Algorithmic randomness and splitting of supermartingales

Abstract

Randomness in the sense of Martin-Löf can be defined in terms of lower semicomputable supermartingales. We show that such a supermartingale cannot be replaced by a pair of supermartingales that bet only on the even bits (the first one) and on the odd bits (the second one) knowing all preceding bits.

The same text in English (13 pages) and in Russian (14 pages). Journal version