Generalization of van Lambalgen's theorem and blind randomness for conditional probabilities

TL;DR

This paper explores the extension of van Lambalgen's theorem to blind randomness without requiring the computability of conditional probabilities, providing new conditions and examples where the theorem holds.

Contribution

It introduces new sufficient conditions for the generalization of Lambalgen's theorem in the context of blind randomness without assuming computable conditional probabilities.

Findings

Finiteness of martingale for blind randomness established

Classification of blind randomness via likelihood ratio test

Example demonstrating Lambalgen's theorem with non-computable conditional probabilities

Abstract

Generalization of the Lambalgen's theorem is studied with the notion of Hippocratic (blind) randomness without assuming computability of conditional probabilities. In [Bauwence 2014], a counter-example for the generalization of Lambalgen's theorem is shown when the conditional probability is not computable. In this paper, it is shown that (i) finiteness of martingale for blind randomness, (ii) classification of two blind randomness by likelihood ratio test, (iii) sufficient conditions for the generalization of the Lambalgen's theorem, and (iv) an example that satisfies the Lambalgen's theorem but the conditional probabilities are not computable for all random parameters.