Harmonizable Stochastic Processes

M.M. Rao, along with other notable researchers, have made significant contributions to the theory of harmonizable processes. Some of the fundamental theorems and results one might find in a comprehensive textbook on this topic are:

1. Loève's Harmonizability Theorem:

A complex-valued stochastic process {X(t), t ∈ R} is harmonizable if and only if its covariance function C(s,t) can be represented as:

C(s,t) = ∫∫ exp(iλs - iμt) dF(λ,μ)

where F is a complex measure of bounded variation on R² (called the spectral measure).

2. Characterization of Harmonizable Processes:

A process X(t) is harmonizable if and only if it admits a representation:

X(t) = ∫ exp(iλt) dZ(λ)

where Z(λ) is a process with orthogonal increments.

3. Cramér's Representation Theorem for Harmonizable Processes:

For any harmonizable process X(t), there exists a unique (up to equivalence) complex-valued orthogonal random measure Z(λ) such that:

X(t) = ∫ exp(iλt) dZ(λ)

4. Karhunen-Loève Theorem for Harmonizable Processes:

A harmonizable process X(t) has the representation:

X(t) = ∑ₖ √λₖ ξₖ φₖ(t)

where λₖ and φₖ(t) are eigenvalues and eigenfunctions of the integral operator associated with the covariance function, and ξₖ are uncorrelated random variables.

5. Rao's Decomposition Theorem:

Any harmonizable process can be uniquely decomposed into the sum of a purely harmonizable process and a process harmonizable in the wide sense.

6. Spectral Representation of Harmonizable Processes:

The spectral density f(λ,μ) of a harmonizable process, when it exists, is related to the spectral measure F by:

dF(λ,μ) = f(λ,μ) dλdμ

7. Continuity and Differentiability Theorem:

A harmonizable process X(t) is mean-square continuous if and only if its spectral measure F is continuous in each variable separately. It is mean-square differentiable if and only if ∫∫ (λ² + μ²) dF(λ,μ) < ∞.

8. Prediction Theory for Harmonizable Processes:

The best linear predictor of a harmonizable process X(t) given its past {X(s), s ≤ t} can be expressed in terms of the spectral measure F.

9. Sampling Theorem for Harmonizable Processes:

If a harmonizable process X(t) has a spectral measure F supported on a bounded set, then X(t) can be reconstructed from its samples at a sufficiently high rate.

10. Rao's Theorem on Equivalent Harmonizable Processes:

Two harmonizable processes are equivalent if and only if their spectral measures are equivalent.

11. Stationarity Conditions:

A harmonizable process is (wide-sense) stationary if and only if its spectral measure is concentrated on the diagonal λ = μ.

12. Gladyshev's Theorem:

A process X(t) is harmonizable if and only if for any finite set of times {t₁, ..., tₙ}, the characteristic function of (X(t₁), ..., X(tₙ)) has a certain specific form involving the spectral measure.

These theorems form the core of the theory of harmonizable processes, providing a rich framework for analyzing a wide class of non-stationary processes. M.M. Rao's contributions, particularly in the areas of decomposition and characterization of harmonizable processes, have been instrumental in developing this field.