Tail probability via tube formula when the critical radius is zero

Abstract

It has recently been established that 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--Lo\`eve expansion and the index set has positive critical radius. We show that the positiveness of the critical radius is an essential condition. When the critical radius is zero, we prove that only the main term is valid and that other higher-order terms are generally not valid in the formal asymptotic expansion based on the tube formula. This is done by first establishing an exact tube formula and comparing the formal tube formula with the exact formula. Furthermore, we show that the equivalence of the formal tube formula and the Euler characteristic method no longer holds when the critical radius is zero. We conclude by applying our results to some specific examples.