Hypothesis testing with explosive time series. An approach to the theory of the functional central limit theorem

Dealing with uncertainty about the true data generating process requires a differentiated perspective of the distributions in hypothesis testing. In particular, the realizations, or the observed data, generated by interactions that are naturally ordered in time, posits a need for a differentiated analysis with respect to the standard statistics available for hypothesis testing. The Functional Central Limit Theorem provides a framework that enables the researcher to build a statistic that fits his data and hypothesis at hand. In this paper I show some of the necessary conditions under which the popular t-statistic properly condenses the information of the underlying distribution as well as the additional tools available when the t distribution is not suitable for hypothesis testing.