Correction to: On maximum of Gaussian random fields having unique maximum point of its variance

High excursion probabilities for Gaussian fields on smooth manifolds

“…3) to use Bonferroni inequalities with estimating of the double sum of the joint probabilities; and 4) to use back passage from the informative manifold M ε to whole S. Remark that our results here agree with assertions of Theorem 3, [4], where d = 1, and Theorem 1 in [5], [6] for dim M = 0.…”

“…Remark 1 The case r = 0 is considered in [5], see also [6], so that we assume further that r ≥ 1.The case r = d, in particular σ(t) is a constant on S, is called a locally homogeneous (stationary) case, provided the below Condition 3 is fulfilled. We shall consider this case as well.…”

“…In case of processes, it was described in [4]. Then in [5], with corrections in [6], this have been generalized on Gaussian fields with single maximum variance point.…”

“…Turning to (34), we have, using again Bonferroni inequalities, with some additional standard techniques for probabilities of large excursions, see [5], [6], we get the following. Remark 6 Remark that the summands in (39) can have different orders in u depending on the dimension of the corresponding component K 0i , on behavior of q i,t j (u)s and on behavior of σ 2 (t).…”

“…This work is a continuation of papers [5], [6] and [4], where asymptotic behavior of large excursuion probabilities were evaluated for Gaussian nonhomogeneous fields given on Euclidean space. An essential condition in these and many previous works is that the variance of a considered Gaussian field reaches its maximum at unique point in the considered parameter set.…”