Algorithmic randomness and splitting of supermartingales

TL;DR

This paper investigates the limitations of splitting supermartingales into separate betting strategies for even and odd bits, revealing fundamental constraints in algorithmic randomness.

Contribution

It demonstrates that a single lower semicomputable supermartingale cannot be effectively replaced by a pair betting solely on even or odd bits, highlighting a key limitation in the structure of supermartingales.

Findings

A supermartingale cannot be decomposed into two supermartingales betting on even and odd bits separately.

The result impacts the understanding of Martin-Löf randomness and supermartingale-based definitions.

Shows inherent limitations in splitting supermartingales for randomness tests.

Abstract

Randomness in the sense of Martin-L\"of 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.