Periodically Correlated Stochastic Process

• Periodically correlated processes are stochastic processes with periodic mean and covariance functions, enabling applications in communications, finance, and climatology.
• Spectral analysis and vectorization techniques transform these nonstationary processes into stationary multivariate frameworks for efficient filtering and prediction.
• Minimax estimation methods provide robust linear filtering by solving saddle-point problems, ensuring accurate performance under model uncertainty.

A periodically correlated stochastic process (often abbreviated as PC process, also called cyclostationary process in some literature) is a stochastic process whose finite-dimensional distributions—or, more specifically, its mean and second-order statistics—are periodic functions of time (or, in multi-parameter settings, periodic under group shifts). These processes generalize stationarity, allowing for periodic nonstationarity, and have deep implications in inference, prediction, and robust filtering when periodic structures manifest in observed data.

1. Definition and Structure of Periodically Correlated Processes

A stochastic process $\zeta(t)$ is periodically correlated with period $T > 0$ if its correlation function $$K(t+u, u) = \mathbb{E}[\zeta(t + u)\,\overline{\zeta(u)}]$$ satisfies

$K(t + u, u) = K(t + u + T, u + T)$

for all $t, u \in \mathbb{R}$ (Dubovets'ka et al., 15 Oct 2025). The mean function $\mu(t) = \mathbb{E}[\zeta(t)]$ also exhibits periodicity: $\mu(t + T) = \mu(t)$.

For stochastic sequences indexed by integers (discrete-time setting), the corresponding definition is

$$\mathbb{E}[\xi(n + T)\,\overline{\xi(m + T)}] = \mathbb{E}[\xi(n)\,\overline{\xi(m)}]$$

for all $n, m \in \mathbb{Z}$ (Golichenko et al., 2021).

The periodic structure extends to more elaborate settings:

• Multi-parameter fields: This includes processes indexed over $G = \mathbb{Z}^n \times \mathbb{R}^m$ (Dehay et al., 2013). The covariance $C_X(t, s)$ then satisfies $C_X(t + u, s + u) = b_X(t, s; ι(u))$ where $ι$ is the canonical projection onto the quotient $G/K$ for a closed subgroup $K$.
• Vector representation: The process can be recast as an infinite-dimensional vector-valued stationary sequence by partitioning time into intervals of length $T$ and defining, e.g., $\zeta_j(u) = \zeta(u + jT)$ for $u \in [0, T)$, $j \in \mathbb{Z}$ (Dubovets'ka et al., 15 Oct 2025).

Key properties:

• Stationary processes are special cases (with $T$ arbitrary).
• Any periodic function is almost periodic; almost periodicity further generalizes exactly periodic structure (see (Lenart et al., 2012) for the APC extension).
• Cyclostationarity and periodic correlation are equivalent concepts in much of the literature (Dubovets'ka et al., 19 Oct 2025).

2. Spectral Representation and Harmonic Analysis

The periodicity in second-order structure leads to rich spectral properties:

• Blocking (vectorization): A univariate period-$T$ PC sequence $\{\xi(j)\}$ can be mapped into a $T$-variate stationary sequence via $\eta(v, k) = \xi(kT + v),\ v=1,\ldots,T$ (Golichenko et al., 2021). The covariance and spectrum of the originally nonstationary process are encoded in the covariance and spectral density matrix of this stationary vector sequence.
• Spectrum decomposition: For $K$-periodically correlated fields (over LCA groups), spectral mass is concentrated over a set $\Lambda_K$ in the dual group, and the spectral covariance $$a_\lambda(t) = \int_{G/K} (\lambda, x)\,C_X(t;x)\,d\mu_{G/K}(x)$$ can be defined for each $\lambda \in \Lambda_K$ (Dehay et al., 2013).
• Harmonic sum/decomposition: Any $K$-PC field can be expressed as a sum (or integral) over stationary "harmonic modes": $$X(t) = \int_{\Lambda_K} \overline{(\lambda, t)}\,X^\lambda(t)\,d\nu(\lambda)$$ where $X^\lambda$ are stationary fields (Dehay et al., 2013).

3. Filtering, Prediction, and Estimation: Spectral Certainty and Uncertainty

PC processes pose unique challenges for optimal linear estimation, e.g., prediction, interpolation, filtering, and minimax estimation. The key steps, unified across several studies (Golichenko et al., 2020, Golichenko et al., 2021, Luz et al., 2023, Luz et al., 2023, Luz et al., 2023, Luz et al., 2 Feb 2024, Dubovets'ka et al., 15 Oct 2025, Dubovets'ka et al., 19 Oct 2025), are:

• Hilbert space projection: The optimal linear estimate of a functional $A\zeta = \int_0^{\infty} a(t)\zeta(t)\,dt$ (or sum in the discrete setting) is the orthogonal projection onto the closed span of observations, utilizing the vector (blocked) stationary representation.
• Spectral characteristic: The estimator is given, in the frequency domain, via:

$$\hat{A} = \int_{-\pi}^\pi h^\top(e^{i\lambda})\,Z^{(\zeta + \theta)}(d\lambda)$$

where $h$ is the spectral characteristic function, $\zeta$ is the signal, $\theta$ is noise (possibly also PC), and $Z^{(\zeta + \theta)}$ is the associated spectral process (Dubovets'ka et al., 19 Oct 2025).

• Formulas for $h$ and mean square error:

$$h^\top(f, g) = [A^\top(e^{i\lambda})f(\lambda) - C^\top(e^{i\lambda})]\,[f(\lambda) + g(\lambda)]^{-1}$$

The mean square error is then

$$\Delta(f, g) = \langle a, R a \rangle + \langle c, B c \rangle$$

with $R$ and $B$ matrices built from (cross-)Fourier coefficients (Dubovets'ka et al., 19 Oct 2025).

• Spectral certainty vs. uncertainty: If the spectral densities $f(\lambda)$, $g(\lambda)$ are known (certainty), the above formulas solve the problem. Under uncertainty (densities in admissible classes), minimax (robust) estimation is used, with solutions for "least favorable" densities prescribed by constrained optimization and saddle-point conditions (Dubovets'ka et al., 15 Oct 2025, Dubovets'ka et al., 19 Oct 2025).

4. Minimax Estimation and Least Favorable Processes

Under model uncertainty, the minimax estimation minimizes the maximal mean square error over all processes in a given admissible class (e.g., bounded energy, prescribed spectral class):

• Minimax error: For the functional $A\zeta$ and estimation set $\Lambda$,

$$\min_{\hat{A} \in \Lambda} \max_{\zeta \in \mathbb{Y}} \Delta(\zeta, \hat{A}) = P \cdot \nu^2$$

where $\nu^2$ is the maximal eigenvalue of the self-adjoint operator $Q$ defined by the structure of the periodic decomposition (specific matrix elements given in (Dubovets'ka et al., 15 Oct 2025)).

• Least favorable process: The process $\zeta_j$ attaining maximal error is a one-sided moving average of orthogonal innovations, with kernel given by the eigenvector associated with $\nu^2$ (Dubovets'ka et al., 15 Oct 2025).

Similarly, for estimation under spectral uncertainty, least favorable spectral densities $(f^0, g^0)$ solve a constrained supremum problem. The minimax-robust spectral characteristic $h^0$ is defined accordingly, giving the estimator with best worst-case error properties (Dubovets'ka et al., 19 Oct 2025).

5. Practical Applications and Significance

Periodically correlated processes are fundamental in fields where periodic structures arise in data:

• Communications and radar: Signals are often cyclostationary due to modulations.
• Econometrics and finance: Seasonal and cyclical effects, business cycles extracted via PC and APC analyses (Lenart et al., 2012).
• Signal processing: Filtering and prediction of signals or noise with periodic/cyclical second-order statistics.
• Environmental and climatological modeling: Seasonal time series and geophysical fields (e.g., when considering fields indexed over $\mathbb{Z}^n \times \mathbb{R}^m$ (Dehay et al., 2013)).

Minimax estimation theory for PC processes ensures robust filter and estimator design even when the precise correlation or spectral structure is not known, with explicit error guarantees tied to the solution of operator eigenvalue problems (Dubovets'ka et al., 15 Oct 2025).

Explicit representations, both in time and frequency, are possible due to lifting to stationary vector frameworks, and the associated operator-theoretic and spectral methods are readily deployed for both classical and robust inference. The transformation techniques and explicit saddle-point criteria underlie the practical calculation of minimax-robust filters for cyclic, seasonal, or engineered periodic signals (Dubovets'ka et al., 19 Oct 2025).

6. Summary

Periodically correlated stochastic processes model nonstationary random phenomena exhibiting periodicity in mean and correlation. Key analytical tools include vectorization through blocking, harmonic analysis, explicit spectral decompositions, and minimax robust estimation. For both observation- and inference-driven applications, these processes enable modeling and filtering in the presence of cycles or engineered periodic structures. The minimax framework, developed in recent literature (Dubovets'ka et al., 15 Oct 2025, Dubovets'ka et al., 19 Oct 2025), provides comprehensive tools for optimal and robust estimation in realistic, uncertainty-prone environments.