Source
Filip (2017), “Notes on the Multiplicative Ergodic Theorem”, Ergodic Theory and Dynamical Systems 39(5), LIT-093, read in full in NOTE-078 from arXiv v1, every proof followed at the level of its argument.
What was actually shown
Oseledets (Thm 2.2.6). Let (Ω, µ, T) be ergodic and probability preserving, and T_ω : V_ω → V_{Tω} a measurable cocycle of invertible linear maps with ∫ log⁺‖T_ω‖ dµ < ∞. Then there are exponents λ₁ > ⋯ > λ_k and an invariant filtration V^{≤λ_k} ⊊ ⋯ ⊊ V^{≤λ₁} = V such that (1/N) log‖T^N v‖ → λ_i for v in V^{≤λ_i} but not in V^{≤λ_{i+1}}. The proof avoids Kingman. It rests on a growth dichotomy (Lemma 2.3.1) and an explicit invariant splitting when one subbundle grows faster than its quotient (Lemma 2.3.3), which a reader with ergodic-theory basics can check (C2).
Karlsson–Ledrappier (Thm 4.3.1). Let X be a proper metric space and ω ↦ g_ω a measurable map into its isometries with ∫ d(g_ω x₀, x₀) dµ < ∞. The drift l = lim (1/N) d(g_ω⋯g_{T^{N−1}ω} x₀, x₀) exists almost everywhere, and there is a measurable family of horofunctions h_ω with −(1/N) h_ω(g_ω⋯g_{T^{N−1}ω} x₀) → l, lying on the boundary when l > 0. No curvature is assumed. NOTE-078 checked the step from the integral bound to the almost-everywhere limit (C5).
The geometric reading (§3). The metrics ‖v‖_N = ‖T^N v‖ form a sequence in the symmetric space GLₙ/Oₙ that tracks a geodesic: the rate vector is the Lyapunov spectrum and the endpoint is the Oseledets flag (Thm 3.5.1, under two-sided log-integrability).
A by-product (Cor. 4.5.1). Applied to ℝ with a concave metric, the horofunction theorem gives a Marcinkiewicz–Zygmund-type law S_N/N^{1/p} → 0 for f ∈ Lᵖ, 0 < p < 1.
What could have come out otherwise. Growth rates could have varied continuously across directions, or drift could have needed curvature to be seen at the boundary. Neither is the case.
What this does not say
- That the geometric form is proved here in general. Kaimanovich’s regularity criterion (Thm 3.3.9) is proved for the hyperbolic plane only, and Prop. 3.4.8(iii) is mis-stated: it needs a logarithm, and a one-dimensional cocycle with CLT-size fluctuations is a counterexample as printed (NOTE-078, corrections).
- That the notes’ tracking proof is sound. The §5.3 rendition of Karlsson–Margulis misapplies Lemma 5.1.5. The theorem is not in doubt; cite Karlsson and Margulis (1999) for it.
- That “without Kingman” covers every proof here. Only §2 avoids it; §§3.5 and 4 invoke it.
- That it applies to trained deep networks. The hypotheses are ergodic, stationary sampling of the matrices and N → ∞. They fit i.i.d. random products at initialisation, where Furstenberg–Kesten already suffices, not the layer-dependent finite-depth Jacobian products of trained networks. The notes make no ML claim.
Connections
- THEORY-325 is about growth of non-normal products under time dependence at finite times. The exponents here are asymptotic rates and say nothing about transient growth.