Limit theorems for a strongly irreducible product of independent random matrices under optimal moment assumptions

The paper’s pivotal-extraction framework rigorously delivers the required block-structure, high-probability alignment, exponential tails, and conditional moment domination via an explicit pivot/acceptance mechanism and Markov-bundle extraction (see Theorem 1.1 and its construction via Theorem 4.9 and Lemmas 4.13–4.22 ). By contrast, the model’s Step 3 crucially asserts the existence of fixed left/right compact sets L_j^⋄, R_i^⋄ so that for every core c in a compact C of strongly contracting blocks, the top singular directions of every product ℓ c r land in prescribed small projective neighborhoods indexed by (j,i). This uniform steering of top singular directions across all c by a finite family of right/left factors independent of c is not justified and, in general, contradicts the stability behavior of singular directions under multiplication by a strongly contracting middle block. The paper avoids this pitfall by using a data-dependent acceptance/pivot scheme that guarantees Schottky alignment and conditional control without requiring such uniform directional nets, and it also provides the two-sided/middle alignment mechanisms (Lemma 5.4 and the pivotal concatenation used in Theorem 5.13) with exponential tail controls .