Growing-memory log-wealth average (Cover–Thomas): the time average up to horizon n of the per-epoch log returns of the stagewise conditional log-optimal portfolios bstar i along the shift orbit of T. At epoch i the causal strategy uses the i-past optimal portfolio bstar i, so growingMemoryLogAvg X bstar T n ω = (1/(n+1)) log S*_n where S*_n = ∏ᵢ (bstar i · Xᵢ) is the growing-memory wealth. This is the object whose limsup is bounded above by W_∞ (upper half of the Algoet–Cover sandwich) and whose liminf is bounded below by each finite-memory rate.

Upper half of the growing-memory W_∞ AEP (Cover–Thomas): the growing-memory log-wealth average is eventually below W_∞ = condOptGrowthInfty up to any margin ε, almost surely. The proof decomposes growingMemoryLogAvg n = (1/n) log Mₙ + (1/n) ∑ᵢ log(bstarInf · Xᵢ): the first term is eventually below any positive threshold (wealthRatio_logAvg_eventually_le, from the supermartingale integral bound) and the second converges to ∫ log(bstarInf · X) = W_∞ by Birkhoff's ergodic theorem (birkhoff_ergodic_ae) and the gateway identity condOptGrowthInfty_eq_integral_infPast. The eventual-upper-bound form is the honest content of limsup ≤ W_∞; combined with the Birkhoff lower half it yields the almost-sure convergence (the ℝ-limsup value is junk on paths where the growing memory underperforms to −∞, which only the lower half rules out). bstar/bstarInf and their conditional-dominance properties (hbstar_dom/hInf_dom) are received as the stagewise/infinite-past conditional log-optimal selections (constructed separately, e.g. via exists_condLogOptimalSeq / exists_infPast_condLogOptimal); the remaining hypotheses are market-regularity/ergodicity preconditions. hcoh is the shift/past coherence (measurability only) letting the wealth-ratio supermartingale bound pull the growing history out of the epoch-k+1 conditional expectation; it holds for the concrete past-filtration/shift instantiation.

Lower half of the growing-memory W_∞ AEP (Cover–Thomas): the growing-memory log-wealth average is eventually above W_∞ = condOptGrowthInfty down to any margin ε, almost surely. For a fixed finite memory K, the growing-memory returns dominate the fixed-K strategy up to the lower wealth ratio: growingMemoryLogAvg n = (1/n) ∑ᵢ log(bstar K · Xᵢ) + (head)/n − (1/n) log Nₙ^{(K)}, where the first term converges to ∫ log(bstar K · X) = condOptGrowth K = W*_K by Birkhoff's ergodic theorem, the head (a finite ω-constant) vanishes, and (1/n) log Nₙ^{(K)} is eventually below any positive threshold (logAvg_eventually_le_of_lintegral_le_one, from the lower-ratio supermartingale bound lowerRatioProcess_lintegral_le_one). Since W*_K ↑ W_∞ (condOptGrowth_monotone + condOptGrowth_bddAbove, the monotone convergence for this bstar), choosing K with W*_K > W_∞ − ε/2 yields growingMemoryLogAvg n ≥ W_∞ − ε eventually. W_∞ is not received as a hypothesis: it is pinned constructively as the supremum of the W*_K, and the K for each ε is chosen deterministically (ω-independent) from the monotone limit, so the almost-sure set is the countable intersection over K ∈ ℕ of the Birkhoff and lower-ratio a.e. sets. hcoh is the shift/past coherence (measurability only). Combined with the upper half (growingMemory_eventually_le_condOptGrowthInfty) this gives almost-sure convergence to W_∞.

Growing-memory W_∞ AEP (Cover–Thomas): the growing-memory log-wealth average converges almost surely to the infinite-past optimal growth rate W_∞ = condOptGrowthInfty. This is the Algoet–Cover sandwich: the eventual upper bound ≤ W_∞ + ε (growingMemory_eventually_le_condOptGrowthInfty, from the wealth-ratio supermartingale against the infinite-past optimal bstarInf) and the eventual lower bound ≥ W_∞ − ε (growingMemory_eventually_ge_condOptGrowthInfty, from the fixed-memory Birkhoff rates rising to W_∞) pinch the average to W_∞. bstar/bstarInf are the stagewise/infinite-past conditional log-optimal selections (constructed separately via exists_condLogOptimalSeq / exists_infPast_condLogOptimal); the remaining hypotheses are market-regularity/ergodicity preconditions and the two measurability-only shift/past coherences (hcoh_inf at ⨆ⱼℱⱼ for the upper half, hcoh at each ℱ (k+1) for the lower).