Furstenberg-Type Theorem: Non-Stationary Matrix Products

• The theorem extends classical Furstenberg results to non-stationary settings, analyzing asymptotic exponential growth in products of independent, non-identically distributed matrices.
• It employs finite moment, no common invariant measure, and no finite-union-of-subspaces conditions to ensure effective mixing and exponential norm growth.
• Sharp quantitative bounds and large deviation principles are derived, with applications ranging from Anderson localization to robust random dynamical systems.

A Furstenberg-type theorem for non-stationary matrix products concerns the almost-sure asymptotic behavior of random products of independent, not necessarily identically distributed, matrices. The theory extends the classical Furstenberg theorem—originally formulated for i.i.d. random matrices—to highly non-stationary sequences, encompassing arbitrary (possibly Markov-dependent) randomness, additive noise, and parameter families. The modern theory develops a framework for exponential norm growth, almost-sure approximation by deterministic sequences, large deviation principles, and spectral gap estimates, with sharp quantitative control under explicit irreducibility and moment conditions.

1. Genericity and Hypotheses for Non-stationary Products

Let $K$ be a compact set of probability measures on $\SL(d, \mathbb{R})$. Consider a sequence of independent random matrices $A_i \sim \mu_i$, where each $\mu_i \in K$ may vary with $i$. The main Furstenberg-type results assert exponential behavior under three uniform “genericity” or non-degeneracy conditions (Gorodetski et al., 2022):

1. Finite-moment condition: There exist $\gamma>0$, $C < \infty$ such that

$$\forall \mu \in K, \quad \int_{SL(d, \mathbb{R})} \|A\|^\gamma\, d\mu(A) \leq C.$$

This ensures bounded exponential moments for all $\mu \in K$.

1. No common invariant measure: For every $\mu \in K$, no probability measures $\nu_1, \nu_2$ on $\mathbb{R}P^{d-1}$ satisfy $(f_A)_*\nu_1 = \nu_2$ for $\mu$-almost every $A$.
2. No finite-union-of-subspaces invariance: For every $\mu \in K$, there is no pair of finite unions of proper subspaces $U, U' \subset \mathbb{R}^d$ with $A(U) = U'$ for $\mu$-almost every $A$.

These hypotheses provide non-stationary analogues of Furstenberg’s strong irreducibility and proximality conditions, guaranteeing effective mixing on projective space and precluding algebraic obstructions to exponential growth.

2. Main Theorem and Quantitative Statements

Given the above data, the non-stationary Furstenberg theorem quantifies the almost-sure exponential growth of the product norm, and provides sharp large deviation control (Gorodetski et al., 2022):

• Matrix products: $T_n = A_n \cdots A_1$.
• Expectation sequence: $L_n := \mathbb{E}[ \log \|T_n\| ]$.

Theorem:

There exists $\lambda > 0$ such that for all $n$ and all sequences $\mu_i \in K$,

$L_n \geq n \lambda.$

Moreover, almost surely under the random matrix law, one has

$$\lim_{n \to \infty} \frac{1}{n} \Big(\log \|T_n\| - L_n\Big) = 0,$$

that is,

$\log \|T_n\| = L_n + o(n).$

Additionally, for every $\varepsilon > 0$ there exist $\delta(\varepsilon)>0$, $C(\varepsilon)<\infty$ such that for all large $n$,

$$\mathbb{P}\left\{\, |\log \|T_n\| - L_n| > \varepsilon n\, \right\} \leq C e^{-\delta n},$$

with the same bound valid for any fixed $v \in \mathbb{R}^d$.

This result applies to highly non-stationary settings and also admits substantially quantitative forms, including explicit large deviation rates that depend on $\gamma$ and block size (Gorodetski et al., 2022). The theory extends to Markov-dependent sequences, with appropriate spectral gap and non-degeneracy requirements (Goldsheid, 2020).

3. Probabilistic and Structural Tools

The proofs exploit probabilistic block decompositions, entropy additivity, and fine control of projective measures:

• Key sequences:

For a block-length $k \gg 1$, set $B_j = A_{jk}\cdots A_{(j-1)k+1}$ (so that $T_{mk} = B_m \cdots B_1$). Define $\xi_j = \log \|B_j\|$, $R_j = \log \|B_j\| - \log |B_j v_{j-1}|$, and recursively normalize $v_j$ as $v_j = B_j v_{j-1}/|B_j v_{j-1}|$. Then

$$\log \|T_{mk}\| = \sum_{j=1}^m \xi_j - \sum_{j=1}^m R_j.$$

The $\xi_j$ are i.i.d. (in blocks), and possess uniform exponential moments.

• Atom-dissolving and subspace-avoidance:

The atom-dissolving theorem asserts that under the genericity conditions, the largest atomic weight in any convolution on $\mathbb{R}P^{d-1}$ decays exponentially, ensuring that projective randomness spreads effectively (Gorodetski et al., 2022). This controls the $R_j$ correction terms and eliminates the effect of initial alignment (subspace trapping) in the norm.

• Entropy additivity and Furstenberg entropy:

The lower bound $L_n \geq n\lambda$ is proved using a non-stationary additivity property of Furstenberg (Kullback–Leibler) entropy, which quantifies the exponential contraction in projective space under non-degenerate convolutions.

• Large deviation principles:

Blockwise i.i.d. structure, subadditivity, and exponential moment control allow application of Chernoff bounds and classical large deviations to the sums $\sum_j \xi_j$, with all error terms dominated for suitable block size.

A schematic of the block decomposition and error control employed in the proof is as follows:

Term	Role	Estimate/Method
$\sum \xi_j$	Leading exponential sum	i.i.d., Chernoff/LDP
$\sum R_j$	Misalignment penalty	Atom-dissolving
Blocks	Sufficient mixing in each block	Quantitative bounds

4. Extensions: Markov Dependence, Additive Noise, and Parameter Families

The non-stationary framework generalizes to several advanced contexts:

• Non-stationary Markov chains:

In (Goldsheid, 2020), for matrix products $S_n = G_n G_{n-1} \cdots G_1$ where each $G_n$ depends Markovly on a time-inhomogeneous chain, exponential norm growth is proved under a spectral gap for the chain and uniform non-degeneracy for the associated Furstenberg group. The proof uses operator-theoretic contractions on $X \times S^{m-1}$, quantifying failure of projective invariance.

• Additive small noise:

(Bednarski et al., 5 Jul 2025) establishes quantitative Furstenberg-type singular value separation for non-stationary products $B^n = \prod_{i=1}^n (A_i + \epsilon E_i)$, where $A_i$ are bounded, deterministic matrices and $E_i$ are i.i.d. noise with absolutely continuous law. The gap between log-singular values grows linearly in $n$ at rate $\Theta(\epsilon^2 n)$, with explicit entropy-based lower bounds.

• Parameter-dependent products and Anderson localization:

(Gorodetski et al., 24 Mar 2024, Zieber, 20 Dec 2025) show continuity and positivity of parameter-dependent Lyapunov exponents associated to Schrödinger cocycles with independent, non-identically distributed random potentials. Large deviation estimates and Hölder continuity in the spectral parameter establish spectral and dynamical localization.

• Nonconventional (long-range) products:

(Kifer et al., 2018) handles products $X_n = F(\xi_{q_1(n)}, \dots, \xi_{q_\ell(n)})$ for (possibly) Markov, long-range dependent $\xi_i$ and arbitrary time-shifts $q_j(n)$. Subadditive ergodic theorem and large deviation methods ensure existence of Lyapunov exponents as long as the Markov chain of $\Xi_n$ is time-homogeneous (linear shifts), under strong irreducibility and proximality.

5. Topological and Ergodic Generalizations

A topological version of the Furstenberg–Kesten theorem for matrix cocycles $(A(\omega))$ over ergodic, non-stationary dynamical bases is established in (Fan et al., 2022). Given any continuous cocycle $A: \Omega \to M_d(\mathbb{R}_{\geq 0})$ over a subshift $(\Omega, \sigma)$ and an ergodic measure $\nu$, if there are uniform positivity blocks and lower bounds on matrix entries, then for $\nu$-generic points $\omega$,

$$\lambda(\omega) = \lim_{n\to\infty} \frac{1}{n} \log \|A(\omega) \cdots A(\sigma^{n-1} \omega)\|$$

exists and equals the space average except in the exceptional case of finite-time vanishing. This result highlights the role of orbit decomposition, local Kingman subadditivity, and quasi-multiplicativity from cone contraction rather than invertibility.

Applications include weighted ergodic averages, unique ergodicity scenarios, and analytic dependence in parameter families, unifying probabilistic and topological aspects of random matrix product theory.

6. Applications and Impact

Furstenberg-type theorems for non-stationary matrix products underlie rigorous analysis in several areas:

• Spectral and dynamical localization: Control of Lyapunov exponents for general random Schrödinger operators under minimal stationarity assumptions yields Anderson localization, dynamical localization, and sharp exponential localization of eigenfunctions (Gorodetski et al., 24 Mar 2024, Zieber, 20 Dec 2025).
• Robustness to dependence and inhomogeneity: Exponential growth persists under non-stationarity, inhomogeneous Markov dependence, or long-range dependencies as long as explicit irreducibility, moment, and mixing properties are maintained (Goldsheid, 2020, Kifer et al., 2018).
• Separation of Lyapunov exponents: Quantitative, entropy-based results give order-$\epsilon^2 n$ separation between the log-singular values of noisy matrix products, directly generalizing classical multiplicative ergodic theorems to non-stationary, non-elliptic regimes (Bednarski et al., 5 Jul 2025).
• Topological dynamical systems: Existence of Lyapunov exponents, large deviation principles, and variational formulas extend to continuous matrix cocycles over non-stationary subshifts (Fan et al., 2022).

The non-stationary theory thus provides a comprehensive quantitative framework that supports a variety of localization and stability phenomena, with broad applicability in random dynamical systems, random operators, and ergodic theory.

7. Table: Summary of Principal Non-stationary Furstenberg-type Theorems

Setting	Key Hypotheses	Main Result/Behavior
Arbitrary i.i.d. sequence	Moment, non-degeneracy/irreducibility	$\log\|T_n\| = L_n + o(n)$, $\|T_n\| \gtrsim e^{\lambda n}$ (Gorodetski et al., 2022)
Markov-dependent	Uniform spectral gap, Furstenberg group non-deg.	Exponential norm growth, robust to non-stationarity (Goldsheid, 2020)
Additive noise	Compactly supported, smooth noise law	Singular value gaps $\sim \epsilon^2 n$ (Bednarski et al., 5 Jul 2025)
Parameter cocycle	Uniform moment, Furstenberg-type irreducibility	Continuity/positivity of Lyapunov exponent (localization) (Gorodetski et al., 24 Mar 2024, Zieber, 20 Dec 2025)
Topological subshift	Cone-contraction, positivity, ergodicity	Existence of $\lim_n n^{-1}\log\|A^{(n)}(\omega)\|$ (Fan et al., 2022)
Nonconventional (long-range)	Homogeneity, strong irred./proximality	$n^{-1}\log\|\Pi_n\| \to \lambda$ a.s. (Kifer et al., 2018)

In all these regimes, the non-stationary theory recovers and extends the classical exponential growth law, with explicit quantification and robust error bounds. The key ingredients—irreducibility, finite moment, and entropy or atom-dissolving techniques—enable transfer of much of the stationary random matrix product theory to highly non-homogeneous settings.