Abstract
Causal thinking is deeply embedded in scientific understanding of the problems of applied statistics. This can not always be done by experiments and the researcher is restricted to observing the system he wants to describe. This is the case in many fields, for example, in economics, demography, neuroscience, et cetera.
In this paper we give different concepts of causality between σ- algebas and between Hilbert spaces, using conditional independence and conditional orthogonality, respectively, that can be applied on both stochastic processes and events. These definitions are based on Granger’s definition of causality which has great applications in economics (see Florens, Mouchart, 1982; Florens, Fougère, 1996; McCrorie, Chambers, 2006) and also in some other disciplines; for example, see a recent application in neuroscience (see Valdes-Sosa, Roebroeck, Daunizeau, Friston, 2011). The study of Granger’s causality has been mainly preoccupied with discrete time processes (i.e. time series). We shall instead concentrate on continuous-time processes. Many of systems to which it is natural to apply tests of causality, take place in continuous time. For example, this is generally the case within economy, demography, finance. The given definitions of causality extend the ones already given in the case of discrete-time processes.
This paper represents a comprehensive survey of causality concepts between flows of information represented by filtrations and by Hilbert spaces. Also, there are given some new results in Section 4 and Section 5.