Asymptotic formulae for sequences of approximating positive linear operators play an important role in the investigation of the corresponding saturation classes. As a consequence of the pioneering work of the first author it has been shown that asymptotic formulae are useful also in studying the representation of some semigroups of operators in terms of iterates of positive linear operators and, hence, in the study of qualitative properties of the solutions of the corresponding evolution equations. In this paper some asymptotic formulae are used in order to give a purely functional-analytic proof of the Central Limit Theorem, a fundamental result in Probability Theory. The authors establish some asymptotic formulae for positive operators acting on a function space defined on a real (not necessarily bounded) interval, with respect to (extended) weighted norms. Some applications are given by considering particular sequences of positive linear operators. Then the functional-analytic proof of the Central Limit Theorem is furnished. Extensions to double sequences of positive linear operators are given and, as an application, the more general Central Limit Theorem due to Lindeberg and Feller is proved.