i-Gaussian Approach: Unified Stochastic Modeling

• The i-Gaussian approach is a unified modeling framework for stochastic processes that extends classical Gaussian methods using operator theory and Lévy-driven innovations.
• It decouples correlation and sparsity by employing a whitening operator L alongside a customizable Lévy measure, enabling control over smooth and jump-dominated behaviors.
• The framework’s rigorous foundation in spectral analysis and functional calculus supports advanced applications in signal processing, Bayesian inference, and robust stochastic modeling.

The i-Gaussian approach denotes a set of mathematical and modeling techniques that utilize Gaussian processes, generalized Gaussian structures, or Gaussian-based transforms and analysis as fundamental components for representing, estimating, or processing random phenomena in continuous or discrete domains. Across signal processing, statistics, stochastic modeling, information theory, and mathematical analysis, i-Gaussian methodologies provide a unified framework for bridging classical Gaussian models with more general, often sparse or heavy-tailed, random structures. The approach is deeply rooted in operator theory, Lévy process analysis, and generalized functional calculus, and is crucial both for theoretical development and for practical algorithm design.

1. Unified Innovation Model for Stochastic Processes

The i-Gaussian framework formalizes stochastic processes $s$ as filtered generalized white noise $w$ via a linear, often differential, whitening operator $L$. This model is succinctly described as

$$s = L^{-1}w \quad \text{or equivalently} \quad Ls = w$$

where the "innovation" $w$ has a characteristic functional determined by a Lévy exponent $f$: $$\widehat{\mathscr{P}}_w(\varphi) = \exp\left( \int_{\mathbb{R}} f(\varphi(t)) dt \right).$$ This innovation model encompasses both the classical Gaussian case (where $f(u) = -\frac{|u|^2}{2}$ and $w$ is Gaussian white noise) and a spectrum of non-Gaussian processes (Laplace, compound Poisson, $\alpha$-stable, etc.), enabling the construction of processes ranging from fully dense (Gaussian) to highly sparse (e.g., heavy-tailed or impulsive) regimes. The process' correlation properties are imposed entirely by $L$, while its sparsity is dictated by the Lévy measure underlying $f$.

2. Role of the Whitening Operator and Decoupling of Structure

The operator $L$ serves as a "decorrelator," transforming the process $s$ into a generalized innovation $w$. Its inverse $L^{-1}$ confers the desired autocorrelation or smoothness properties onto $w$. In the frequency domain, the spectral density is

$$\Phi_s(\omega) = \frac{\sigma_0^2}{|\widehat{L}(-\omega)|^2},$$

demonstrating that smoothness and spectral decay are entirely governed by the operator's structure. Operator $L$ can be decomposed into first-order factors, realized via convolution with fundamental solutions (Green's functions), supporting explicit construction even in unstable or nonstationary cases (e.g., when poles lie on the imaginary axis or at the origin).

3. Lévy Measure: Controlling Sparsity and Innovation Law

The choice of Lévy measure $V$ in the Lévy–Khintchine formula defines the innovation's law and hence the stochastic or jump structure of $w$. For Gaussian processes, $V \equiv 0$, yielding dense, smooth processes. For compound Poisson, $V$ is finite and localized, modeling discrete jumps. Heavy-tailed or $\alpha$-stable innovations entail infinite $V$ (not in $L^1$), resulting in extremely sparse, jump-dominated behavior. Thus, the i-Gaussian framework provides a parameterization bridging the Gaussian and super-sparse regimes, with the Lévy measure as a dial.

4. Mathematical Foundations: Characteristic Functionals and Existence Criteria

The general construction is rigorous within the Gelfand–Vilenkin theory of generalized stochastic processes. Rather than pointwise definition, the process is described by its characteristic functional: $$\widehat{\mathscr{P}}_s(\varphi) = \mathbb{E}\left\{ \exp\left[j\langle s, \varphi \rangle\right] \right\}, \quad \varphi \in \mathcal{S}.$$ Well-posedness is achieved if the adjoint inverse operator $L^{-1*}$ is continuous on Schwartz space $\mathcal{S}$ and a $p$-admissibility condition on $f$ holds: $|f(u)| + |u|\,|f'(u)| \leq C |u|^p,$ with $p \geq 1$. This ensures positive-definiteness and normalization, guaranteeing the measure's existence by Minlos–Bochner’s theorem.

5. Operator-Based Solution Methods and Generalizations

A central methodological advance is the operator-based decomposition of $L$ into first-order operators, each inverted by convolution with (possibly causal or anti-causal) Green's functions:

• For $\mathrm{Re}(\alpha_n)<0$, the causal solution $\rho_{\alpha_n}(t) = 1_+(t) e^{\alpha_n t}$ is used.
• For $\mathrm{Re}(\alpha_n)>0$, an anti-causal kernel applies.
• For critical cases ($\mathrm{Re}(\alpha_n) = 0$), e.g., integration or oscillatory terms, an inverse is constructed with polynomial or sinusoidal correction terms to enforce boundary conditions, ensuring invertibility even when $L$ is not BIBO stable.

For processes with unstable or marginally stable dynamics, boundary condition enforcement yields generalized processes with random polynomial trends, further extending the class of admissible models beyond stationary CARMA processes.

6. Stationarization via Finite Differences and Multiscale Decomposition

In unstable (e.g., integrated or random walk-like) settings, stationarity and decorrelation are recovered by applying finite difference operators. The increments

$\Delta_0 s[k] = s[k] - s[k-1]$

recover the innovation $w$ on a discrete grid. Discretized increments are independent and preserve the innovation’s law, fully decoupling correlation. Moreover, an orthogonal multiscale (wavelet-like) representation can be achieved by constructing matched wavelets from the system's smoothing kernels, achieving additional sparsity and decorrelation at multiple scales.

7. Applications and Implications Across the Gaussian–Sparse Spectrum

The i-Gaussian formulation subsumes stationary processes (e.g., CARMA driven by Gaussian or Lévy noise), unstable and non-stationary generalized Lévy processes (with polynomial or fractal behavior), and supports construction of models possessing both prescribed spectral laws and targeted sparsity. This unification permits systematic design of continuous-domain priors for stochastic modeling, image processing, machine learning (via sparse and heavy-tailed stochastic priors), and signal reconstruction tasks. It also facilitates the analysis of stochastic differential equations driven by arbitrary Lévy noise with explicit operator-theoretic tools.

8. Summary Table of Structural Decoupling

Model Ingredient	Controls	Mathematical Representation
Whitening operator $L$	Correlation/Spectral	$L s = w$, $$\Phi_s(\omega) = \sigma_0^2/|\widehat{L}(-\omega)|^2$$
Lévy measure $V$	Sparsity/Innovation law	$f$ in Lévy–Khintchine, jump statistics
Boundary corrections	Stationarity/Trends	Correction terms in $L^{-1}$, Finite difference $\Delta_0$

9. Connections and Extensions

The i-Gaussian approach is compatible with classical Gaussian process theory and extends naturally to non-Gaussian, sparse, or heavy-tailed signals by virtue of the Lévy-driven innovation. It opens avenues for precise, operator-theoretic control over statistical, spectral, and sparsity properties, enabling advanced modeling, inference, and synthesis across engineering and applied mathematics. The formalism is robust to inclusion in Bayesian inverse problems, compressed sensing, and modern high-dimensional statistics, where blending spectral shape (via $L$) and distributional sparsity (via $V$) is essential.

In conclusion, the i-Gaussian approach furnishes a mathematically rigorous, structurally decoupled, and highly generalizable framework for the continuous-domain formulation of stochastic processes, capturing the entire spectrum from dense Gaussian fields to highly sparse Lévy-driven signals (Unser et al., 2011).