Hermite Process: Theory & Extensions

• Hermite process is a family of self-similar stochastic processes defined via multiple Wiener–Itô integrals with stationary increments and non-Gaussian dependence.
• It generalizes fractional Brownian motion and the Rosenblatt process, effectively modeling long-range dependence and rough sample paths with Hölder continuity.
• Variants such as generalized, tempered, and multifractional Hermite processes extend its applicability, aiding in advanced estimation techniques like quadratic and wavelet variations.

A Hermite process is a family of real-valued, self-similar stochastic processes characterized by non-Gaussian dependence structure, stationary increments, and representations as multiple Wiener–Itô integrals of order $q \ge 1$ with a parameterized Hurst exponent %%%%1%%%%. This class generalizes fractional Brownian motion (first-order, Gaussian) and the Rosenblatt process (second-order, non-Gaussian) and naturally arises as scaling limits in non-central limit theorems for long-range dependent stationary sequences. The Hermite process framework also extends to multifractional, tempered, and generalized kernels.

1. Mathematical Definition and Integral Representations

The canonical Hermite process of order $q \in \mathbb{N}_+$ and Hurst parameter $H \in (\tfrac12,1)$ is defined as

$$X^{(q,H)}(t) = c_{q,H} \int_{\mathbb{R}^q} \left( \int_0^t \prod_{j=1}^q (s - x_j)_+^\gamma ds \right) dW(x_1) \cdots dW(x_q),$$

where $(u)_+ = \max(u,0)$, $\gamma = H - 1 - 1/q$, and $c_{q,H}$ is chosen so that $\operatorname{Var}(X^{(q,H)}(1))=1$ (Loosveldt, 2023). This process lives in the $q$th Wiener chaos. Special cases include:

• $q=1$: fractional Brownian motion (Gaussian), integral kernel reduces to $(s-x)^{H-3/2}_+$.
• $q=2$: Rosenblatt process (non-Gaussian), integral kernel in $(s-x_1)^{H-1}_+(s-x_2)^{H-1}_+$.

Hermite processes admit several equivalent representations, including frequency-domain integrals (Bai, 2019). The norming and exclusion of diagonal terms (denoted by the prime on the integral) are essential for higher-order chaos.

2. Key Properties: Self-similarity, Stationarity, and Covariance

Hermite processes are $H$-self-similar and have stationary increments: $$\{ X^{(q,H)}(a t)\}_{t\ge0} \overset{f.d.d.}{=} \{ a^H X^{(q,H)}(t)\}_{t\ge0},$$ with $\operatorname{Var}(X^{(q,H)}(t)) = t^{2H}$ for all $t \ge 0$. The covariance structure mirrors that of fractional Brownian motion: $$\operatorname{Cov}(X^{(q,H)}(t), X^{(q,H)}(s)) = \frac12( t^{2H} + s^{2H} - |t-s|^{2H} ),$$ despite the non-Gaussian nature when $q \ge 2$ (Loosveldt, 2023, Clausel et al., 2014). The increments exhibit long-range dependence: for large lags $k$,

$$\operatorname{Cov}(X_{t+1} - X_t, X_{s+1} - X_s) \sim c |t-s|^{2H-2}.$$

Sample paths are almost surely Hölder continuous of any order $\alpha<H$ but nowhere differentiable (Ayache, 2019). The Hausdorff (and box) dimension of the graph is $2-H$.

3. Generalized, Tempered, and Multifractional Extensions

Generalized Hermite Processes

A generalized Hermite process is constructed with a homogeneous kernel $g:\mathbb{R}^k \to \mathbb{R}$ (degree $\alpha$). The process is

$$Z(t) = I_k(h_t), \quad h_t(x) = \int_0^t g(s-x_1,\dots,s-x_k) \prod_{j=1}^k 1_{s>x_j} ds,$$

with $H = \alpha + k/2 + 1 \in (\tfrac12,1)$ (Bai et al., 2013, Araya, 2022). When $g(x) = \prod x_j^{d-1} 1_{(0,\infty)}(x_j)$, the classical Hermite process is recovered. Generalized Hermite processes include fractionally-filtered variants capable of $H \in (0,1)$.

Tempered Generalized Hermite Processes

Tempering modifies the kernel by an exponential factor: $$h_{t,\lambda}(x) = \int_0^t g(s-x) \exp\left(-\lambda \sum_{j=1}^k (s-x_j)\right) \prod_{j=1}^k 1_{s>x_j} ds,$$ yielding stationary increments and a modified scaling relation (Araya, 2022). The process exhibits controlled (short-to-medium range) dependence due to exponential decay of correlations, while retaining non-Gaussianity and small-scale self-similarity.

Multifractional Hermite Processes

Multifractional Hermite processes replace the constant Hurst parameter by a function $H(t)$. The generator field is

$$X_q(t,h) = I_q \left( \int_0^t \prod_{j=1}^q (s-x_j)_+^{(h-1)/q-1/2} ds \right ),$$

and the process is $X_q^{H(\cdot)}(t) = X_q(t, H(t))$ (Loosveldt, 2023). The pathwise regularity, modulus of continuity, and fractal dimensions are governed by $H(\cdot)$, with local asymptotic self-similarity at each $t_0$ to a standard Hermite process of order $q$ and Hurst $H(t_0)$. For $q=2$ (Rosenblatt), advanced Malliavin calculus shows a continuous, bounded increment density, yielding sharp fractal dimension results.

4. Limit Theorems, Statistical Estimation, and Quadratic/Wavelet Variations

Hermite processes arise as non-Gaussian limits in sums of nonlinear functionals of long-range dependent stationary Gaussian sequences (non-central limit theorems), with scaling exponent determined by the memory structure (Clausel et al., 2014, Bai et al., 2013).

Quadratic Variation and Estimation

For $q \ge 2$, standard quadratic variation estimators of the Hurst parameter do not admit Gaussian CLTs except in specific regimes. Ayache–Tudor (Ayache et al., 2023) introduce a modified quadratic variation based on sparsely selected increments that become i.i.d. in the limit: $$V_N = 2^{2NH} \frac{1}{|L_{N,\beta,\eta}|} \sum_{\ell \in L_{N,\beta,\eta}} \{ (\Delta Z_{\ell,N})^2 - \mathbb{E}[(\Delta Z_{\ell,N})^2]\},$$ which admits a Gaussian CLT at an explicit geometric rate under the Wasserstein distance (using Stein–Malliavin calculus). The associated estimator for $H$ is strongly consistent and asymptotically normal for all $q$.

Modified Wavelet Variations

Loosveldt–Tudor (Loosveldt et al., 2024) achieve an asymptotically normal, strongly consistent wavelet-based estimator for $H$ in any Hermite process. By partitioning wavelet coefficients so dominant terms are independent, they derive a multidimensional CLT for the normalized wavelet variations with an explicit bound on the Wasserstein distance.

5. Sample Path Regularity, Oscillations, and Persistence

Hermite process sample paths are almost surely nowhere differentiable. Ayache (Ayache, 2019) provides quasi-optimal lower bounds for the local oscillations, showing in particular that for any $\tau>0$,

$$\liminf_{r \to 0} r^{-\kappa} |\log_2 r|^{\kappa} S(|\log_2 r|)^{-\kappa} \operatorname{Osc}(X^{N,H},\tau,r) \ge c_{N,H}^*$$

for suitable $S$, where $\kappa = N(H-1)+1$. This formalizes the inherent roughness and precludes differentiability.

Persistence exponents, i.e., probabilities that the process remains below a barrier on $[0,T]$, have been established for the Rosenblatt process with exponent $1-H$ via a decorrelation inequality reminiscent of Slepian's lemma (Aurzada et al., 2016). For general Hermite processes of order $m\geq 2$, only sharp upper and lower polynomial bounds exist for persistence; the exact exponent is conjectured to be $1-H$ in all cases, but proofs for $m \ge 3$ remain an open problem.

6. Simulation and Series Expansions

While wavelet series representations for fractional Brownian motion ($q=1$) and the Rosenblatt process ($q=2$) have long been known, Ayache–Hamonier–Loosveldt (Ayache et al., 2023) and (Ayache et al., 31 Mar 2025) developed constructive wavelet-type random series expansions for arbitrary order $q$ Hermite and generalized Hermite processes. These expansions leverage Meyer wavelets and fractional scaling functions to provide pathwise convergence with explicit almost sure error bounds: $$\| X_H^{(m)} - S_{h,J}^{(m)} \|_{I,\infty} = O(J^{m/2} 2^{-J(H - 1/2)}),$$ facilitating precise numerical simulation even for $q \ge 3$. Python routines implementing these schemes compute discretized sample paths for the Rosenblatt ($m=2$) and order-3 Hermite processes ($m=3$).

7. Local-Time and Regenerative Set Representations

Bai (Bai, 2019) revealed a structural link between Hermite processes and local times of intersections of stationary stable regenerative sets. For order $m$ and memory parameter $\beta \in (1-1/m,1)$, the process can be expressed as

$$Z(t) = c_{m,\beta} \int_{E^m}' L_t\left(\bigcap_{i=1}^m \overline{R}(x_i)\right) W(dx_1) \cdots W(dx_m),$$

where $L_t$ is Kingman's local time functional and $\overline{R}(x)$ denotes a stationary $\beta$-stable regenerative set. This probabilistic viewpoint, proven via covering and spectral techniques, provides an alternative and universally valid construction of Hermite processes.

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References:

• "Multifractional Hermite processes: definition and first properties" (Loosveldt, 2023)
• "Asymptotic normality for a modified quadratic variation of the Hermite process" (Ayache et al., 2023)
• "Generalized Hermite process: tempering, properties and applications" (Araya, 2022)
• "Numerical simulation of Generalized Hermite Processes" (Ayache et al., 31 Mar 2025)
• "Modified wavelet variation for the Hermite processes" (Loosveldt et al., 2024)
• "Generalized Hermite processes, discrete chaos and limit theorems" (Bai et al., 2013)
• "Persistence probabilities and a decorrelation inequality for the Rosenblatt process and Hermite processes" (Aurzada et al., 2016)
• "Lower bound for local oscillations of Hermite processes" (Ayache, 2019)
• "Wavelet-Type Expansion of Generalized Hermite Processes with rate of convergence" (Ayache et al., 2023)
• "Asymptotic behavior of the quadratic variation of the sum of two Hermite processes of consecutive orders" (Clausel et al., 2014)
• "Representations of Hermite processes using local time of intersecting stationary stable regenerative sets" (Bai, 2019)