Homogeneity criterion for agreement between population TDMI estimators

Let $\bar{I}(\tau)$ and $\hat{I}(\tau)$ denote the two population time-delayed mutual information estimators, and define their difference by

$$\delta I(\tau)=\left|\bar{I}(\tau)-\hat{I}(\tau)\right|.$$

A population is statistically homogeneous temporally when the probability density functions representing its individuals are identical, as are the probability density functions under temporal evolution. Homogeneity criterion. In the circumstance where $\bar{I}(\tau)$ can be accurately estimated, $\delta I(\tau)\sim 0$ if and only if the population used to estimate $\bar{I}(\tau)$ and $\hat{I}(\tau)$ is statistically homogeneous temporally. The reverse direction asserts that if the population represents a single, stationary, homogeneous distribution, then $\delta I(\tau)\sim 0$; in this case all $\epsilon$'s are zero, so $\bar{I}(\tau)$ and $\hat{I}(\tau)$ represent a homogeneous source and are equivalent up to bias. The forward direction, that nonzero $\delta I(\tau)$ implies a heterogeneous population, is described as more complicated to prove and is only briefly discussed in the paper.