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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.07771 |
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| _version_ | 1866913433459359744 |
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| author | Kanazawa, Shu Trinh, Khanh Duy Yogeshwaran, D. |
| author_facet | Kanazawa, Shu Trinh, Khanh Duy Yogeshwaran, D. |
| contents | We prove normal approximation bounds for statistics of randomly weighted (simplicial) complexes. In particular, we consider the complete $d$-dimensional complex on $n$ vertices with $d$-simplices equipped with i.i.d. weights. Our normal approximation bounds are quantified in terms of stabilization of difference operators, i.e., the effect on the statistic under addition/deletion of simplices. Our proof is based on Chatterjee's normal approximation bound and is a higher-dimensional analogue of the work of Cao on sparse Erdős-Rényi random graphs but our bounds are more in the spirit of `quantitative two-scale stabilization' bounds by Lachièze-Rey, Peccati, and Yang. As applications, we prove a CLT for nearest face-weights in randomly weighted $d$-complexes and give a normal approximation bound for local statistics of random $d$-complexes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_07771 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Normal approximation for statistics of randomly weighted complexes Kanazawa, Shu Trinh, Khanh Duy Yogeshwaran, D. Probability 60F05, 05E45, 60B99, 55U10 We prove normal approximation bounds for statistics of randomly weighted (simplicial) complexes. In particular, we consider the complete $d$-dimensional complex on $n$ vertices with $d$-simplices equipped with i.i.d. weights. Our normal approximation bounds are quantified in terms of stabilization of difference operators, i.e., the effect on the statistic under addition/deletion of simplices. Our proof is based on Chatterjee's normal approximation bound and is a higher-dimensional analogue of the work of Cao on sparse Erdős-Rényi random graphs but our bounds are more in the spirit of `quantitative two-scale stabilization' bounds by Lachièze-Rey, Peccati, and Yang. As applications, we prove a CLT for nearest face-weights in randomly weighted $d$-complexes and give a normal approximation bound for local statistics of random $d$-complexes. |
| title | Normal approximation for statistics of randomly weighted complexes |
| topic | Probability 60F05, 05E45, 60B99, 55U10 |
| url | https://arxiv.org/abs/2312.07771 |