A limiting Gaussian process in entropic perspective

Entropic statistics is a collection of statistical methodologies with a special perspective: the underlying random system is described exclusively based on label-independent parameters, also known as entropies. Statistical models based on entropies are increasingly popular inspired by needs in modern data science. However the general theoretical support for statistical inference based on these models is porous and needs attention. In this article, several basic entropic objects are first introduced on a countable alphabet, including entropic parameters, general entropies, entropic moments, and an entropic moment-generating function. The main objective of this article is to establish the fact that a sample version of the entropic moment-generating function, properly normed, converges to a Gaussian process. The asymptotic Gaussian process may serve as an anchor point in probability for deriving entropy-based statistical inferential tools.