粗糙广义Hermite过程的二次变差：隐藏长记忆与一个普适高斯边界场
Quadratic variations of rough generalized Hermite processes: hidden long memory and a universal Gaussian boundary field
AI总结：
本文研究分数阶滤波广义Hermite过程的二次变差，证明其归一化极限由可见粗糙度决定，而记忆由潜在奇异指数决定，并揭示普适高斯边界场及各向同性极限行为。
AI中文摘要：
我们研究了分数阶滤波广义Hermite过程的中心化二次变差。对于每个混沌阶$q\ge2$、每个允许的各向异性单项核$g_{\boldsymbol\gamma}$以及每个固定的$0<h\le1/2$，我们证明了$N^{2h+\theta-1}V_N$收敛到一个自相似指数为$1-\theta$的Rosenblatt过程，无需减去任何混沌投影。因此，可见粗糙度$h$决定了归一化，而潜在奇异指数$\theta$决定了极限的记忆性。证明使用了尖锐的配对滤波器阈值和带电图幂计数。在$h=0$时，裸核具有对数发散的能量。方差归一化的$h$-正则化和硬端点截断具有相同的残差，并在有限维分布上收敛到普适高斯场$G_t=(\zeta_t-\zeta_0)/\sqrt2$。尽管$G$没有随机连续修正，迭代的边界优先二次能量收敛到$\sqrt3B$。在各向同性情形下，同时极限$h_N\downarrow0$由$\lambda_N=h_NN^{1/2-\theta}$控制：$\lambda_N\to0$产生布朗极限，$\lambda_N\to\lambda\in(0,\infty)$产生独立的布朗-Rosenblatt和，$\lambda_N\to\infty$在除以$\lambda_N$后产生Rosenblatt极限。我们还得到了固定Malliavin-Sobolev范数中的显式多项式速率以及little-Hölder空间中的函数收敛。
英文摘要：
We study centered quadratic variations of fractionally filtered generalized Hermite processes. For every chaos order $q\ge2$, every admissible anisotropic monomial kernel $g_{\boldsymbolγ}$, and every fixed $0<h\le1/2$, we prove that $N^{2h+θ-1}V_N$ converges to a Rosenblatt process with self-similarity exponent $1-θ$, without subtracting any chaos projection. Thus the visible roughness $h$ determines the normalization, whereas the latent singularity exponent $θ$ 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 $G_t=(ζ_t-ζ_0)/\sqrt2$. Although $G$ has no stochastically continuous modification, the iterated boundary-first quadratic energies converge to $\sqrt3B$. In the isotropic case, simultaneous limits $h_N\downarrow0$ are governed by $λ_N=h_NN^{1/2-θ}$: $λ_N\to0$ yields a Brownian limit, $λ_N\toλ\in(0,\infty)$ yields an independent Brownian-Rosenblatt sum, and $λ_N\to\infty$ yields a Rosenblatt limit after division by $λ_N$. We also obtain an explicit polynomial rate in fixed Malliavin-Sobolev norms and functional convergence in little-Hölder spaces.
