The Theory of Khinchin Families

Abstract

The theory of Khinchin families connects Probability Theory and Complex Analysis. Along this PhD thesis, we exploit this connection to obtain, using a variety of local central limit theorems, asymptotic formulas for the coefficients of analytic functions with non-negative coefficients. We give criteria for a power series to have certain specific, and in most cases, computable, asymptotic formula. The theory of Khinchin families was initiated and is strongly influenced by the work of Walter K. Hayman, Paul C. Rosembloom, and Luis B\'aez-Duarte. We can assign to each power series $f$ in the class $\mathcal{K}$, having radius of convergence $R>0$, a family of random variables $(X_t)_{[0,R)}$. This theory examines the behavior of this family of random variables, and also of its normalized version, as $t \uparrow R$. This thesis aims to harmonize the various asymptotic formulas for combinatorial or probabilistic objects within a single framework. With this aim we have extended some of the classes of functions and also studied the coefficients of large powers of analytic functions. One of our main goals was to consolidate and develop some aspects of a comprehensive theory: providing a guidebook for combinatorialists or probabilists, a series of results, or basic criteria, where they can follow a step-by-step process, verify if certain straightforward conditions are met, and then derive an asymptotic formula for the object of study. However, this is not the sole primary goal; we are also profoundly interested in the functional theoretical properties of these classes of functions and their connections to the respective Khinchin families (such as the case of $f \in \mathcal{K}$ being an entire function).