Using Vector Representations of Characteristic Functions and Vector Logarithms When Proving Asymptotic Statements

In this methodological-technical note, in addition to the well-known concepts of logarithms of positive real numbers and operators, we open a path for a mathematical treatment of the mathematical concept of the logarithm of a vector. We prove the most basic arithmetic operations for this new logarithm concept and demonstrate how it is applies to characteristic functions and limit theorems of probability theory. As a side result, we revise a formula for $i^i$ that is known from the literature.