Quadratic variations of rough generalized Hermite processes: hidden long memory and a universal Gaussian boundary field

Abstract

We study centered quadratic variations of fractionally filtered generalized Hermite processes. For every chaos order $q\ge 2$, every admissible anisotropic monomial kernel $g_{\mathbf{\gamma }}$, and every fixed $0<h\le 1/2$, we prove that $N^{2h+\theta -1}{V}_N$ converges to a Rosenblatt process with self-similarity exponent $1-\theta$, without subtracting any chaos projection. Thus the visible roughness $h$ determines the normalization, whereas the latent singularity exponent $\theta$ determines the memory of the limit. The proof uses a sharp paired-filter threshold and charged diagram power counting. At $h=0$, the bare kernel has logarithmically divergent energy. Variance-normalized $h$-regularization and a hard endpoint cutoff have the same residue and converge in finite-dimensional distributions to the universal Gaussian field $G_t=({\zeta }_t-{\zeta }_0)/\sqrt{2}$. Although $G$ has no stochastically continuous modification, the iterated boundary-first quadratic energies converge to $\sqrt{3}B$. In the isotropic case, simultaneous limits $h_N\downarrow 0$ are governed by ${\lambda }_N=h_N{N}^{1/2-\theta }$: ${\lambda }_N\to 0$ yields a Brownian limit, ${\lambda }_N\to \lambda \in (0,\infty )$ yields an independent Brownian-Rosenblatt sum, and ${\lambda }_N\to \infty$ yields a Rosenblatt limit after division by ${\lambda }_N$. We also obtain an explicit polynomial rate in fixed Malliavin-Sobolev norms and functional convergence in little-Hölder spaces.