Information dependency: Strong consistency of Darbellay–Vajda partition estimators

•A strong consistency for the information dependency's estimation of the Darbellay–Vajda partition estimator is proven in the bivariate case.•The basic technique of this work is the Lugosi and Nobel inequality, a consequence of the Vapnik–Chervonenkis inequality.•The key technique of this work is exploiting a property of supremum in the definition of information dependency.•This result is an extension of the Silva and Narayanan's works (results about Gessaman's partition estimator and tree-quantization estimator).

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 Narayanan, 2010a and Silva and Narayanan, 2010b work which gives some simple conditions to confirm that the Gessaman's partition estimator and the tree-quantization partition estimator, other estimators in the class of data-dependent partition estimators, are strongly consistent.