Complex analytic proofs of two probabilistic theorems

TL;DR

This paper employs complex analytic methods to prove probabilistic theorems, including a generalized Phragmén-Lindelöf principle and a series expansion for Green's function, linking classical analysis with probabilistic results.

Contribution

It introduces purely complex analytic proofs for probabilistic theorems previously proven only probabilistically, expanding the analytical toolkit for such results.

Findings

Proved a generalized Phragmén-Lindelöf principle using complex analysis.

Derived a series expansion for Green's function of a disk.

Connected classical infinite product expansions to probabilistic theorems.

Abstract

In this paper, we use purely complex analytic techniques to prove two results of the first author which were hitherto given only probabilistic proofs. A general form of the Phragm\'en-Lindel\"of principle states that if the $p$\textsuperscript{th} Hardy norm of the conformal map from the disk to a simply connected domain is finite, then an analytic function on that domain is either bounded by its supremum on the boundary or else goes to $\ff$ along some sequence more rapidly than $e^{|z|^{p}}$. We will prove this and discuss a number of special cases. We also derive a series expansion for the Green's function of a disk, and show how it leads to an infinite product identity. The celebrated infinite product expansions for sine and cosine are realized as special cases.