اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدError bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion
127
0
0.0
(
0
)
تأليف
Jean-Christophe Breton
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $qgeq 2$ be a positive integer, $B$ be a fractional Brownian motion with Hurst index $Hin(0,1)$, $Z$ be an Hermite random variable of index $q$, and $H_q$ denote the Hermite polynomial having degree $q$. For any $ngeq 1$, set $V_n=sum_{k=0}^{n-1} H_q(B_{k+1}-B_k)$. The aim of the current paper is to derive, in the case when the Hurst index verifies $H>1-1/(2q)$, an upper bound for the total variation distance between the laws $mathscr{L}(Z_n)$ and $mathscr{L}(Z)$, where $Z_n$ stands for the correct renormalization of $V_n$ which converges in distribution towards $Z$. Our results should be compared with those obtained recently by Nourdin and Peccati (2007) in the case when $H<1-1/(2q)$, corresponding to the situation where one has normal approximation.
قيم البحث
اقرأ أيضاً
We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet $W^{alpha, beta}$ with Hurst parameter $(alpha, beta) in (0,1)^2$. When $0<alpha leq 1-frac{1}{2q}$ or $0<beta leq 1-frac{1}{2q}$
a central limit theorem holds for the renormalized Hermite variations of order $qgeq 2$, while for $1-frac{1}{2q}<alpha, beta < 1$ we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time $(1,1)$.
Nils Tongring (1987) proved sufficient conditions for a compact set to contain $k$-tuple points of a Brownian motion. In this paper, we extend these findings to the fractional Brownian motion. Using the property of strong local nondeterminism, we sho
w that if $B$ is a fractional Brownian motion in $mathbb{R}^d$ with Hurst index $H$ such that $Hd=1$, and $E$ is a fixed, nonempty compact set in $mathbb{R}^d$ with positive capacity with respect to the function $phi(s) = (log_+(1/s))^k$, then $E$ contains $k$-tuple points with positive probability. For the $Hd > 1$ case, the same result holds with the function replaced by $phi(s) = s^{-k(d-1/H)}$.
We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion $B$ with Hurst index $H=1/4$. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C. A. Tudor. Moreover, as an appli
cation, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to $B$.
To extend several known centered Gaussian processes, we introduce a new centered mixed self-similar Gaussian process called the mixed generalized fractional Brownian motion, which could serve as a good model for a larger class of natural phenomena. T
his process generalizes both the well known mixed fractional Brownian motion introduced by Cheridito [10] and the generalized fractional Brownian motion introduced by Zili [31]. We study its main stochastic properties, its non-Markovian and non-stationarity characteristics and the conditions under which it is not a semimartingale. We prove the long range dependence properties of this process.
The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power law shape function and non-stationary
noises with a power-law variance function. In this paper we study sample path properties of the generalized fractional Brownian motion, including Holder continuity, path differentiability/non-differentiability, and functional and local Law of the Iterated Logarithms.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
