Ergodic Theorems for PSPACE functions and their converses

TL;DR

This paper explores effective pointwise ergodic theorems within resource-bounded computational settings, establishing convergence results for PSPACE functions and transformations, and characterizing randomness through ergodic behavior.

Contribution

It introduces resource-bounded ergodic theorems for PSPACE functions and transformations, and characterizes randomness via convergence and non-convergence of ergodic averages.

Findings

Ergodic averages converge for PSPACE randoms and equal space averages on EXP randoms.

Non-randomness in PSPACE can be characterized by non-convergence of ergodic averages.

SUBEXP-space randomness precisely characterizes when the ergodic theorem holds for a point.

Abstract

We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow. In contrast, we show that for a class of PSPACE L1 functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists for all PSPACE randoms and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.