Central limit theorem for an additive functional of the fractional Brownian motion II

We prove a central limit theorem for an additive functional of the $d$-dimensional fractional Brownian motion with Hurst index $H\in(\frac{1}{2+d},\frac{1}{d})$, using the method of moments, extending the result by Papanicolaou, Stroock and Varadhan in the case of the standard Brownian motion

Density convergence in the Breuer-Major theorem for Gaussian stationary sequences

Consider a Gaussian stationary sequence with unit variance $X=\{X_k;k\in {\mathbb{N}}\cup\{0\}\}$. Assume that the central limit theorem holds for a weighted sum of the form $V_n=n^{-1/2}\sum^{n-1}_{k=0}f(X_k)$, where $f$ designates a finite sum of Hermite polynomials. Then we prove that the uniform convergence of the density of $V_n$ towards the standard Gaussian density also holds true, under a mild additional assumption involving the causal representation of $X$.Comment: Published at http://dx.doi.org/10.3150/14-BEJ646 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Kernel entropy estimation for long memory linear processes with infinite variance

Let $X=\{X_n: n\in\mathbb{N}\}$ be a long memory linear process with innovations in the domain of attraction of an $\alpha$-stable law $(0<\alpha<2)$. Assume that the linear process $X$ has a bounded probability density function $f(x)$. Then, under certain conditions, we consider the estimation of the quadratic functional $\int_{\mathbb{R}} f^2(x) \,dx$ by using the kernel estimator $$T_n(h_n)=\frac{2}{n(n-1)h_n}\sum_{1\leq j<i\leq n}K\left(\frac{X_i-X_j}{h_n}\right).$$ The simulation study for long memory linear processes with symmetric $\alpha$-stable innovations is also given

Limit theorems for functionals of long memory linear processes with infinite variance

Let $X=\{X_n: n\in\mathbb{N}\}$ be a long memory linear process in which the coefficients are regularly varying and innovations are independent and identically distributed and belong to the domain of attraction of an $\alpha$-stable law with $\alpha\in (0, 2)$. Then, for any integrable and square integrable function $K$ on $\mathbb{R}$, under certain mild conditions, we establish the asymptotic distributions of the partial sum $$\sum\limits_{n=1}^{N}\big[K(X_n)-\mathbb{E} K(X_n)\big]$$ as $N$ tends to infinity