Markov kernel: Difference between revisions

Revision as of 09:40, 19 April 2019

Revision as of 09:41, 19 April 2019

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===[[Galton–Watson process]]===

As a less obvious example, take <math>X = \N, \mathcal A = \mathcal P(\N),</math>, and <math>(Y, \mathcal B)</math> the real numbers <math>\R</math> with the standard sigma algebra of [[Borel set]]s. Then

:<math>\kappa(B|n)=\begin{cases} \mathbf{1}_B(0) & n=0\\ \Pr(\xi_1 + \cdots + \xi_x \in B) & n \neq 0 \\ \end{cases} </math>

with [[Independent and identically distributed random variables|i.i.d.]] [[random variable]]s <math>\xi_i</math> (usually with mean 0) and where <math>\mathbf{1}_B</math> is the indicator function. For the simple case of [[Bernoulli distribution|coin flips]] this models the different levels of a [[Galton board]].