Information dependency: Strong consistency of Darbellay–Vajda partition estimators

The Darbellay–Vajda partition scheme is a well known method to estimate the information dependency. This estimator belongs to a class of data-dependent partition estimators. We would like to prove that with some simple conditions, the Darbellay–Vajda partition estimator is a strong consistency for the information dependency estimation of a bivariate random vector. This result is an extension of Silva and Narayananʹs (2010a,b) work which gives some simple conditions to confirm that the Gessamanʹs partition estimator and the tree-quantization partition estimator, other estimators in the class of data-dependent partition estimators, are strongly consistent.