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The law of large numbers for non-additive capacities without continuity
Abstract
We consider a capacity on a general measurable space associated with a sequence of identically distributed random variables. With a weak assumption concerned with the independence and very few restrictions on the capacity, we show that, on a set with full lower-capacity, any cluster point of empirical averages lies in the mean interval with the lower and upper Choquet integrals of the random variables being its bounds, and, under a Fubini-independent product capacity introduced by Ghirardato (1997), cluster points fill the whole mean interval with full upper-capacity. The results give answers to two open questions of Terán (2014).
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BibTeX
@misc{zhang2026large,
title = {{The law of large numbers for non-additive capacities without continuity}},
author = {Li-Xin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27717},
eprint = {2610.27717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27717},
}
RIS
TY - UNPB
TI - The law of large numbers for non-additive capacities without continuity
AU - Li-Xin Zhang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27717
UR - https://arxiv.org/abs/2610.27717
AB - We consider a capacity on a general measurable space associated with a sequence of identically distributed random variables. With a weak assumption concerned with the independence and very few restrictions on the capacity, we show that, on a set with full lower-capacity, any cluster point of empirical averages lies in the mean interval with the lower and upper Choquet integrals of the random variables being its bounds, and, under a Fubini-independent product capacity introduced by Ghirardato (1997), cluster points fill the whole mean interval with full upper-capacity. The results give answers to two open questions of Terán (2014).
ER -