Gaussian Limiting Distribution

• Gaussian limiting distribution is a concept that explains how normalized nonlinear functionals of Gaussian processes converge to a normal law, highlighting long-range dependence and spectral singularities.
• The framework employs Hermite expansions, spectral convolutions, and weight normalization to derive explicit covariance structures under precise regularity conditions.
• This theory is vital for applications in nonlinear regression and time series analysis, offering robust methods for inference in models with long memory and periodic behaviors.

A Gaussian limiting distribution refers to the appearance of the (possibly multivariate) normal law as the limiting distribution of a properly normalized sequence of random quantities arising from structured probabilistic, stochastic process, or functional-analytic frameworks. In the context of Gaussian stationary processes, nonlinear functionals, or statistical models with singular spectra, rigorous limit theorems establish the emergence of such distributions, typically under normalization and specific assumptions on dependence, nonlinearity, and weighting schemes.

1. Gaussian Stationary Processes and Singular Spectra

A real-valued, zero-mean stationary Gaussian process $\xi(t)$ (with $t \in \mathbb{R}$ or $\mathbb{Z}$) is specified by its covariance function $B(t)$, which is translation invariant: $B(t) = E[\xi(0)\xi(t)]$. In the singular spectrum regime, the covariance admits a finite decomposition: $$B(t) = \sum_{j=0}^r A_j B_{\alpha_j, \kappa_j}(t) \quad \text{with} \quad B_{\alpha_j, \kappa_j}(t) = \frac{\cos(\kappa_j t)}{(1 + t^2)^{\alpha_j / 2}}$$ for nonnegative weights $A_j$ summing to one, singularity parameters $\alpha_j > 0$, and spectral frequencies $\kappa_j$. The corresponding spectral density $f(\lambda)$ displays algebraic singularities at each $\pm\kappa_j$, typically diverging like a power law as $\lambda \to \kappa_j$. These singularities induce long-range dependence features and encode periodic or seasonalities in the process (Ivanov et al., 2013).

2. Nonlinear Functionals and Hermite Expansions

Gaussian limiting behavior is not restricted to linear functionals. The analysis extends to nonlinear functionals of the form $\psi(\xi(t))$ for measurable functions $\psi$ in $L_2(\mathbb{R}, \varphi(x) dx)$ (with $\varphi$ the standard normal density). Such nonlinearities are expanded using the orthogonal Hermite polynomial system: $$\psi(x) = \sum_{k=m}^\infty \frac{C_k(\psi)}{k!} H_k(x)$$ where Hermite rank $m$ is the lowest $k$ such that $C_k(\psi) \neq 0$. This expansion crucially enables functional central limit theorems via decomposition of the process into independent Wiener chaoses, with primary contributions arising from the leading chaos indexed by $m$.

The functional of interest is often a normalized weighted sum or integral: $$\zeta_T = W_T^{-1} \int_0^T w(t) \psi(\xi(t))\,\nu(dt)$$ where $w(t)$ is a deterministic weight, and $W_{i,T}^2 = \int_0^T w_i^2(t)\,\nu(dt)$ provides normalization. The large-$T$ limiting behavior of $\zeta_T$ encodes the interplay between the nonlinearity, the spectral structure of $\xi$, and the weight function (Ivanov et al., 2013).

3. Main Limit Theorem: Gaussian Limiting Distribution

The core result is that, under regularity assumptions on the process, spectra, nonlinearity, and weights (see assumptions (A1)-(A4), (B1)-(B3), and convolution condition (C) in (Ivanov et al., 2013)), the weighted functional $\zeta_T$ converges in distribution as $T \to \infty$ to a multivariate Gaussian: $\zeta_T \xrightarrow{d} \mathcal{N}(0, \Xi)$ with explicit covariance

$$\Xi = 2\pi\, \sum_{j=m}^\infty \frac{C_j(\psi)^2}{j!} \int_\Lambda f^{(*j)}(\lambda)\, \mu(d\lambda)$$

where:

• $f^{(*j)}(\lambda)$ is the $j$-fold convolution of the spectral density $f$,
• $\mu$ is a limiting (weakly convergent) matrix-valued measure derived from the weights $w(t)$,
• $m$ is the Hermite rank of $\psi$.

The limiting covariance depends essentially on both the Hermite expansion of the nonlinearity and the spectral structure, particularly through convolution powers that magnify singularities and create intricate limit covariances (see equation (lim14) in (Ivanov et al., 2013)).

Normalization of the weight functions—specifically, controlling their spectral measures—prevents divergence or degeneracy of the limit covariance. Convergence in nonlinear functionals is analyzed with tools such as contraction norms showing vanishing higher-order contractions in Wiener chaos decompositions (cf. Proposition 2 in (Ivanov et al., 2013)).

4. Statistical Applications: Nonlinear Regression and Asymptotic Inference

A key application is nonlinear regression with Gaussian stationary noise of singular spectrum: $x(t) = g(t, \theta) + \psi(\xi(t))$ where $g(t, \theta)$ models the deterministic trend (potentially with multiple frequencies), and the error is the nonlinear functional of the process. Analysis of the weighted gradient integral

$$\zeta_T = d_T^{-1}(\theta)\int_0^T \nabla g(t,\theta) \, \psi(\xi(t))\, \nu(dt)$$

yields the asymptotic distribution for the least squares estimator (LSE). The main theorem guarantees that, under suitable normalization, the LSE converges to a Gaussian law, even in the presence of long memory and periodic dependencies from the underlying noise. This supports valid construction of confidence intervals and hypothesis testing for regression parameters in dependent contexts (Ivanov et al., 2013).

The framework applies broadly where periodicities and long-range dependencies are present (e.g., time series from climatology, finance, or telecommunications), supplying a rigorous route to asymptotic normality beyond standard mixing or short-memory models.

5. Technical Conditions and Covariance Identification

Emergence of the Gaussian limit hinges on several technicalities:

• The weight functions' spectral measures must converge weakly (as $T \to \infty$) to a nondegenerate limit $\mu$, ensuring that the quadratic forms in the frequency domain remain invertible.
• Boundedness near spectral singularities is imposed to prevent blow-up of variances.
• The singularities in the spectral density must be separated (no overlap) for the stated CLT; otherwise, normalization or the limiting law may need adjustment, possibly leading to non-central limit behaviors.

The explicit limiting covariance matrix $\Xi$ is sensitive to the order and nature of the singularities, the Hermite rank, and the weighting. Computation of $\Xi$ requires careful analysis of convolution powers of singularly supported spectral densities and integration against limiting measures derived from the weights.

6. Further Directions: Extensions and Open Problems

Several research avenues remain open or conjectured:

• The Gaussian limit is established for minimal singularity parameter $\alpha = \min\{\alpha_j\} > 1/2$, sufficient to control variance, but it is conjectured that similar results extend to the broader range $\alpha_j \in (0,1)$, potentially with modified centering or normalization.
• Overlapping singularities (e.g., when regression and noise possess spectral atoms at coinciding frequencies) can induce noncentral limit theorems or require alternative normalization, especially in discrete time.
• Generalization to multidimensional stationary random fields with singular spectra and extension to spatial or spatiotemporal models are natural directions.
• The limit theory has yet to be fully exploited to construct finite-sample procedures; statistical methods based on the derived CLTs (e.g., for constructing confidence regions for nonlinear parameters under dependent noise) merit further exploration.

7. Summary Table: Major Elements in the Gaussian Limit Setting

Component	Description	Role in Limiting Law
$\psi(\cdot)$	Nonlinear transformation, expanded in Hermite polynomials, rank $m$	Determines chaos level, coefficients $C_m$
$f(\lambda)$	Spectral density of Gaussian process $\xi(t)$, with singularities at specific frequencies	Governs long-range dependence, convolutions
$w(t)$	Weight function, normalized; vector-valued in multivariate case	Impacts limiting measure $\mu$
$f^{(*j)}$	$j$-fold convolution of spectral density	Enters covariance expression
$\mu$	Weak limit of matrix-valued spectral measures of weights	Defines limit covariance structure
Normalization	$W_T^{-1}$, scaling the weighted nonlinear functional appropriately	Ensures finite and nondegenerate limits
Wiener chaos	Orthogonal decomposition of nonlinear functionals in terms of Hermite polynomials, isolating the dominant chaos	Facilitates CLT proof methodology

The synthesis and general framework developed in (Ivanov et al., 2013) comprehensively characterize how Gaussian limiting distributions arise for nonlinear functionals of Gaussian stationary processes exhibiting both long memory and singular spectral behavior, with explicit covariance formulas for the limits, structural dependence on nonlinearity and weights, and broad implications for regression, time series inference, and theoretical stochastic process analysis.