Mod-phi convergence for random variables with cyclotomic generating functions

Accepted for publication in Probability and Mathematical Statistics on August 19, 2026.

We consider a sequence of discrete random variables whose generating functions are polynomials only having roots on the unit circle in the complex plane. For such sequences we prove, under mild assumptions, mod-$\phi$ convergence with explicit rate and limiting function. This allows us to recover, in a unified way, classical central limit theorems as well as finer asymptotic expansions. The general theory is applied to random permutations and random partitions.

Citation: C. Thäle and P. Tuchel (2026). "Mod-phi convergence for random variables with cyclotomic generating functions." Probability and Mathematical Statistics, accepted for publication. arXiv:2608.14143.
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