Multivariate central limit theorems for averages of fractional Volterra processes and applications to parameter estimation

Abstract

Abstract. The purpose of this paper is to establish the multivariate normal convergence for the average of certain Volterra processes constructed from a fractional Brownian motion with Hurst parameter H > . Some applications to parameter estimation are then discussed.

Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process

“…They also obtained uniform convergences in the continuous topology for a restrictive class of functions G (assuming sufficiently fast decay of the coefficients in the Wiener chaos expansion). This was extended in [30] to vector valued X ε , when each component of X ε falls in the Brownian case, with convergence understood in the sense of finite dimensional distributions. The result in [30] was improved in [11], where the fast chaos decay restrictions on G k , for G k ∈ L p for p > 2, are removed with techniques from Malliavin calculus.…”

“…This was extended in [30] to vector valued X ε , when each component of X ε falls in the Brownian case, with convergence understood in the sense of finite dimensional distributions. The result in [30] was improved in [11], where the fast chaos decay restrictions on G k , for G k ∈ L p for p > 2, are removed with techniques from Malliavin calculus. In the continuous long-range-dependent case Taqqu, [40], obtained convergence in the continuous topology.…”

Statistical inference for Vasicek-type model driven by Hermite processes

“…Along the proof of Theorem 1.3, we will also make use of another result for the Gaussian case (q = 1), which we take from [16]. Proposition 2.6 Let Y be given by (2.9), with q = 1 and H ∈ 1 2 , 3 4 .…”

“…Our paper is relevant to the literature on parameter estimation for processes with Gaussian and non-Gaussian long-memory processes, including [1,5,6,7,8,9,12,13,14,16,27]. In the finance context, the highly cited paper [11] investigates the high-frequency behavior of volatility, drawing on ideas in the paper [21] on long-memory parameter estimation, and before this, the 1997 paper [14], and the 2001 paper [7].…”

Functional limit theorems for the fractional Ornstein-Uhlenbeck process

“…They also obtained uniform convergences in the continuous topology for a restrictive class of functions G (assuming sufficiently fast decay of the coefficients in the Wiener chaos expansion). This was extended in [NNZ16] to vector valued X ε , when each component of X ε falls in the Brownian case, with convergence understood in the sense of finite dimensional distributions. The result in [NNZ16] was improved in [CNN20], where the fast chaos decay restrictions on G k , for G k ∈ L p for p > 2, are removed with techniques from Malliavin calculus.…”