On limit theorems under the Shilkret integral

Abstract

The Shilkret integral or idempotent expectation is a sublinear functional which is very close to being a sublinear expectation since it satisfies all the required properties but its domain is not a linear space. In this paper, we prove that it admits a law of large numbers which is structurally similar to Peng’s LLN for sublinear expectations although significant differences exist. As regards the central limit theorem, the situation is radically different as the $\sqrt{n}$ normalization can lead to a trivial limit and other normalizations are possible for variables with a finite second moment or even bounded.

Keywords

Law of large numbers / Possibility measure / Shilkret integral / Sublinear expectation

Acknowledgements

Nian Yao was supported in part by the Natural Science Foundation of China(Grant No. 12071361), the Natural Science Foundation of Guangdong Province(Grant No. 2020A1515010822) and Shenzhen Natural Science Fund (the Stable Support Plan Program 20220810152104001).

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