Periodically Stationary Increments

• Periodically stationary increments are defined by increments that have a T-periodic mean and covariance, extending classical stationarity to nonstationary contexts.
• The process is embedded into an infinite-dimensional stationary vector by block-wise projection, allowing the use of classic spectral analysis tools.
• Robust estimation via the Wiener–Kolmogorov framework and minimax techniques provides optimal prediction even under spectral uncertainty.

A stochastic process or sequence exhibits periodically stationary increments when its increments—over a fixed period—are, in distribution, periodic in mean and covariance, but not necessarily stationary overall. Periodically stationary increments (also called periodically correlated or cyclostationary increments) generalize classical stationarity and arise naturally in processes combining nonstationarity, multi-seasonality, integration, and long memory. Recent developments, especially those of Luz & Moklyachuk, have unified the theory for both discrete and continuous time, provided a robust spectral estimation framework, and addressed optimal inference under spectral uncertainty.

1. Definition and Characterization

Let $\{X(t): t \in \mathbb{R}\}$ be a real-valued, mean-square continuous, second-order stochastic process. Fix $T > 0$ (the period) and an integer $d \geq 1$ (the order of differencing). The $d$-th difference operator of step $T$ is

$$\Delta^d_T X(t) = \sum_{\ell=0}^d (-1)^\ell \binom{d}{\ell} X(t - \ell T).$$

$X(t)$ is said to have periodically stationary (periodically correlated) $d$‑th increments of period $T$ if:

• The increment mean,

$c^{(d)}(t, T) = E[\Delta^d_T X(t)],$

is $T$-periodic in $t$, i.e. $c^{(d)}(t+T,T) = c^{(d)}(t,T)$ for all $t$;

• The increment covariance,

$$D^{(d)}(t, s; T_1, T_2) = \mathrm{Cov}(\Delta^d_{T_1} X(t), \Delta^d_{T_2} X(s)),$$

is jointly $T$-periodic, i.e. $D^{(d)}(t+T,s+T;T_1, T_2) = D^{(d)}(t,s;T_1, T_2)$ for all $t, s$ and integer multiples $T_1, T_2 \in T\mathbb{Z}$.

In the case $d=1$, this reduces to first-difference increments with periodically varying mean and covariance. The theory readily generalizes to higher-order, seasonal or fractional differencing, and to vector-valued processes by stacking one period into a $T$-vector to induce stationarity in the vector sequence (Luz et al., 2023).

2. Embedding and Spectral Representation

The nonstationary process with periodically stationary increments can be mapped (embedded) into an infinite-dimensional, vector-valued stationary increment sequence as follows:

• For each integer $j$, define

$$\xi_j^{(d)}(u) = \Delta^d_T X(u + jT), \qquad u \in [0, T).$$

• Select an orthonormal basis $\{e_k(u)\}_{k\geq 1}$ in $L^2([0, T])$. The projected sequence,

$$\xi_{k, j}^{(d)} = \int_0^T \xi_j^{(d)}(u) \overline{e_k(u)} du,$$

forms the infinite-dimensional vector $$\boldsymbol{\xi}_j^{(d)} = (\xi_{1, j}^{(d)}, \xi_{2, j}^{(d)}, \ldots)^{\top} \in \ell^2$$, indexed by discrete “block” time $j$.

The resulting $\ell^2$-valued stationary increment sequence $\{\boldsymbol{\xi}^{(d)}_j : j \in \mathbb{Z}\}$ admits a spectral representation

$$R^{(d)}(\ell) = \mathrm{Cov}(\boldsymbol{\xi}_j^{(d)}, \boldsymbol{\xi}_{j-\ell}^{(d)}) = \int_{-\pi}^{\pi} e^{i\ell\lambda} (1 - e^{-iT\lambda})^d (1 - e^{iT\lambda})^d F(\lambda)\, d\lambda,$$

where $F(\lambda)$ is an operator-valued, positive semidefinite spectral density (“spectral density matrix”) on $[-\pi, \pi)$ and the factors $(1 - e^{-iT\lambda})^d$ encapsulate the $d$-th differencing structure and period.

This block-wise vectorization allows all classical spectral-theoretic machinery for stationary vector processes to be employed, reducing the periodically stationary increment process to a well-posed problem in $\ell^2$ (Luz et al., 2023).

3. Optimal Linear Estimation: Spectral Characteristic and Mean-Square Error

For estimation—typically, predicting linear functionals constructed from unobserved values (e.g., predicting $$L = \sum_{j=0}^{\infty} \langle b_j, \boldsymbol{\xi}_{-j}^{(d)} \rangle_{\ell^2}$$ from observed past increments)—the Hilbert-space projection paradigm yields explicit spectral solutions:

• The transfer vector in the frequency domain is

$$C(\lambda) = \sum_{j=0}^{\infty} b_j e^{i j \lambda} (1 - e^{iT\lambda})^d.$$

• For any $\ell^2$-valued spectral characteristic $H(\lambda)$, the MSE is

$$\mathrm{MSE}[H] = \int_{-\pi}^{\pi} H(\lambda)^* F(\lambda) H(\lambda) d\lambda.$$

• The unique optimal spectral characteristic is the Wiener–Kolmogorov formula:

$H_{\rm opt}(\lambda) = F(\lambda)^{-1} C(\lambda),$

and the minimum mean-square error is

$$\min_{H}\,\mathrm{MSE}[H] = \int_{-\pi}^{\pi} C(\lambda)^* F(\lambda)^{-1} C(\lambda) d \lambda.$$

These formulae require the invertibility and summability of the underlying spectral density matrix and are valid provided the process is nondegenerate (minimality holds) (Luz et al., 2023).

4. Minimax Robust Estimation and Least-Favorable Spectra

When the spectral density $F(\lambda)$ is uncertain (known only to belong to a convex, weakly compact set $\mathcal D$), minimax theory seeks a spectral density $F_0 \in \mathcal D$ such that

$$\max_{F \in \mathcal D} \min_H \mathrm{MSE}[H; F] = \min_H \max_{F \in \mathcal D} \mathrm{MSE}[H; F].$$

$F_0$ (the least-favorable spectral density) satisfies the operator Euler–Lagrange/KKT equations,

$$- F_0(\lambda)^{-1} C(\lambda) C(\lambda)^* F_0(\lambda)^{-1} = \sum_\alpha \gamma_\alpha M_\alpha(\lambda),$$

where $M_\alpha(\lambda)$ encodes the constraints (e.g., on total variance, power, trace, or operator norm) and $\gamma_\alpha$ are Lagrange multipliers. The robust spectral characteristic is again

$$H_{\rm robust}(\lambda) = F_0(\lambda)^{-1} C(\lambda).$$

A variety of constraint classes (energy, band, norm, etc.) can be handled, and the solution structure is entirely analogous to finite-dimensional convex robust estimation (Luz et al., 2023).

5. Structural and Regularity Conditions

The validity of this periodic increments/spectral embedding theory rests on several regularity assumptions:

• The process $X(t)$ is mean-square continuous.
• Periodic mean and covariance of increments: $c^{(d)}(t+T,T)=c^{(d)}(t,T)$, $D^{(d)}(t+T,s+T)=D^{(d)}(t,s)$.
• Spectral summability: $\sum_{k=1}^\infty \|R^{(d)}(0) e_k\|^2 < \infty$ ensures $\ell^2$-valued processes.
• Minimality: $$\int_{-\pi}^\pi C(\lambda)^* F(\lambda)^{-1} C(\lambda) d\lambda > 0$$.
• Convexity/compactness of the admissible spectral class $\mathcal D$ for minimax theory.
• The Hilbert-space (Wiener-Kolmogorov) projection theorem holds in the subspace generated by past increments (Luz et al., 2023).

6. Context and Applications

Periodically stationary increments formalize a natural class of nonstationary processes, including cyclostationary phenomena with periodic regime-switching or seasonal trends, and generalize both stationary and cyclostationary processes. Their utility is most evident in:

• Optimal and robust estimation (prediction or filtering) for signals with periodic trend or periodic volatility structures, e.g., engineering, geophysics, econometrics, or biological rhythms.
• Modeling and inference for multi-seasonal, integrated (long-memory) or fractionally integrated time series, when trends and cycles are superimposed.
• The robust predictive framework admits explicit spectral formulas for practical filters even under spectral uncertainty or “model risk,” providing critical performance guarantees (minimax optimality).

By reducing the estimation problem to the stationary vector setting via the periodic differencing and block embedding, all classical Hilbert-space spectral projection machinery, including Toeplitz operator inversion and canonical factorization, becomes available (Luz et al., 2023).

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These innovations establish periodically stationary increments as a core unifying principle for modern inference in periodic, multi-seasonal, and nonstationary stochastic systems and provide a fully explicit, operator-theoretic framework for robust, optimal estimation under both spectral certainty and uncertainty.