An asymptotic expansion of the norm of $e^{-|{t-s}|}{1}_{\{0\le s,t\le T\}}$ in the canonical Hilbert space of fractional Brownian motion
Yong Chen
TL;DR
This paper derives a detailed asymptotic expansion for the squared norm $||f_T||^2$ of the kernel $f_T(s,t)=e^{-|t-s|}\mathbf{1}_{\{0\le s,t\le T\}}$ in the canonical Hilbert space of fractional Brownian motion with Hurst parameter $H\in(0,1)$. Using two inner-product representations, the authors obtain an explicit expansion up to $O(T^{4H-4})$ for $H\neq 3/4$, and a logarithmic correction when $H=3/4$, with constants expressed in terms of $a=H\Gamma(2H)$ and $\sigma^2_H=(4H-1)(1-1/\cos(2\pi H))$. They derive corollaries on the existence of an oblique asymptote for $\tfrac{1}{2}||f_T||^2$ when $H\in(0,1/2]$, and a sharp bound for $\left|\tfrac{1}{2T}||f_T||^2-\sigma^2\right|$ for $H\in(0,3/4)$, with implications for ergodic fractional Ornstein–Uhlenbeck processes. The paper also applies these results to bound second-chaos functionals like $W_T$ and to quantify convergence rates in OU-type estimators, offering concrete rates that depend on $H$. Overall, the work provides precise quantitative tools for parameter estimation and ergodic analysis in Gaussian rough-path settings driven by fBm.
Abstract
Using the inner product formula of the canonical Hilbert space of fractional Brownian motion on an interval $[0,T]$ with Hurst parameter $H\in (0,1)$ given by Alazemi et al., we show the asymptotic expansion of the norm of $f_T(s,t):=e^{-|t-s|}\mathbf{1}_{\{0\le s,t\le T\}}$ up to the term $T^{4H-4}$. As applications, we show that the existence of the oblique asymptote of the norm $\frac12\|f_T\|^2_{\mathfrak{H}^{\otimes2}}$ if and only if $H\in (0,\frac12]$ and that we obtain a sharp upper bound of the difference $\left|\frac{1}{2 {T}} \|f_T\|_{\mathfrak{H}^{\otimes 2}}^2-σ^2\right|$ for $H\in (0,\frac34)$ which implies two significant estimates concerning to an ergodic fractional Ornstein-Uhlenbeck process, where $σ^2$ is the slope of the oblique asymptote.