On the local limit theorem for densities in case of a limit law of the class L (Q582680)

Let H be a real functional of Hermite index k, \({\mathcal X}=\{X_ t\), \(t\in {\mathbb{R}}\}\) a stationary Gaussian process with covariance R verifying: (1) \(1-R(t)=t^{\beta}L(t)\) with L slowly varying at \(t=0\) and \(0<\beta <2.\) (2) There exists \({\mathbb{R}}''(t)\) if \(t\neq 0\), \({\mathbb{R}}''(t)=t^{\beta - 2}L_ 1(t)\) where \(L_ 1\) is the variation at 0, continuous on (0,\(\infty)\). \(| {\mathbb{R}}''(t)|\), \(t>0\), is non-increasing at a neighbourhood of 0. (3) \(L_ 1(t)=\beta (1-\beta)L(t)(1+o(1))\) at a neighbourhood of 0. Let \(\Pi_ n\) be a regular partition of [0,1]: \[ \Pi_ n:\quad \{0<x^ n_ 1<x^ n_ 2<...<x^ n_ n=1\},\quad x^ n_ j=j/n,\quad j=1,...,n, \] and \(\Delta^ nX=\{\Delta^ n_ jX=X(j/n)- X((j-1)/n)\), \(j\in {\mathbb{Z}}\}\) the process with discrete time and covariance \(r_ n\) \((\sigma^ 2_ n=r(0))\) with correlation \(S_ n\). The H-variation of X along \(\Pi_ n\) is defined by \[ V_{H,n}=\sum^{n}_{j=1}H(\Delta^ n_ jX/\sigma_ n). \] If \(k(2- \beta)<1\), then under (1)-(3) the H-variation weakly converges to the k- th Wiener chaos. If \(k(2-\beta)>1\) and \[ \limsup_{0<x\to 0}(\sup_{y:x\leq y\leq x^ b}| L(y)/L(x)|)<\infty,\quad for\quad some\quad 0<b<1, \] then \(\{V_{H,n}\), \(n\geq 1\}\) weakly converges to a Gaussian process.