The universal measure of nonstochastic objects
Abstract
The usual settings for discussing the notion of stochastic objects are very different from Kolmogorov's original structure-free setting, in which there is no predefined external structure on the objects (such as the lengths of binary strings). A natural modification of stochasticity, obtained by replacing finite sets as models with probability measures, is quasi-stochasticity. The structure-free setting is common in studies of quasi-stochasticity, and known results show that the universal machine generates non-quasi-stochastic objects with exponentially small probability. This paper shows that, on the other hand, the universal machine generates nonstochastic objects surprisingly often. It also systematically explores numerous connections between stochasticity and quasi-stochasticity in the structure-free setting. In particular, the definition of quasi-stochasticity can be modified in a natural way modelled on Kolmogorov's definition so that non-quasi-stochastic objects are produced as often as nonstochastic ones; the results for this modification imply all the results of this paper showing that nonstochastic objects are frequent.
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BibTeX
@misc{vovk2026universal,
title = {{The universal measure of nonstochastic objects}},
author = {Vladimir Vovk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.27098},
eprint = {2608.27098},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27098},
}
RIS
TY - UNPB
TI - The universal measure of nonstochastic objects
AU - Vladimir Vovk
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2608.27098
UR - https://arxiv.org/abs/2608.27098
AB - The usual settings for discussing the notion of stochastic objects are very different from Kolmogorov's original structure-free setting, in which there is no predefined external structure on the objects (such as the lengths of binary strings). A natural modification of stochasticity, obtained by replacing finite sets as models with probability measures, is quasi-stochasticity. The structure-free setting is common in studies of quasi-stochasticity, and known results show that the universal machine generates non-quasi-stochastic objects with exponentially small probability. This paper shows that, on the other hand, the universal machine generates nonstochastic objects surprisingly often. It also systematically explores numerous connections between stochasticity and quasi-stochasticity in the structure-free setting. In particular, the definition of quasi-stochasticity can be modified in a natural way modelled on Kolmogorov's definition so that non-quasi-stochastic objects are produced as often as nonstochastic ones; the results for this modification imply all the results of this paper showing that nonstochastic objects are frequent.
ER -