Talk:Campbell's theorem (probability)

Different Campbell's theorems
Campbell's theorem comes in a few forms, so I'll cite the one from Kingman's book (for Poisson processes), then give the more general Campbell's theorem/formula (for general point processes). I will put the more general formula first since it is, well, more general, and leads to more general formulas and theorems such as Campbell-Mecke theorems and Palm calculus results, which may be incorporate later into this article.Improbable keeler (talk) 21:26, 17 November 2015 (UTC)

Neuroscience application?
I definitely believe that Campbell's theorem (or shot noise) is used in neuroscience, but I need a reference. Also, is it necessary to state when and where higher moments (for which process, incidentally) were done? Improbable keeler (talk) 21:26, 17 November 2015 (UTC)

Neuroscience
The total input electrical current in neurons is the sum of many inputs with similar timecourses. When a model based on Poisson point process is used, the mean current and variance are given by Campbell theorem. When the synaptic timecourse is known, this model can be applied to infer the input rate.

Extension to higher moments is given in.

Another common extension is to consider a sum with random amplitudes
 * $$ S =\sum_{x\in {N}} a_n f(x) $$

In this case the cumulants $$ \kappa_i $$ of $$ S $$ equal
 * $$ \kappa_i= \lambda \overline{a^i} \int f(x) dx $$

where $$ \overline{a^i} $$ are the raw moments of the distribution of $$ a $$.