# A Development of Continuous-Time Transfer Entropy

## TL;DR

This paper develops a rigorous mathematical framework for defining and analyzing transfer entropy in continuous time stochastic processes, extending the discrete-time concept and illustrating it with Poisson processes.

## Contribution

It introduces a novel continuous-time transfer entropy definition using advanced measure theory and provides conditions linking it to discrete-time TE, with applications to Poisson processes.

## Key findings

- Continuous-time TE is defined via Radon-Nikodym derivatives.
- Necessary and sufficient conditions relate discrete and continuous TE.
- Stationarity implies a constant transfer entropy rate.

## Abstract

Transfer entropy (TE) was introduced by Schreiber in 2000 as a measurement of the predictive capacity of one stochastic process with respect to another. Originally stated for discrete time processes, we expand the theory in line with recent work of Spinney, Prokopenko, and Lizier to define TE for stochastic processes indexed over a compact interval taking values in a Polish state space. We provide a definition for continuous time TE using the Radon-Nikodym Theorem, random measures, and projective limits of probability spaces. As our main result, we provide necessary and sufficient conditions to obtain this definition as a limit of discrete time TE, as well as illustrate its application via an example involving Poisson point processes. As a derivative of continuous time TE, we also define the transfer entropy rate between two processes and show that (under mild assumptions) their stationarity implies a constant rate. We also investigate TE between homogeneous Markov jump processes and discuss some open problems and possible future directions.