# Mean and variance of level crossings times gradient in an interval

by quenderin   Last Updated October 10, 2019 05:20 AM

Consider a Gaussian random process $$x(t)$$ with some two-point correlation function $$\xi$$. Suppose $$x(t_i) = u, t_i \in I$$ for some set of points $$\{t_i\}$$. Define the function

$$M_I(x,u) = \sum_{i} |x'(t_i)|$$

What can be said about the mean and variance of $$M_I$$? This is very close to Rice's formula and all the work on level crossings. I think I know how to compute the mean following the proof of Rice's formula, but the variance seems much harder.

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