Integrated functionals of normal and fractional processes

TL;DR

This paper investigates the limiting behavior of integrated functionals of normal and fractional processes, especially at the critical dependence level where classical results do not apply, revealing a nonstandard Brownian limit.

Contribution

It extends existing limit theorems to include the critical dependence case for fractional processes, showing a Brownian limit with nonstandard normalization.

Findings

Limiting process is Brownian motion with nonstandard norming at critical dependence.

Results apply to fractional Brownian motion functionals for all Hurst indices.

Existing results are limited to H<3/4 or H>3/4, our work covers H=3/4.

Abstract

Consider $Z^f_t(u)=\int_0^{tu}f(N_s) ds$, $t>0$, $u\in[0,1]$, where $N=(N_t)_{t\in\mathbb{R}}$ is a normal process and $f$ is a measurable real-valued function satisfying $Ef(N_0)^2<\infty$ and $Ef(N_0)=0$. If the dependence is sufficiently weak Hariz [J. Multivariate Anal. 80 (2002) 191--216] showed that $Z_t^f/t^{1/2}$ converges in distribution to a multiple of standard Brownian motion as $t\to\infty$. If the dependence is sufficiently strong, then $Z_t/(EZ_t(1)^2)^{1/2}$ converges in distribution to a higher order Hermite process as $t\to\infty$ by a result by Taqqu [Wahrsch. Verw. Gebiete 50 (1979) 53--83]. When passing from weak to strong dependence, a unique situation encompassed by neither results is encountered. In this paper, we investigate this situation in detail and show that the limiting process is still a Brownian motion, but a nonstandard norming is required. We apply our result to some functionals of fractional Brownian motion which arise in time series. For all Hurst indices $H\in(0,1)$, we give their limiting distributions. In this context, we show that the known results are only applicable to $H<3/4$ and $H>3/4$, respectively, whereas our result covers $H=3/4$.