Law of large numbers for the spectral radius of random matrix products

Cagri Sert; Richard Aoun

Law of large numbers for the spectral radius of random matrix products

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Pith's one-line read For i.i.d. products of complex matrices, the spectral radius has a law of large numbers: almost sure growth at the first Lyapunov exponent under finite second moment, convergence in probability under finite first moment, with no…

desk verdict A genuine and clean result: the spectral radius of i.i.d. matrix products satisfies the same law of large numbers as the norm, with no irreducibility assumption and essentially optimal moment conditions. read the letter →

arxiv 1908.07469 v3 pith:M65PDEZL submitted 2019-08-20 math.DS math.PR

Richard Aoun , Cagri Sert This is my paper · ORCID

classification math.DSmath.PR MSC 37H1560F1515A1860B15

keywords lawoflargenumbersspectralradiusrandommatrixproductsLyapunovexponentsdeviationsnoirreducibilityassumptionprojectivespaceMarkovianincrements

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that the spectral radius of an i.i.d. random walk on $\mathrm{GL}_d(\mathbb{C})$ satisfies a law of large numbers: if $\mu$ has finite second moment, then $(1/n)\log\rho(L_n)$ converges almost surely to the first Lyapunov exponent $\lambda_1(\mu)$, and if $\mu$ has only finite first moment, the same quantity converges in $L^1$ and hence in probability. The main advance is that no irreducibility assumption is put on the group generated by the support of $\mu$, in contrast with earlier strong laws that required strong irreducibility and an exponential moment. The proof controls, with failure probabilities summable in $n$, the projective distance between the attracting point and the repelling hyperplane of $L_n$, then converts that distance into a lower bound for $\rho(L_n)/\|L_n\|$ and combines it with the classical norm law. A section of counterexamples shows the i.i.d. assumption cannot be weakened to ergodic stationary Markovian increments: the same normalized spectral radius then has almost sure liminf $0$ and limsup $\lambda_1(\mu)$.

What carries the argument

The machinery is organized around the first Lyapunov gap---the integer $s$ with $\lambda_1(\mu)=\cdots=\lambda_s(\mu)>\lambda_{s+1}(\mu)$---and around the subspace $L_\mu$ of vectors whose growth rate is strictly below $\lambda_1(\mu)$. Projecting the walk to $V/L_\mu$ makes the cocycle have a unique average, and Proposition 2.5 imports a large-deviation estimate: with summable failure probability, uniformly in $v\notin L_\mu$, $\|L_n v\|/\|v\|$ is at least $\delta([v],[L_\mu])e^{n(\lambda_1(\mu)-\epsilon)}$ and at most $e^{n(\lambda_1(\mu)+\epsilon)}$. The geometric heart is the pair $(x^+_{L_n},H^<_{L_n})$ in projective space---the fastest-expanding direction of $L_n$ and the hyperplane it repels---and the proof shows by a chain of large-deviation estimates (Lemmas 3.2--3.6) that this pair stays at distance at least $e^{-\epsilon n}$ with high probability when the top exponent is simple. Lemma 2.2 then turns that distance into the ratio bound $\rho(L_n)/\|L_n\|\ge \delta(x^+_{L_n},H^<_{L_n})/2$, reducing the problem to the known norm law; larger Lyapunov gaps are handled by applying the simple-gap result to the $s$-th exterior power, and the no-gap case by the inequality $a_d(g)\le \rho(g)\le a_1(g)$.

What would settle it

Run the probabilities in Proposition 2.5 numerically for a concrete non-irreducible i.i.d. measure on $\mathrm{GL}_2(\mathbb{C})$ with $L_\mu\neq\{0\}$ and a Lyapunov gap, scanning a fine grid of directions $[v]$; if some sequence $v_n\to [L_\mu]$ gives non-summable probabilities of violating the lower bound, the proposition is false and Theorem 1.1 does not follow. Simulating the Section 4 Markov chain with i.i.d. factors drawn from its stationary marginal, while checking that the spectral-radius law then holds, would further confirm that dependence, not the marginal distribution, is what creates the oscillation.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a pair of limit theorems with no irreducibility hypothesis: for i.i.d. factors on $\mathrm{GL}_d(\mathbb{C})$, $(1/n)\log\rho(L_n)\to\lambda_1(\mu)$ almost surely when the measure has finite second moment, and in $L^1$ when it has finite first moment (Theorem 1.1 and Theorem 1.2). The same arguments give convergence of the full vector of eigenvalue moduli to the Lyapunov vector, work for right products as well as left products, and transfer to any local field, since the proofs use only exterior powers and projective geometry. The paper further claims that the i.i.d. condition is doing real work: for an ergodic stationary Markovian sequence of increments, the normalized spectral radius can oscillate, with liminf equal to $0$ and limsup equal to $\lambda_1(\mu)$, even though the norm still obeys the classical law.

Load-bearing premise

Everything rests on Proposition 2.5, which says that once the distance from the starting direction to the slow subspace $L_\mu$ is factored in, the probability of growing far below the top Lyapunov rate is summable in $n$ uniformly over all starting vectors; the paper presents this as a routine extension of a known irreducible-case estimate and does not verify the uniformity line by line, and the strong law's almost-sure conclusion fails if that uniformity breaks.

Editorial extensions

If this is right

If Theorem 1.1 is right, the first-order growth of every eigenvalue modulus of an i.i.d. random matrix product is deterministic and equal to the corresponding Lyapunov exponent, without any irreducibility hypothesis.
Theorem 1.2 shows that the weak law needs no stronger moment hypothesis than the classical norm law, so the spectral-radius result does not require an extra moment price once convergence in probability is enough.
The exterior-power argument shows the same laws hold for all eigenvalue moduli simultaneously, which is Remark 1.3's vector-valued convergence; this also covers products with repeated top exponents.
Because Section 4's Markovian example has spectral-radius oscillations only for stationary dependent increments, any extension to non-i.i.d. walks needs extra hypotheses; the paper identifies this as the natural next problem.
As the paper notes, the almost-sure version under finite first moment remains open; the method's summable large-deviation estimates require the second moment.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One way to extend the paper's conclusion is to test whether the failure of the law for stationary Markovian increments persists when the stationary measure has full support on an irreducible but not strongly irreducible group; if it disappears, strong irreducibility rather than irreducibility is the relevant hypothesis for dependent increments.
A second extension suggested by the exterior-power reduction is to apply the same theorems to the spectral radii of all exterior powers $\bigwedge^k L_n$, which would give a law of large numbers for the entire Lyapunov spectrum; the paper states only the eigenvalue-moduli version explicitly.
The strong-law question with finite first moment may be approachable by replacing the summable large-deviation sequence with a slower decay and using a softer almost-sure argument; nothing in the paper rules this out.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know about 1908.07469. First, it settles a natural question: for i.i.d. products in GL_d(C), (1/n) log rho(L_n) -> lambda_1 almost surely under a finite second moment, and in L^1 under a finite first moment, with no irreducibility condition on the support. That genuinely goes beyond Guivarc'h 1990 and Benoist–Quint 2016, which both needed strong irreducibility and finite exponential moment. Second, the proof is not a black box; it builds on Benoist–Quint's large deviation estimates and splits into three cases by Lyapunov gap, and the main work is in the gap case.

What the paper does well: the exterior-power reduction in case (ii) is elegant, and the Markov-chain counterexample in Section 4 is a nice touch, showing the i.i.d. assumption cannot be relaxed to stationary ergodic increments (you get liminf = 0 while limsup = lambda_1). The moment conditions match what the norm law would suggest, so the statements feel like the right ones.

Soft spots, in proportion. Proposition 2.5 is the load-bearing piece and its proof is compressed: it extends [BQ16a, Prop 4.1] to the non-irreducible case in a paragraph. I read it carefully and the reduction to the quotient V/L_mu does work; the identities lambda_1(pi_*mu) = lambda_1(mu) and L_{pi_*mu} = {0} follow from Furstenberg–Kifer's results, and the stress-test note confirms the details. So this is not a gap, but in a final version I'd want a few more lines there. Second, in the proof of Theorem 1.1, case (ii), the text says 'Applying Theorem 1.2' where it should say the case already proved (Theorem 1.1, case (i)); that's a typo-level slip but worth fixing since the a.s. conclusion depends on it. Third, Proposition 3.8 is presented as a sketch ('we limit ourselves to indicate the proof'); it's clear how the lemmas adapt from summable error probabilities to probabilities tending to zero, but the details should appear in print.

The central argument holds up. This paper deserves a serious referee—I'd send it to an expert in random matrix products without hesitation. My own verdict would be accept after minor revision; the typos and compressed passages are fixable and don't threaten the main theorems. I'd cite this if I worked in the area.

Referee Report

0 major / 5 minor

Summary. The paper proves strong and weak laws of large numbers for the spectral radius of products of i.i.d. random matrices in GL_d(C). Theorem 1.1 shows that, under a finite second moment assumption, (1/n) log rho(L_n) converges almost surely to the first Lyapunov exponent lambda_1(mu); Theorem 1.2 shows that, under only a finite first moment, the same convergence holds in L^1 and hence in probability. No irreducibility assumption is imposed. The proof splits into three cases: a simple top Lyapunov exponent, a higher Lyapunov gap handled via exterior powers, and the case of equal exponents. The central estimates are transferred large-deviation bounds for the projective action (Proposition 2.5), together with geometric lemmas controlling the distance between the attracting point and the repelling hyperplane (Propositions 3.1 and 3.8). The paper also gives an ergodic stationary Markovian example in which the spectral-radius law of large numbers fails, showing that the i.i.d. assumption is essential.

Significance. If correct, the paper is a significant and clean contribution: it removes the strong-irreducibility assumption that was present in the earlier strong laws of Guivarc'h and Benoist--Quint, and it lowers the moment hypothesis to the optimal finite-first-moment range for the weak law. The proof is transparently structured and makes good use of known large-deviation results rather than introducing heavy new machinery. The counterexample in Section 4 is valuable because it demonstrates that the i.i.d. assumption, not just stationarity and ergodicity, is needed for the spectral-radius law. The paper also correctly credits the norm law of Furstenberg--Kesten and the large-deviation results of Benoist--Quint, and the reduction to the quotient space V/L_mu is a natural and effective device.

minor comments (5)

[Section 4, proof of Theorem 1.1, case (ii)] In the sentence 'Applying Theorem 1.2 to eta', the reference should be to the already-proved case (i) applied to the measure eta, since Theorem 1.2 is the weak law and would not provide the almost sure convergence that the strong-law proof requires. This appears to be a typo, but it should be corrected.
[Section 3, Proposition 3.8] The proof of Proposition 3.8 is only sketched with 'one readily checks' and 'the proof of Proposition 3.1 applies verbatim.' Because this proposition is the key step leading to the weak law in Theorem 1.2, please expand at least the modifications in Lemmas 3.5 and 3.6, in particular the preservation of uniformity over H in H_{n,epsilon} when the failure probabilities D_n only tend to 0 rather than being summable.
[Section 2.3, proof of Proposition 2.5] The notation in (2.4) is ambiguous: the same symbol v denotes the original vector and its class in V/L_mu, and the displayed inequality is easy to misread. Please introduce separate notation for the class and the quotient norm, and state explicitly that the sequence D_n in (2.2) is uniform in v in V without 0; also record the convention that delta([v],[L_mu])=1 when L_mu={0}.
[Section 3, proof of Proposition 3.1] The deduction of the estimates (3.10) and (3.11) from Lemma 3.3 is not immediate, because Lemma 3.3 is stated for the doubling comparison R_{2n} versus R_n while (3.10) compares lengths n and floor(n/2) and (3.11) compares lengths n and n-floor(n/2). Please add a sentence explaining the one-step or dyadic adjustment; the necessary estimate follows by the same arguments but should not be left implicit.
[Section 3, Lemma 3.6] The phrase 'a fortiori for every epsilon>0' is correct but deserves a short explanation: since the event delta(x^+_{L_n},H) <= e^{-epsilon n} shrinks as epsilon grows, the probability bound for small epsilon automatically gives the bound for all larger epsilon.

Circularity Check

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No significant circularity: the spectral-radius law of large numbers is reduced to the external Furstenberg-Kesten norm law and Benoist-Quint large-deviation estimates, not to its own conclusion.

full rationale

The derivation is self-contained against external benchmarks. The target statement, (1/n) log rho(L_n) -> lambda_1(mu), is never used as an input; lambda_1(mu) is defined by the Furstenberg-Kesten norm law [FK60] and the proof reduces the spectral-radius statement to that norm law plus the Benoist-Quint large-deviation theorem (Theorem 2.4 = [BQ16a, Prop. 3.2]) and the Furstenberg-Kifer/Hennion reduction to the quotient V/L_mu. Proposition 2.5 obtains the transferred lower bound (2.2) by pushing the cocycle to V/L_mu; the delta-factor comes from the definitional identity (2.4) and the equality lambda_1(pi_*mu)=lambda_1(mu) is cited to the external [FK83, Lemma 3.6], not to the present theorem. The Section 3 estimates (Lemmas 3.2, 3.3, 3.5, 3.6) are all consequences of (2.2), and the final Borel-Cantelli plus Furstenberg-Kesten argument does not reintroduce the spectral-radius conclusion. The 'Theorem 1.2' reference in case (ii) is a typo for the already-proved case (i) applied to the exterior power, so the gap case is not circular. The only self-citations ([AG19] for the Lyapunov-gap approach and Remark 3.7; [PS19] near Lemma 2.6) are contextual and are not load-bearing: Proposition 3.1 is proved from Lemma 3.6 and the Benoist-Quint estimates without invoking [AG19]. No fitted parameter is renamed as a prediction, and the Section 4 Markov-chain example is an independent counterexample. Hence no circular step.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests on the standard machinery of random matrix products: the Furstenberg-Kesten and Kingman laws for the norm, the Furstenberg-Kifer and Hennion theory of the lower-growth subspace L_mu and formula (2.1), Oseledets' theorem for the filtrations, the Benoist-Quint large deviation estimates (Theorem 2.4), and two geometric transfer lemmas (BQ16b Lemma 14.14 and BM11 Lemma 4.1). The paper's own contributions are the assembly of these into Proposition 2.5 with a proof, Lemma 2.6 with a proof, the Section 3 estimates, the three-case reduction in Section 4, and the counterexample. There are no fitted constants, no ad hoc hypotheses beyond the stated moment conditions, and no invented objects. The largest external debt is [BQ16a, Props 3.2 and 4.1], which supplies the exponential probability tails on which every Section 3 estimate is built.

assumptions (8)

standard math Furstenberg-Kesten theorem and Kingman's subadditive ergodic theorem: (1/n) log a_k(L_n) -> lambda_k(mu) almost surely, defining the Lyapunov exponents.
Used in the introduction and at equation (2.5) to set up the target constant lambda_1(mu) and the Lyapunov vector.
standard math Furstenberg-Kifer [FK83] and Hennion [Hen84]: the subspace L_mu of vectors with lower growth exists, is G_mu-invariant, and lambda_1(mu) equals the supremum over stationary measures of the cocycle average (2.1); also lambda_1(pi_*mu) = lambda_1(mu) and L_{pi_*mu} = {0}.
Invoked in Section 2.3 to justify the quotient reduction in Proposition 2.5; the equality of exponents on the quotient is the step that replaces irreducibility.
standard math Oseledets' multiplicative ergodic theorem: the subspaces F^<_n in Lemma 2.6 converge almost surely to a filtration F^< satisfying: lim (1/n) log ||L_n u|| = lambda_1 iff u is not in F^<.
Core of Lemma 2.6, which generalizes [BL85, Corollary 3.4] to the non-irreducible case.
standard math Benoist-Quint large deviation theorem (stated as Theorem 2.4, [BQ16a, Prop 3.2]): for a cocycle with finite second moment and upper and lower averages sigma^+ and sigma^-, sample cocycle averages stay in [sigma^- - eps, sigma^+ + eps] with failure probability summable in n.
The external engine behind Proposition 2.5 and hence all exponential estimates in Section 3; the paper's central technical debt.
standard math Lemma 2.2 = [BQ16b, Lemma 14.14]: if delta(x^+_g, H^<_g) > 2 sqrt(a_2(g)/a_1(g)) then rho(g)/||g|| > delta(x^+_g, H^<_g)/2.
The bridge from projective separation to the spectral radius lower bound in the Borel-Cantelli step of Theorem 1.1.
standard math Inequality (3.7) = [BM11, Lemma 4.1]: ||g^* u||/(||g^*|| ||u||) <= delta(x^+_g, (Cu)^perp) + a_2(g)/a_1(g) for every g and every unit-direction u.
Powers Lemma 3.6, which controls returns of the attractive point to hyperplanes.
domain assumption Domain hypotheses: mu a probability measure on GL_d(C), increments i.i.d., finite first or second moment as stated in Theorems 1.1 and 1.2.
The moment conditions are quantitative hypotheses; the i.i.d. assumption is essential, as the Section 4 Markov-chain example shows.
domain assumption Convention that the factor delta([v],[L_mu]) in (2.2) is read through (2.4) as ||v||/||v|| = 1 when L_mu = {0}, so the lower bound survives the quotient by L_mu.
Needed so that Proposition 2.5 applies in the non-irreducible case with L_{pi_*mu} = {0}; implicit in the proof and never stated as a formal convention.

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Pith. "Pith review of Law of large numbers for the spectral radius of random matrix products." pith.science (2026). https://pith.science/paper/M65PDEZL

@misc{pith2026190807469,
  author       = {Pith},
  title        = {Pith review of: Law of large numbers for the spectral radius of random matrix products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M65PDEZL}},
  note         = {Machine review of arXiv:1908.07469}
}

abstract

We prove that the spectral radius of an i.i.d.\ random walk on $\GL_d(\C)$ satisfies a strong law of large numbers under finite second moment assumption and a weak law of large numbers under finite first moment. No irreducibility assumption is supposed.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Aoun and Y

R. Aoun and Y. Guivarc'h. Random matrix products when the top lyapunov exponent is simple. J. Eur. Math. Soc. , 22(7): 2135--2182, 2020

work page 2020

[2] [2]

Avila and J

A. Avila and J. Bochi. A formula with some applications to the theory of Lyapunov exponents. Israel J. Math. , 131: 125--137, 2002

work page 2002

[3] [3]

odinger operators . Volume 8 of Progress in Probability and Statistics . Birkh\

P. Bougerol and J. Lacroix. Products of random matrices with applications to S chr\"odinger operators . Volume 8 of Progress in Probability and Statistics . Birkh\"auser Boston Inc., Boston, MA, 1985

work page 1985

[4] [4]

Furman, E

A. Furman, E. Lindenstrauss, J. Bourgain and S. Mozes. Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus. J. Amer. Math. Soc. , 24(1):231--280, 2011

work page 2011

[5] [5]

Benoist and J-F

Y. Benoist and J-F. Quint. Central limit theorem for linear groups. Ann. Probab. , 44(2):1308--1340, 2016

work page 2016

[6] [6]

Benoist and J-F

Y. Benoist and J-F. Quint. Random walks on reductive groups . Volume 62 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] . Springer, Cham, 2016

work page 2016

[7] [7]

C. Cuny, J. Dedecker and F. Merlev\` e de. Large and moderate deviations for the left random walk on _d( R ) . ALEA Lat. Am. J. Probab. Math. Stat. , 14, no. 1, 503--527, 2017

work page 2017

[8] [8]

Furstenberg and H

H. Furstenberg and H. Kesten. Products of random matrices. Ann. Math. Statist. , 31:457--469, 1960

work page 1960