Central Limit Theorem in the Functional Approach

Abstract

The central limit theorem is proved within the framework of the functional approach for signal analysis. In this framework, a signal is modeled as a single function of time rather than a stochastic process. Distribution function, expectation, and all the familiar probabilistic parameters are built starting from this single function of time by resorting to the concept of relative measure. Furthermore, the concept of independence among functions of time can be introduced. In the paper it is shown that if a sequence of independent signals fulfills some mild regularity assumptions, then the asymptotic distribution of the appropriately scaled average of such signals has a limiting normal distribution. The approach is shown to be useful when only one realization of a signal is available and no ensemble of realizations is observed or exists. The obtained results also allow one to rigorously justify stochastic models for signals and channels that up to now have been derived starting from a deterministic description of phenomena and for which the inferred stochastic model is built invoking a not proved ergodicity property. An application to the statistical characterization of the output signal of a multipath Doppler channel is presented.