Tail probability via tube formula and Euler characteristic method when critical radius is zero

In Takemura and Kuriki (1999b) we have established that the tube formula and the Euler characteristic method give identical and valid asymptotic expansion of tail probability of the maximum of Gaussian random field when the random field has finite Karhunen-Loève expansion and the index set has positive critical radius. The purpose of this paper is to show that the positiveness of the critical radius is an essential condition. Namely, we prove that if the critical radius is zero, only the main term is valid and other higher order terms are generally not valid in the formal asymptotic expansion based on the tube formula or the Euler characteristic method. Our examples show that index sets with zero critical radius are commonly used in statistics.