Modular aggregation as a debiasing method for non-stationary discrete sources: Convergence and numerical validation

We analyze modular aggregation —summing N independent outcomes modulo m —as a post-processing method for extracting nearly uniform randomness from biased discrete sources. Using discrete Fourier analysis over the cyclic group Z m , we prove exponential convergence of the output distribution to uniformity, with a rate determined by the largest non-trivial Fourier modulus. The result applies to independent non-stationary (non-IID) sources under a uniform spectral-gap condition on the non-trivial Fourier modes. Numerical simulations under several bias regimes, including cyclic drift and extreme cyclic bias, are used as finite-sample diagnostics and illustrate the theoretical predictions in comparison with Peres extraction and SHA-256 post-processing. The robustness of modular aggregation comes at a retention cost of order 1 / N , yielding an explicit trade-off between statistical quality and throughput.