Modern Stochastics: Theory and Applications

The so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein–Uhlenbeck (mmfOU) processes are studied. These processes are constructed by mixing by superimposing or mixing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein–Uhlenbeck processes (fOU), respectively. Their existence as ${L^{2}}$ processes is proved, and their path properties, viz. long-range and short-range dependence, Hölder continuity, p-variation, and conditional full support, are studied.

We show that every multiparameter Gaussian process with integrable variance function admits a Wiener integral representation of Fredholm type with respect to the Brownian sheet. The Fredholm kernel in the representation can be constructed as the unique symmetric square root of the covariance. We analyze the equivalence of multiparameter Gaussian processes by using the Fredholm representation and show how to construct series expansions for multiparameter Gaussian processes by using the Fredholm kernel.

We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian–fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter H of the fractional part satisfies $H\in (3/4,1)$, the central limit theorem holds. In the nonsemimartingale case, that is, where $H\in (1/2,3/4]$, the convergence toward the normal distribution with a nonzero mean still holds if $H=3/4$, whereas for the other values, that is, $H\in (1/2,3/4)$, the central convergence does not take place. We also provide Berry–Esseen estimates for the estimator.