Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability

TL;DR

This paper explores how the convergence rate of the strong law of large numbers relates to the divergence rate of Bayesian prior densities in game-theoretic probability, linking prior choice to classical probabilistic laws.

Contribution

It establishes a connection between the convergence rate of the SLLN and the divergence rate of the Bayesian prior density, providing conditions for the law of the iterated logarithm.

Findings

Prior densities can ensure the validity of the law of the iterated logarithm.

The convergence rate of the SLLN is related to the divergence rate of the prior.

Conditions for the prior density to guarantee classical probabilistic laws.

Abstract

We study the behavior of the capital process of a continuous Bayesian mixture of fixed proportion betting strategies in the one-sided unbounded forecasting game in game-theoretic probability. We establish the relation between the rate of convergence of the strong law of large numbers in the self-normalized form and the rate of divergence to infinity of the prior density around the origin. In particular we present prior densities ensuring the validity of Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm.