  Causal inference seeks to identify cause-and-effect interactions in coupled
systems. A recently proposed method by Liang detects causal relations by
quantifying the direction and magnitude of information flow between time
series. The theoretical formulation of information flow for stochastic
dynamical systems provides a general expression and a data-driven statistic for
the rate of entropy transfer between different system units. To advance
understanding of information flow rate in terms of intuitive concepts and
physically meaningful parameters, we investigate statistical properties of the
data-driven information flow rate between coupled stochastic processes. We
derive relations between the expectation of the information flow rate statistic
and properties of the auto- and cross-correlation functions. Thus, we elucidate
the dependence of the information flow rate on the analytical properties and
characteristic times of the correlation functions. Our analysis provides
insight into the influence of the sampling step, the strength of
cross-correlations, and the temporal delay of correlations on information flow
rate. We support the theoretical results with numerical simulations of
correlated Gaussian processes.
