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Limit Laws of the Iterated Logarithm Under Sub-linear Expectations

  • Li-Xin Zhang
  • Yongsheng Song
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Abstract

Let {Yn; n≥ 1} be a sequence of independent and identically distributed random variables with mean zero in Peng's framework of the sub-linear expectation space (Ω,ℋ,𝔼̂), and Sn=∑i=1nYi. In this paper, we establish a limit law of limn→ ∞maxk≤ nSk/(√(2k loglog n)). Different from the result obtained by Chen (2015) in which the limit is a constant, it is shown that under the upper capacity the limit may be prescribed as a given function of Y1,Y2,…, taking values in the standard deviation interval. As a result, it is also shown that the set of limit points in the compact law of the iterated logarithm can be a symmetric random interval. This paper (Chinese version) has been submitted to Special Issue of Science in China-Mathematics in Celebration of Professor Peng Shige's 80th Birthday. In Theorem 2.2 of the original paper, an additional condition (2.6) is needed.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026limit,
  title = {{Limit Laws of the Iterated Logarithm Under Sub-linear Expectations}},
  author = {Li-Xin Zhang and Yongsheng Song},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.30848},
  eprint = {2608.30848},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30848},
}
RIS
TY  - UNPB
TI  - Limit Laws of the Iterated Logarithm Under Sub-linear Expectations
AU  - Li-Xin Zhang
AU  - Yongsheng Song
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.30848
UR  - https://arxiv.org/abs/2608.30848
AB  - Let {Y_n; n≥ 1} be a sequence of independent and identically distributed random variables with mean zero in Peng's framework of the sub-linear expectation space (Ω,ℋ,𝔼̂), and S_n=∑_i=1^nY_i. In this paper, we establish a limit law of lim_n→ ∞max_k≤ nS_k/(√(2k loglog n)). Different from the result obtained by Chen (2015) in which the limit is a constant, it is shown that under the upper capacity the limit may be prescribed as a given function of Y_1,Y_2,…, taking values in the standard deviation interval. As a result, it is also shown that the set of limit points in the compact law of the iterated logarithm can be a symmetric random interval. This paper (Chinese version) has been submitted to Special Issue of Science in China-Mathematics in Celebration of Professor Peng Shige's 80th Birthday. In Theorem 2.2 of the original paper, an additional condition (2.6) is needed.
ER  -