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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