Keywords

• Khinchin family
• gaussianity

Abstract Generating functions lie at the crossroads of combinatorics, complex analysis, and probability. Given a power series $f(z) = \sum_{n \ge 0} a_n z^n$ with non-negative coefficients and positive radius of convergence $R > 0$, we can turn the coefficients into a Khinchin family by declaring, for each $t \in [0, R)$, the mass function [ \mathbb{P},!\bigl(X_t = n\bigr) = \frac{a_n, t^n}{f(t)}, \qquad \text{for any } n \geq 1. ] This simple prescription simultaneously encodes three objects: the combinatorial numbers $a_n$, the analytic function $f$, and a one‑parameter family of probability laws $(X_t)_{t \in [0, R)}$. This prompts natural questions: How do analytic properties of $f$ influence the probabilistic behaviour of $X_t$? and, conversely, what can the probabilistic properties of $X_t$ tell us about the coefficients?

Beyond providing a probabilistic interpretation of combinatorial sequences, the Khinchin family approach also serves as a powerful tool for asymptotic analysis: under appropriate regularity conditions, probabilistic limit theorems for $(X_t)$ can be translated into precise asymptotic formulas for the coefficients $a_n$. These regularity conditions are captured by the class of strongly Gaussian power series and the class of Hayman functions. Both classes are contained within the broader class of Gaussian power series, so understanding when a power series is Gaussian is highly relevant for the theory of Khinchin families.

This talk will focus primarily on analytic criteria for Gaussianity in Khinchin families. Starting from the general framework that assigns to any power series with non-negative coefficients a family of probability laws $(X_t)_{t \in [0, R)}$, we investigate when the normalized variables converge in distribution to a standard Gaussian. In particular, we present explicit and verifiable analytical conditions, expressed in terms of derivatives of the fulcrum function associated with $f$, that ensure this convergence. These results allow us to deduce Gaussian behavior using only the behavior of the analytic function along the positive real axis, yielding a flexible toolkit applicable to a wide range of settings from classical partition functions to canonical products and exponentials of entire functions of finite order.