Berry–Esseen bounds and almost sure CLT for the quadratic variation of the bifractional Brownian motion

Abstract

Abstract Let B be a bifractional Brownian motion with parameters H ∈ (0,1) and K ∈ (0,1]. For any n ≥ 1, set Z n = ∑ i = 0 n - 1 [ n 2 H K ( B ( i + 1 ) / n - B i / n ) 2 - 𝔼 ( ( B i + 1 - B i ) 2 ) ] ${Z_n = \sum _{i=0 }^{n-1} [n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\mathbb {E} ((B_{i+1}-B_{i})^2 ) ]}$ . We use Malliavin calculus and the so-called Stein's method on Wiener chaos introduced by Nourdin and Peccati [11] to derive, in the case when 0 < H K ≤ 3/4, Berry–Esseen-type bounds for the Kolmogorov distance between the law of the correct renormalization V n of Z n and the standard normal law. Finally, we study almost sure central limit theorems for the sequence V n .

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