By M. Hušková, R. Beran, V. Dupac

Hájek was once unquestionably a statistician of huge strength who, in his rather brief lifestyles, contributed basic effects over quite a lot of topics...

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Edu 37 38 C. Wijeratne and H. 1 Fractional Brownian Motion Fractional Brownian motion (fBm) was first introduced by Kolmogorov in 1940 by the name “Wiener spiral” [26] within a Hilbert space setting. The name “fractional Brownian motion” is due to Mandelbrot and Van Ness as they gave a representation to the fractional Brownian motion through a fractional integral with respect to the standard Brownian motion in 1968 [31]. Definition 1 Let B H = {BηH , η ≥ 0} be a stochastic process, and H ∈ (0, 1).

1 (30) Consider the first term on the right side of the above equality, ξ t |Yξ (t)| ≤ |φ(ξ )| + 0 0 A(Yη (s)) dg(s, η) ds. From Lemma (2) we can obtain |Yξ (t)| ≤ |φξ | + t M2 + M1 ||Y (s)||α,∞ ds . 1−α α (g)t 0 (31) Consider the second term. We have ξ |Yξ (t) − Yη (t)| dη (ξ − η)α+1 0 ξ = 1 (ξ − η)α+1 0 0 ξ ≤ 0 A(Yγ (s)) dgγ ds − φ(η) + 1 (ξ − η)α+1 0 ξ ≤ ξ t φ(ξ ) + |φξ − φη | + |φξ − φη | dη + (ξ − η)α+1 0 ξ t 0 0 ξ t 0 η η t 0 0 A(Yγ (s)) dgγ ds A(Yγ (s)) dgγ dsdη 1 (ξ − η)α+1 ξ η A(Yγ (s)) dgγ dηds.

Taking into account that φ(x) = E[1{Z >x} Z ], where Z has the N (0, 1) distribution, it suffices to estimate the difference E[1{F>x} F] − E[1{Z >x} Z ], which can be done by Stein’s method and Malliavin calculus. ∞ Example 8 Let q = 2 and F = i=1 λi (X (ei )2 −1), where {ei , i ≥ 1} is a complete orthonormal system in H and λi is a decreasing sequence of positive numbers such An Introduction to the Malliavin Calculus and Its Applications that obtain ∞ 2 i=1 λi 33 < ∞. Suppose E[F 2 ] = 1. Then, if λ N = 0 for some N > 4, we ∞ sup | p F (x) − φ(x)| ≤ C N ,λ N λi4 .