Stationarity Vector: Theory and Applications

• Stationarity vector is defined as an invariant element or sequence that characterizes stationary regimes across diverse domains.
• It appears in contexts ranging from higher-order Markov chains and stationary random fields to Killing vector fields and set theory sequences.
• Practical insights include its role in spectral analysis, stability criteria in time series, and diagnostic measures for departures from stationarity.

In the literature represented here, “stationarity vector” is not a single canonical term. It appears in several technically distinct settings, each centered on an invariance property: a probability vector fixed by a higher-order Markov operator, a random vector distributed according to an invariant law, a stationary Killing vector field for wave analysis on Kerr–de Sitter spacetime, a sequence of stationary sets in set theory, and vector-like diagnostics for departures from stationarity in plasma physics and vector quantization. This suggests that the term is best understood contextually: the relevant “vector” is stationary because it is fixed by, compatible with, or used to parameterize the stationary regime of the model under study (Li et al., 2013, Blanchet et al., 2016, Petersen et al., 2023, Ben-Neria, 2017).

1. Terminological scope

The principal usages represented in the cited literature are summarized below.

Domain	Object called or functioning as a stationarity vector	Formal marker
Higher-order Markov chains	Probability vector $x$	$x = P x^{(m)}$ (Li et al., 2013)
Multidimensional RBM	Stationary random vector $\mathbf{Y}(\infty)$	$\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$ (Blanchet et al., 2016)
Kerr–de Sitter spacetime	Stationary Killing vector field $T$	$T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$ (Petersen et al., 2023)
Set theory	Sequence $S=(S_n\mid n<\omega)$ of stationary sets	Tight stationarity via $\sup(M\cap\kappa_n)\in S_n$ (Ben-Neria, 2017)
Vector quantization	Drift-measuring stationarity vector	Deviation from the stationary encoder setting (Lu et al., 21 Feb 2026)
Space plasmas	Vector-like entropic characterization	Entropic deviation ratio and SDI (Livadiotis et al., 17 Feb 2025)

A common feature across these usages is that stationarity is never purely nominal. It is encoded by a fixed-point equation, an invariant distribution, a symmetry generator, or an asymptotic stability criterion. The “vector” may therefore be an element of a simplex, a random state in $\mathbb{R}^d_+$, a vector field, or an indexed sequence, depending on the ambient theory.

2. Stationary probability vectors in higher-order Markov theory

For an $m$-th order Markov chain on $x = P x^{(m)}$0 states with transition probability tensor $x = P x^{(m)}$1, a stationary probability vector is a probability vector $x = P x^{(m)}$2 satisfying

$x = P x^{(m)}$3

or, in vector notation, $x = P x^{(m)}$4 (Li et al., 2013). This is the direct higher-order analogue of the invariant distribution equation for first-order chains, but the tensor structure makes the stationary set far more flexible.

A central result for second-order chains is a complete characterization of those chains for which every probability distribution vector is stationary. Writing the stationary equation as

$x = P x^{(m)}$5

with each $x = P x^{(m)}$6 column-stochastic, the characterization states that this happens if and only if there exist vectors $x = P x^{(m)}$7, with entries in $x = P x^{(m)}$8 and $x = P x^{(m)}$9, such that

$\mathbf{Y}(\infty)$0

for each $\mathbf{Y}(\infty)$1 (Li et al., 2013). In this class, stationarity is maximally non-unique: the entire simplex is stationary.

The same paper shows that the geometry of the stationary set can be prescribed very flexibly. For second-order and higher-order chains, the set of stationary vectors can have arbitrary affine dimension between $\mathbf{Y}(\infty)$2 and $\mathbf{Y}(\infty)$3; it can be the whole simplex, a face such as $\mathbf{Y}(\infty)$4, exactly the set of basis vectors $\mathbf{Y}(\infty)$5, or a disconnected union such as a simplex face together with isolated points (Li et al., 2013). A notable rigidity phenomenon also appears: if a $\mathbf{Y}(\infty)$6-dimensional face contains two interior stationary points, then the entire edge is stationary (Li et al., 2013).

These constructions extend to arbitrary order. If an order $\mathbf{Y}(\infty)$7 transition tensor satisfies $\mathbf{Y}(\infty)$8 for all $\mathbf{Y}(\infty)$9, then an order $\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$0 tensor can be built by block concatenation of $\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$1 copies of $\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$2, and the property $\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$3 is preserved for all $\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$4. Column permutations corresponding to indistinguishable monomials in $\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$5, and convex combinations of such constructions, further enlarge the family (Li et al., 2013).

A complementary two-dimensional classification is available for symmetric transition probability tensors. For an order $\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$6, dimension $\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$7 symmetric transition probability tensor, there are exactly two special tensors that have and only have two stationary probability vectors; every other symmetric transition probability tensor of order $\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$8, dimension $\mathbf{Y}(t)\Rightarrow \mathbf{Y}(\infty)$9, has a unique stationary probability vector. As a byproduct, any symmetric transition probability tensor of order $T$0, dimension $T$1, has a unique positive stationary probability vector, and any symmetric irreducible transition probability tensor of order $T$2, dimension $T$3, has a unique stationary probability vector (Huang et al., 2018). In that setting, the higher-order fixed-point problem collapses to a one-variable equation $T$4 on $T$5, which yields the complete classification.

3. Stationary vector processes and spectral representations

In stochastic process theory, the relevant object is often a stationary vector-valued field or time series rather than a stationary point of a nonlinear map. On a compact connected two-point homogeneous space $T$6 with temporal domain $T$7, an $T$8-valued random field $T$9 is isotropic in space and stationary in time when its mean is constant and its covariance matrix depends only on the geodesic distance $T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$0 and the time lag $T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$1 (Ma et al., 2018). The covariance matrix function then has the form

$T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$2

where each $T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$3 is itself a stationary covariance matrix function on $T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$4, and the basis functions are Jacobi polynomials determined by the geometry of $T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$5 (Ma et al., 2018). The field admits a corresponding series expansion in independent stationary $T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$6-variate processes $T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$7, which plays the role of a Karhunen–Loève-type decomposition (Ma et al., 2018).

For second-order stationary vector time series, stationarity is the condition that the mean is constant and $T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$8 depends only on $T=\partial_t+\frac{a}{r_0^2+a^2}\partial_\phi$9. This is the setting of time-series PCA for vector processes. The aim is to find a contemporaneous linear transformation

$S=(S_n\mid n<\omega)$0

such that $S=(S_n\mid n<\omega)$1 is segmented into lower-dimensional subseries that are uncorrelated with each other both contemporaneously and serially (Chang et al., 2014). The construction is based on the positive definite matrix

$S=(S_n\mid n<\omega)$2

whose eigenanalysis identifies the transformation up to a permutation and grouping step (Chang et al., 2014). In this usage, stationarity is the structural assumption that makes the segmentation meaningful.

A frequency-domain alternative is the vector exponential model for covariance stationary vector-valued time series. Here the spectral density matrix is modeled as

$S=(S_n\mid n<\omega)$3

or, equivalently, through a Wold filter

$S=(S_n\mid n<\omega)$4

with unconstrained cepstral matrices $S=(S_n\mid n<\omega)$5 or $S=(S_n\mid n<\omega)$6 (Holan et al., 2014). Because the matrix exponential is always well-defined and invertible for finite cepstral coefficients, stationarity and invertibility are automatic for the model class. This avoids the polynomial root constraints that arise in finite-order VAR or VARMA parameterizations (Holan et al., 2014).

4. Stationarity-enforcing parameterizations and stability criteria in multivariate dynamics

In multivariate time-series econometrics, the “vector” is typically the observation vector $S=(S_n\mid n<\omega)$7, while stationarity constrains the coefficient matrices. For a VAR($S=(S_n\mid n<\omega)$8),

$S=(S_n\mid n<\omega)$9

stationarity holds if and only if all roots of

$\sup(M\cap\kappa_n)\in S_n$0

lie outside the unit circle (Heaps, 2020, Binks et al., 2023). The difficulty is that the stationary region $\sup(M\cap\kappa_n)\in S_n$1 is highly complex and non-convex (Binks et al., 2023).

A major solution is reparameterization by partial autocorrelation matrices $\sup(M\cap\kappa_n)\in S_n$2, each belonging to the set of matrices whose singular values are less than $\sup(M\cap\kappa_n)\in S_n$3, followed by the unconstraining transform

$\sup(M\cap\kappa_n)\in S_n$4

This yields a bijection between stationary VAR coefficients and unconstrained matrices $\sup(M\cap\kappa_n)\in S_n$5, which supports standard priors and Hamiltonian Monte Carlo via Stan while enforcing stationarity by construction (Heaps, 2020, Binks et al., 2023). When the order is unknown, a multiplicative gamma process prior is used to shrink higher-lag partial autocorrelations, and a truncation criterion

$\sup(M\cap\kappa_n)\in S_n$6

decides when a partial autocorrelation matrix is effectively zero, thereby determining the effective order (Binks et al., 2023).

For nonlinear multiregime systems, stationarity is governed by spectral growth across regime sequences rather than by a single companion matrix. In vector STAR models, the key condition is that the joint spectral radius of the regime-dependent companion matrices is below $\sup(M\cap\kappa_n)\in S_n$7. If

$\sup(M\cap\kappa_n)\in S_n$8

then the stacked lag process is a geometrically ergodic Markov chain, and the observed process is strictly stationary, second-order stationary, and $\sup(M\cap\kappa_n)\in S_n$9-mixing with geometrically decaying mixing numbers (Kheifets et al., 2018). The literature emphasized there also shows that checking each regime separately is necessary but not sufficient.

Censored and kinked structural vector autoregressions sharpen this picture. Their stationarity, ergodicity, and weak dependence follow when the deterministic subsystem is globally asymptotically stable, meaning

$\mathbb{R}^d_+$0

for all initial conditions (Duffy et al., 2023). Sufficient criteria are expressed through the joint spectral radius, the constrained joint spectral radius, and the relaxed joint spectral radius, with the constrained and relaxed versions using admissible regime paths and piecewise Lyapunov functions to reduce conservatism (Duffy et al., 2023).

A further multivariate example is DCC-GARCH. The model can be rewritten as a nonlinear Markov chain

$\mathbb{R}^d_+$1

and Tweedie’s criterion is then used to prove existence of a strictly stationary solution (Fermanian et al., 2014). A sufficient condition is

$\mathbb{R}^d_+$2

with simplified explicit conditions

$\mathbb{R}^d_+$3

for $\mathbb{R}^d_+$4 (Fermanian et al., 2014).

The phrase “stationarity vector” also appears in the analysis of an unrestricted AR($\mathbb{R}^d_+$5) process with geometrically decaying coefficients. There the main condition for convergence to stationarity is

$\mathbb{R}^d_+$6

and the spectral analysis is carried by a “stationarity matrix” $\mathbb{R}^d_+$7, whose eigenvalues bound the admissible region (Kulkarni, 2021). In that usage, the vector notion is tied to the eigenstructure controlling stationarity rather than to an invariant probability vector.

5. Equilibrium random vectors, convergence rates, and identification from stationarity

For multidimensional reflected Brownian motion, the stationary object is an equilibrium random vector $\mathbb{R}^d_+$8. The process solves the Skorokhod problem

$\mathbb{R}^d_+$9

with $m$0, and has a stationary distribution if and only if

$m$1

componentwise (Blanchet et al., 2016). Under uniform contraction of the routing matrix, uniform stability of the drift, and bounded variances, the process converges exponentially fast to stationarity, with relaxation time of order

$m$2

as $m$3 (Blanchet et al., 2016). Here the stationarity vector is a genuine invariant random vector in $m$4, and the emphasis is quantitative mixing.

A different equilibrium usage appears in the identifiability theory of VAR(1) models from stationary distributions. If

$m$5

has spectral radius of $m$6 less than $m$7, then $m$8 converges in distribution to

$m$9

where $x = P x^{(m)}$00 solves the discrete-time Lyapunov equation

$x = P x^{(m)}$01

and

$x = P x^{(m)}$02

(Liu, 4 Apr 2025). The support of $x = P x^{(m)}$03 is interpreted as a directed influence graph, and the stationary covariance matrix $x = P x^{(m)}$04 encodes its maximal classes: $x = P x^{(m)}$05 if and only if $x = P x^{(m)}$06 and $x = P x^{(m)}$07 are not in the same maximal class (Liu, 4 Apr 2025). Different maximal classes, or different model dimensions, then yield generic identifiability from stationary data alone.

These two lines of work illustrate two complementary roles for a stationarity vector. In one, it is the target of convergence. In the other, it is the equilibrium object from which hidden dynamic structure may be inferred.

6. Geometric, set-theoretic, and quantization meanings

In Kerr–de Sitter analysis, the relevant object is a stationary Killing vector field rather than a probability vector. The admissible family is

$x = P x^{(m)}$08

and every such choice is shown to be compatible with the Fredholm theory used for quasinormal-mode analysis (Petersen et al., 2023). The key geometric condition is the absence of trapped lightlike geodesics orthogonal to $x = P x^{(m)}$09. With horizon generators $x = P x^{(m)}$10 or $x = P x^{(m)}$11, one ergoregion disappears, which simplifies the analysis (Petersen et al., 2023). In this context, the “stationarity vector” is literally a vector field generating the time direction used in spectral theory.

Set theory employs an even more specialized meaning. A stationarity vector is a sequence

$x = P x^{(m)}$12

where each $x = P x^{(m)}$13 is stationary for an increasing sequence of regular cardinals $x = P x^{(m)}$14 (Ben-Neria, 2017). Such a sequence is tightly stationary if for every algebra $x = P x^{(m)}$15 there exists a tight substructure $x = P x^{(m)}$16 such that

$x = P x^{(m)}$17

(Ben-Neria, 2017). Short-extenders forcing is then used to obtain generic extensions in which every fixed-cofinality sequence of stationary sets is tightly stationary (Ben-Neria, 2017). Here “vector” means a countable indexed family of stationary sets, not a vector in linear algebra.

Vector quantization provides two further meanings. Dual quantization introduces a random splitting operator $x = P x^{(m)}$18 with the intrinsic stationarity property

$x = P x^{(m)}$19

where $x = P x^{(m)}$20 and the quantized value lies on the vertices of Delaunay simplices of the grid (Pagès et al., 2010). Unlike Voronoi quantization, this stationarity holds for non-optimal grids as well (Pagès et al., 2010). By contrast, recent work on codebook collapse in modern VQ systems defines stationarity as the regime in which the encoder output distribution does not drift, and introduces a “stationarity vector” that measures the deviation from this stationary setting through encoder drift (Lu et al., 21 Feb 2026). That shift in meaning reflects the move from fixed-distribution quantization theory to learned representations with nonstationary encoders.

7. Entropic diagnostics and vector-like measures of departure from stationarity

An explicitly diagnostic use appears in space plasma physics. Starting from entropy defect, the cited work develops a measure of stationarity based on the Boltzmann–Gibbs entropy

$x = P x^{(m)}$21

and the thermodynamic kappa parameter $x = P x^{(m)}$22 (Livadiotis et al., 17 Feb 2025). The entropic deviation ratio

$x = P x^{(m)}$23

is bounded in magnitude by $x = P x^{(m)}$24, and the relation between $x = P x^{(m)}$25 and $x = P x^{(m)}$26 follows a power law whose exponent defines the stationarity deviation index,

$x = P x^{(m)}$27

The SDI measures the natural tendency of the system to depart from stationarity (Livadiotis et al., 17 Feb 2025).

The paper itself notes that this is not literally a vector in the linear-algebraic sense. Rather, stationarity is characterized by a tuple-like combination of quantities: the entropic deviation ratio quantifies the instantaneous distance from stationarity, while the SDI quantifies the propensity to depart from it (Livadiotis et al., 17 Feb 2025). A plausible implication is that “stationarity vector” can, in some modern usages, denote a multi-component diagnostic package rather than a single invariant object.

Taken together, these developments show that the phrase has evolved into a cross-disciplinary umbrella for fixed points, invariant laws, symmetry generators, indexed stationary families, and diagnostics of deviation from stationarity. The unifying thread is not the algebraic type of the object but the role it plays in expressing, enforcing, or measuring stationary behavior.