High excursions of Bessel and related random processes

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

“…where δ(u) → 0 as u → ∞. The remaining part of the proof almost coincides with the corresponding part of the proof of Theorem 1, Piterbarg and Rodionov (2020), see Section 3.5 there. We just indicate the differences.…”

“…Additionally, note that we need the second part of Condition 5 * to justify that the sets {B k (u)} k≥0 appearing below satisfy the assumptions of Theorem 1 and to apply the standard argument for evaluating the double sum (see the relation (9) below). It might seem that we can replace the second part of Condition 5 * with the following weaker assumption in spirit of the condition E4, Piterbarg and Rodionov (2020),…”

“…The analogue of this assumption was used to examine the stationary-like case in the proof of Theorem 1, Piterbarg and Rodionov (2020). The exact value of K does not play a role, but its finiteness guarantees the existence of "large" subintervals B k (u) satisfying the assumptions of Theorem 1 of the original paper (see this argument on p. 4868, Piterbarg and Rodionov, 2020), and uniformity allows one to show asymptotic negligence of the double sum. Although this assumption is enough to prove Proposition 3 in one-dimensional case, the distance between two nonneighboring sets B k (u) and B l (u) can be arbitrarily small in multi-dimensional case even if K = 2.…”

High excursion probabilities for Gaussian fields on smooth manifolds

“…Here M = {1} × S d . Using behavior of the covariance function of X(t) near this set, it is shown in [13] that…”

“…A typical situation is the large excursions probability evaluation for norms of Gaussian vector processes, when one uses the duality and passes to Gaussian fields on a cylinder, and theirs variance can reach theirs maximum at a submanifold of the cylinder. Well known examples are χ 2 process and its generalizations, see [9] for examples and references, and Bessel process and bridge, with generalizations to corresponding fracttion processes, [13]. These examples motivated the present work.…”