Granger causality for functional valued random processes

To assess influence or information exchange between functional valued signals, we propose to extend the notion of Granger causality when stochastic processes are series of random variables in functional Hilbert spaces. We give strong definitions for Granger causality and instantaneous coupling using conditional independence. We then discuss a weaker form of the definitions which are valid for second order wide sense stationary processes. This weaker approach relies on the Wold decomposition and its invertibility for these processes. The decomposition holds for discrete time signals that takes their values in infinite dimensional Hilbert spaces. We then discuss a possible framework to apply the theory. We propose causality measures that generalizes the measure proposed by Geweke in 1984 to the Hilbert space case. We illustrate and discuss the approach on the simple example of a trivariate autoregressive of order 1 process taking values in L²([0, 1]).