Lower bounds for sources with infinite Shannon entropy

Determine what lower bound replaces (log p)/H(π) for the expected stopping time of exact fair sampling from a countable-alphabet source whose Shannon entropy H(π) is infinite while every Rényi entropy H_α(π) for α>1 is finite.

Background

The paper proves the lower bound E(T) ≥ log m/H(π) for exact uniform sampling when the source entropy is finite. On countable alphabets, Rényi entropies of orders greater than one remain finite even when Shannon entropy diverges.

When H(π)=∞, the established lower bound becomes vacuous because its right-hand side is zero. Open Problem P6 asks for an informative replacement at this entropy boundary.

References

The open question concerns the boundary. For a source with $H(\pi) = \infty$ but $H_\alpha(\pi) < \infty$ for every $\alpha > 1$, what replaces the lower bound $(\log p)/H(\pi)$ of Theorem~\ref{thm:lower-bound}, which becomes vacuous?

— Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction  (2608.20234 - Gravel, 20 Aug 2026) in Open Problem P6, Section 5; discussed after Theorem 3.2