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I am reading a paper and spot the following concept.

Given a sequence of random probability $p_n \in [0,1], n \in \mathbb N$ of some events, assume there exists a stochastic process $X: \Omega \times [0, \infty) \to \mathbb N$ and a $k \in \mathbb N$ and $$ p_n = \int_{n-1}^n I_{\{X(t)=k\}} dt $$ $X$ is said to be the kernel of $p_n$'s.

$I_{\{X(t)=k\}}$ looks like a density function, but not really.

Is there more information or reference so that I can learn more about the concept? Thanks!

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It isn't exactly the same, but it reminds me of stochastic discount factors/pricing kernels in finance as here, too, the variable $X$ allows you to recover the information $p_n$. With the caveat of it not being exactly the same, there is a wiki article on this. –  gnometorule Mar 3 '13 at 4:07

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