SMB Algoet–Cover sandwich #

The Algoet–Cover sandwich discharges the four hypotheses of shannon_mcmillan_breiman_of_sandwich (liminf ≥ H, limsup ≤ H, a.s. boundedness above and below) to produce the hypothesis-free shannon_mcmillan_breiman theorem. The proofs combine the chain rule log_block_eq_sum_pmfLogCond, Birkhoff for the per-step conditional log-likelihood, a k-Markov approximation with conditional entropy H_k = conditionalEntropyTail μ p k, and a likelihood-ratio + Borel–Cantelli bound to convert expected-value inequalities into a.s. bounds.