The harmonic mean formula for random processes

Abstract

Motivated by the harmonic mean formula in [1], we investigate the relation between the sojourn time and supremum of a random process X(t), t ∈ R d and extend the harmonic mean formula for general stochastically continuous X. We discuss two applications concerning the continuity of distribution of supremum of X and representations of classical Pickands constants.

On the speed of convergence of discrete Pickands constants to continuous ones

“…This, in particular implies that the truncation error of the Dieker-Yakir estimator decays no slower than exp{−CT α } and together with Theorem 2.3 they imply that ξ δ α (T ) has a uniformly bounded sampling error, i.e. (4) sup…”

Shift-invariant homogeneous classes of random fields

“…(B.11)Proof of Lemma B.7 For all z ∈ (0, 1] almost surelyB L,τ (zY ) ≤ B L,τ (Y ) ≤ B L,τ (z −1 Y ). (B.12)Further since Y is quadrant stochastically continuous, then from[45][Thm 2.1, Rem 2.2,iii)] M L (Y ) > 1/z implies B L,τ (zY ) > 0 almost surely for all z > 0. Hence since also P{B L,τ (Y ) > 0} = 1 as shown in(5.8), for all z ∈ (0, 1] by (4.7), (B.12) and the Fubini-Tonelli theoremE 1 B L,τ (Y ) I(B L,τ (Y ) < ∞, M L (Y ) = ∞) ≤ E B L,τ (zY ) B L,τ (Y )B(zY ) I(B(Y ) < ∞, M L (Y ) > 1/z) = T E Y (0) τ B L,τ (Y )B L,τ (zY ) I(zM L (Y ) > 1, z Y (t) > 1) λ(dt) = z α T E Y (−t) τ B L,τ (z −1 Y )B L,τ (Y ) I( Y (−t) > z) λ(dt) = z α E I(B L,τ (z −1 Y ) < ∞) B L,τ (z −1 Y )B L,τ (Y ) T Y (0) τ I( Y (−t) > z)λ(dt) = z α E I(B L,τ (z −1 Y ) < ∞)B L,τ (z −1 Y ) B(z −1 Y )B(Y ) ≤ z α E 1 B L,τ (Y ) < ∞,where the last inequality follows from (4.10).…”