Tail probability via the tube formula when the critical radius is zero
The tube formula and the Euler characteristic method give an identical and valid asymptotic expansion of the tail probability of the maximum of a Gaussian random field when the random field has finite Karhunen-Loève expansion and the index set has positive critical radius. The authors show that the positiveness of the critical radius is an essential condition. When the critical radius is zero, they prove that only the main term is valid and that other higher-terms are generally not valid in the formal asymptotic expansion based on the tube formula. This proof is done by first establishing an exact tube formula and comparing the formal tube formula with the exact one. Furthermore, the authors show that the equivalence of the formal tube formula and the Euler characteristic method does no longer hold when the critical radius is zero. They conclude by applying their results to some specific examples.
scientific article; zbMATH DE number 2075981