Relative-entropy tests of market efficiency: symbolic benchmarks, finite alphabets, and attainable log-growth

Brouty and Garcin (2023) test a symbolic implication of weak-form efficiency by comparing the next-return sign after each finite history with a conditionally fair sign. This paper reconstructs their entropy argument and identifies the statistic as a conditional Kullback–Leibler divergence whose plug-in estimate is a rescaled likelihood-ratio deviance. The representation extends to finite alphabets and gives a test of whether the conditional law of the next symbol, given a history of length L, equals a specified history-free benchmark. A data-processing theorem shows that deterministic coarsening cannot increase population information. Paired simulations show that information and power need not be ordered alike: binary tests benefit from fewer degrees of freedom for sign-level alternatives, whereas four-state tests detect within-sign dependence erased by coarsening. For overlapping histories, finite-order Markov transition-likelihood theory supplies chi-square inference, local power, and resampling. When benchmark probabilities are offered as normalized state prices, the same relative entropy measures maximal expected log growth relative to cash; the complete-market result accommodates zero physical probabilities. In daily S&P 500 data, evidence around the COVID crash is strongest at the one-day, one-lag specification and weakens for longer memory or disjoint two-day returns. A stylized bounded cash/index allocation yields no statistically significant in-sample log-growth improvement over the best history-free exposure in either crisis window.