A new analysis of the Bernard-Letac fair-sampling construction reports that it can turn draws from a non-degenerate source into an exactly uniform result, while stopping with probability one. The source may favor some symbols, but the final choice among the target outcomes is stated to be even and independent of that bias. The paper presents five algorithms with formal correctness guarantees and comprehensive complexity analysis, then examines the prime-modulus case more closely.
In the composite tests, the direct Bernard-Letac procedure was compared with two alternatives that combine the von Neumann method with either rejection or Lumbroso's procedure. The metric was the expected number of biased-source draws needed to produce one uniform output.
A sharper picture of the prime case
For a prime number of target outcomes, the paper gives an exact infinite-product expression for the expected number of draws. The product uses a sequence of source power sums, while the renewal estimate uses only the first one. The authors show that the renewal estimate is strictly higher than the true expected cost for every non-degenerate source, so that estimate is never exact in this setting.
The analysis also treats the prime parameter as if it could vary continuously and asks what happens as it approaches one. The expected draw cost does not fall to one. Instead, it approaches a source-dependent constant that is strictly greater than one and reflects the full distribution of source probabilities. For a symmetric binary source, the reported constant begins at 2.8457, while the reciprocal of the source's Shannon entropy, an information measure of uncertainty, begins at 1.4427. The comparison illustrates why the paper says Shannon entropy alone does not determine the boundary constant.
Entropy still provides a useful benchmark. For any exactly fair procedure in the paper's class, the expected number of draws must be at least the target information divided by the source entropy. The lower bound is never attained for target sizes of at least two and for every non-degenerate source covered by the result. Exact fairness and entropy optimality are therefore separate properties in this analysis.
The useful scale may be somewhere in the middle
The paper's efficiency analysis points away from both extremes. Under its continuous-parameter interpretation, efficiency tends to zero at the two ends of the allowed range and reaches a maximum at an interior scale. Numerical calculations found a single interior peak near 5 for a source with probabilities 0.7 and 0.3, and near 10 for a source with probabilities 0.9 and 0.1. Those patterns suggest, but do not prove, that the maximum is unique.
To handle the assignment step, the study describes a finite transfer automaton for prime moduli and finite alphabets. It has exactly the number of states given by the prime modulus plus the alphabet size plus three, and its kernel span has dimension equal to the alphabet size plus two over the field with that prime number of elements. The representation is also reported to be minimal. For a binary target on a binary alphabet, the resulting assignment cost is on the order of the stopping time times its logarithm, compared with the direct dynamic-program approach. The paper presents this as an asymptotic operation-count improvement, not as a wall-clock timing experiment.
Lower means in the reported composite tests
The numerical benchmark favored direct Bernard-Letac in every listed composite-modulus configuration. Its reported mean draw cost was the lowest in each row, with ratios to the Lumbroso comparator of 0.73, 0.66, 0.45, 0.41, 0.40 and 0.41 across the reported cases. The calculations used 2,000,000 samples per configuration. Values for the two von Neumann-based columns were obtained from exact Markov-chain calculations, while the Bernard-Letac figures were empirical means.
Bernard-Letac had the lowest reported mean in every listed configuration, but empirical uncertainty intervals were not reported. The transfer statement is restricted to prime moduli and finite alphabets, and the one-peak efficiency pattern is numerical rather than a proof of uniqueness.
What the evidence leaves open
The work is an arXiv preprint identified as arXiv:2608.20234v2 and dated 25 September 2026. No external data were used; the numerical results came from the author's implementation of the described algorithms. No specific grant from a funding agency, commercial or not-for-profit, was reported.
Paper data and sources
Original title: Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction
Authors: Claude Gravel
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text