition]] of the [[Lévy process]]es.

			==References==
			*{{cite book \|last=Sato \|first=K. \|year=2010 \|title=Lévy Processes and Infinitely Divisible Distributions \|publisher=Cambridge University Press \|isbn=0-521-55302-4 }}

			[[Category:Probability theory]]

en>Hobbema: /* See also */

2011-06-15T00:18:46Z

Autocorrelator - Revision history

en>Cedar101: /* Details */ fix for mathjax

en>Hobbema: /* See also */