On a general Kac-Rice formula for the measure of a level set

2023/04/14 by Armentano, Diego, Azaïs, Jean-Marc, León, José Rafael · 1 citation
#60GG10 #60GG15 #60GG60 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2304.07424

Abstract

Let X(⋅) be a random field ℝ^D → ℝ^d, D≥ d. We first studied the level set X^-1( u) , u ∈ ℝ^d. In particular we gave a weak condition for this level set to be rectifiable. Then, we established a Kac-Rice formula to compute the D-d Hausdorff measure. Our results extend known results, particularly in the non-Gaussian case where we obtained a very general result. We conclude with several extensions and examples of application, including functions of Gaussian random field, zeroes of the likelihood, gravitational microlensing, shot-noise.

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