Shannon Entropy Estimation in $\infty$-Alphabets from Convergence Results

The problem of Shannon entropy estimation in countable infinite alphabets is revisited from the adoption of convergence results of the entropy functional. Sufficient conditions for the convergence of the entropy are used, including scenarios with both finitely and infinitely supported distributions. From this angle, four plug-in histogram-based estimators are studied showing strong consistency and rate of convergences results for the case of finite and unknown supported distributions and families of distributions with summable tail bounded conditions.