Harmonizable Stochastic Processes

Harmonizable Stochastic Processes: Mathematical Foundations and Key Theorems

The theory of harmonizable stochastic processes extends spectral analysis to non-stationary processes through measure-theoretic frameworks. This report synthesizes fundamental theorems, emphasizing rigorous mathematical formulations. Below, we present the core results with precise definitions, proofs, and applications.

---

1. Spectral Representation and Harmonizability

1.1 Loève’s Harmonizability Theorem

A stochastic process ${X(t), t }$ is harmonizable if and only if its covariance function $C(s,t) = [X(s)]$ admits a spectral representation:

C(s,t) = ∬_ℝ²e^i(λs−μt) dF(λ,μ),

where $F$ is a complex-valued bimeasure of bounded variation on $^2$ . This generalizes the Wiener-Khinchin theorem for stationary processes, where $F$ reduces to a measure on the diagonal $=$.

1.2 Cramér’s Representation Theorem

Every harmonizable process $X(t)$ can be expressed as:

X(t) = ∫_ℝe^iλt dZ(λ),

where $Z()$ is a stochastic process with orthogonally scattered increments satisfying:

$$\mathbb{E}[dZ(\lambda)\overline{dZ(\mu)}] = dF(\lambda, \mu).$$

This representation preserves the Fourier-analytic structure while accommodating non-stationarity.

---

2. Structural Decompositions

2.1 Rao’s Decomposition Theorem

Any harmonizable process decomposes uniquely into:

X(t) = X_p(t) + X_w(t),

where:

• $X_p(t)$ is purely harmonizable (spectral measure absolutely continuous),
• $X_w(t)$ is weakly harmonizable (spectral measure singular).

This separation clarifies the interplay between spectral regularity and pathological components.

2.2 Karhunen-Loève Expansion

For harmonizable processes with continuous covariance, the orthogonal expansion holds:

$$X(t) = \sum_{k=1}^\infty \sqrt{\lambda_k} \xi_k \phi_k(t),$$

where ${_k}$ and ${_k(t)}$ are eigenvalues/eigenfunctions of the covariance operator:

∫_ℝC(s,t)ϕ_k(s) ds = λ_kϕ_k(t),

and $_k$ are uncorrelated random variables with $[_k] = 0$, $[|_k|^2] = 1$ .

---

3. Analytical Properties

3.1 Continuity and Differentiability

A harmonizable process $X(t)$ is:

• Mean-square continuous iff $F$ is continuous in each variable.
• Mean-square differentiable iff:

$$\iint_{\mathbb{R}^2} (\lambda^2 + \mu^2) \, d|F|(\lambda, \mu) < \infty,$$

where $|F|$ denotes the total variation measure.

3.2 Spectral Density and Inversion

When $F$ is absolutely continuous with density $f(, )$, the covariance becomes:

C(s,t) = ∬_ℝ²e^i(λs−μt)f(λ,μ) dλdμ.

The inversion formula recovers $f$ via:

$$f(\lambda, \mu) = \frac{1}{(2\pi)^2} \iint_{\mathbb{R}^2} e^{-i(\lambda s - \mu t)} C(s,t) \, ds dt.$$

---

4. Relationship to Stationarity

4.1 Stationarity as a Special Case

A harmonizable process is wide-sense stationary iff its spectral measure $F$ is concentrated on the diagonal $=$, reducing to:

C(s,t) = ∫_ℝe^iλ(s−t) dF(λ).

This recovers the classical Wiener-Khinchin theorem.

4.2 Gladyshev’s Characterization

A process $X(t)$ is harmonizable iff for any $t_1, , t_n$, the characteristic function of $(X(t_1), , X(t_n))$ satisfies:

$$\phi(\theta_1, \dots, \theta_n) = \exp\left( \iint_{\mathbb{R}^2} \left( \sum_{k=1}^n \theta_k e^{i\lambda t_k} \right) \, dF(\lambda, \mu) \right).$$

This links finite-dimensional distributions to spectral structure.

---

5. Prediction and Sampling

5.1 Optimal Linear Prediction

The best linear predictor $(t)$ given ${X(s) : s t_0}$ is:

X̂(t) = ∫_−∞^t₀h(t,s)X(s) ds,

where the kernel $h(t, s)$ solves the Wiener-Hopf equation:

∫_−∞^t₀C(s,u)h(t,u) du = C(s,t),  ∀s ≤ t₀.

This extends Kalman filtering to non-stationary contexts.

5.2 Sampling Theorem

If $X(t)$ has spectral support in $[-, ]^2$, it is reconstructible from samples ${X(nT)}$ at rate $T /$:

$$X(t) = \sum_{n=-\infty}^\infty X(nT) \frac{\sin[\Omega(t - nT)]}{\Omega(t - nT)}.$$

This generalizes the Nyquist-Shannon theorem.

---

6. Measure-Theoretic Foundations

6.1 Spectral Bimeasures

The spectral bimeasure $F$ is a complex-valued function on $() ()$ satisfying:

1. Hermitian symmetry: $F(A, B) =$.
2. Additivity: $ F(_i A_i, j B_j) = {i,j} F(A_i, B_j) $.
3. Bounded variation: $|F|(, ) <$.

Integration over $F$ requires the strict Morse-Transue integral.

6.2 Operator-Valued Extensions

For Hilbert space-valued processes, the spectral representation generalizes to:

X(t) = ∫_ℝe^iλt dZ(λ),

where $Z$ is an operator-valued measure on . This underpins quantum stochastic processes.

---

7. Computational and Statistical Methods

7.1 Spectral Estimation

Given observations ${X(t_1), , X(t_n)}$, the spectral measure $F$ is estimable via:

$$\hat{F}(\Lambda, \mathrm{M}) = \frac{1}{n} \sum_{j,k=1}^n e^{-i(\lambda t_j - \mu t_k)} X(t_j)\overline{X(t_k)} \mathbf{1}_{\Lambda \times \mathrm{M}}(\lambda, \mu).$$

Consistency requires $n$ and $|t_j - t_{j-1}|$ .

7.2 Simulation Techniques

To simulate $X(t)$ with spectral density $f$:

1. Discretize $f(, )$ on a grid ${_j, _k}$.
2. Generate complex Gaussian variables $_{jk} (0, f(_j, _k) )$.
3. Construct:

X(t) = ∑_j, kξ_jke^{i(λ_j−μ_k)t}.

This approximates the harmonizable synthesis.

---

8. Current Research Frontiers

8.1 Multivariate Extensions

Recent work generalizes harmonizability to matrix-valued processes $(t) ^{n n}$, with spectral representation:

X(t) = ∫_ℝe^iλt dZ(λ),

where $()$ is a matrix-valued orthogonal measure.

8.2 Fractal Harmonizable Processes

Processes with Hurst index $H (0,1)$ admit harmonizable representations:

$$X_H(t) = \int_{\mathbb{R}} \frac{e^{i\lambda t} - 1}{|\lambda|^{H+1/2}} \, dZ(\lambda),$$

linking harmonizability to self-similarity.

---

This report has delineated the mathematical backbone of harmonizable processes, emphasizing their spectral anatomy and analytical power. From foundational theorems to cutting-edge generalizations, the framework remains indispensable for modeling non-stationarity in both theory and practice.