Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets

2019/05/29 by J.M. Azais, Azais, J-M, José R. León +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · doi:10.48550/arxiv.1905.12342

openalex publication_date 2019/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a smooth vectorial stationary Gaussian random field X : Ω× ℝ^d → ℝ^d, we give necessary and sufficient conditions to have a finite second moment for the number of roots of X(t) - u. The results are obtained by using a method of proof inspired on the one obtained by D. Geman for stationary Gaussian processes long time ago. Afterwards the same method is applied to the number of critical points of a scalar random field and also to the level set of a vectorial process X : Ω× ℝ^D → ℝ^d with D > d.

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