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I'm working on a problem where I'm generating a smooth, periodic function of an independent variable, where this function is also a function of a number of random variables. Thus the function itself is very random.

I know the probability distribution of the value of the function at any given point and that this is identical for all points. Indeed the problem is invariant under 'rotations' of the independent variable. I have also calculated the two-point correlation function through integration around one period of the function.

My question is what else do I need to know in order to obtain a complete description of the statistical properties of the function? I'm thinking I might need to calculate higher order correlation functions, but I've no idea if this is enough.

I want to use this to be able to answer questions such as the distribution of the number of peaks and troughs of the function over a period.

Thanks in advance for any help.

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  • $\begingroup$ It's not clear to me what you are asking. What is the function you are looking at? What do you mean by a "complete description of the statistical properties of the function"? Why do you need such a thing when you are interested in the distribution of the number of peaks and troughs? $\endgroup$ Commented Oct 9 at 16:50
  • $\begingroup$ I suspect your intention when saying "I know the probability distribution of the value of the function at any given point and that this is identical for all points" is not the same as how it reads to me, which is as a stationary distribution. $\endgroup$ Commented Oct 9 at 17:19

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Given a random function $f\colon X\to Y$ where $X$ and $Y$ are arbitrary sets that are allowed to be infinite, given any finite subset $S\subseteq X$ the distribution of $f$ restricted to $S$ is known as a finite-dimensional distribution. The field of math in which such random functions are studied is known as stochastic calculus and (when standard assumptions are made involving measurability) it is a theorem that the knowledge of all finite dimensional distributions uniquely determines the distribution of the random function (more precise versions of this question have been answered on this website, see here for example, and for a converse see the Kolmogorov extension theorem).

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