Functional convergence of sequential U-processes with size-dependent kernels

Asymptotic error distribution for the Riemann approximation of integrals driven by fractional Brownian motion

“…If H = 1 2 and if, for all (i, e), x i,e is adapted to B, piecewise continuous and satisfies < +∞ for some β > 1 2 and γ > 0 then, in probability uniformly on [0, T ] (and also in L 2 (Ω) for fixed t), n 2H−1 ν H (n)S n,i,j,e Proof. Even if they are not stated in exactly the same way, the limits (4.15) and (4.16) follow from [5,13] (see especially [13,Sections 4,5,7]) by means of fractional integration techniques. This is why we only concentrate on the case H = 1 2 and the proof of (4.14), not covered by [5,13].…”

The multivariate functional de Jong CLT

“…In [16], the authors used Stein's method in order to establish quantitative uni-and multivariate counterparts to de Jong's theorem. The techniques introduced in [16] put forward several combinatorial quantities that proved useful in further situations: for instance, as demonstrated by the references [13,15,18] the technical findings of [16] allow one to derive quantitative (functional) CLTs for symmetric U -statistics that are expressed in terms of purely analytical quantities, that is, norms of so-called of contraction kernels.…”

“…It is important to notice that such a limit behaviour is in principle not achievable in the framework of symmetric and degenerate U -statistics. Indeed, in [13,Corollary 3.7] it is proved that, given a sequence of symmetric and degenerate U -statistics of order p ≥ 2 verifying asymptotic relations that are roughly equivalent to Conditon 1.3, the corresponding sequence of normalized empirical processes always converges in distribution to a multiple of B(t p ). Remark 1.6 (a) Theorem 1.4 provides a strong functional extension of the finitedimensional de Jong type theorems from [11,16] under very mild additional assumptions.…”

“…3 [14,21,44]), in order to satisfy a FCLT, a symmetric degenerate U -statistic of order p > 1 must have a variable kernel, i.e. one that depends on the sequential parameter m. In the recent work [13] we have proved FCLTs for this situation with sufficient conditions for convergence that are expressed in norms of contraction kernels and, hence, are completely analytical in nature. A more general class of statistics is given by the so-called weighted, degenerate U -statistics of order p. These have the form…”

Functional Gaussian approximations in Hilbert spaces: the non-diffusive case

“…Also, our setting seems to be more general and does not rely on ad-hoc arguments depending on the Gaussian process at hand. Other related references proving functional central limit theorems using Malliavin-Stein techniques are [Kas17,Kas20,DK21,DKP19]. The rest of this paper is organized as follows.…”