Gaussian fluctuations of generalized $U$-statistics and subgraph counting in the binomial random-connection model
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914148808392704 |
|---|---|
| author | Liu, Qingwei Privault, Nicolas |
| author_facet | Liu, Qingwei Privault, Nicolas |
| contents | We derive normal approximation bounds for generalized $U$-statistics of the form \begin{equation*}
S_{n,k}(f):=\sum_{
1 \leq β(1),\dots,β(k) \leq n \atop
β(i)\neβ(j), \ 1\leq i\ne j \leq k}
f\big(X_{β(1)},\dots,X_{β(k)},Y_{β(1),β(2)},\dots,Y_{β(k-1),β(k)}\big),
\end{equation*}
where $\{X_i\}_{i=1}^n$ and $\{Y_{i,j}\}_{1\le i<j\le n}$ are independent sequences of i.i.d. random variables. Our approach relies on moment identities and cumulant bounds that are derived using partition diagram arguments. Normal approximation bounds in the Kolmogorov distance and moderate deviation results are then obtained by the cumulant method. Those results are applied to subgraph counting in the binomial random-connection model, which is a generalization of the Erdős-Rényi model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12338 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gaussian fluctuations of generalized $U$-statistics and subgraph counting in the binomial random-connection model Liu, Qingwei Privault, Nicolas Probability 60F05, 60G50, 05C80 We derive normal approximation bounds for generalized $U$-statistics of the form \begin{equation*} S_{n,k}(f):=\sum_{ 1 \leq β(1),\dots,β(k) \leq n \atop β(i)\neβ(j), \ 1\leq i\ne j \leq k} f\big(X_{β(1)},\dots,X_{β(k)},Y_{β(1),β(2)},\dots,Y_{β(k-1),β(k)}\big), \end{equation*} where $\{X_i\}_{i=1}^n$ and $\{Y_{i,j}\}_{1\le i<j\le n}$ are independent sequences of i.i.d. random variables. Our approach relies on moment identities and cumulant bounds that are derived using partition diagram arguments. Normal approximation bounds in the Kolmogorov distance and moderate deviation results are then obtained by the cumulant method. Those results are applied to subgraph counting in the binomial random-connection model, which is a generalization of the Erdős-Rényi model. |
| title | Gaussian fluctuations of generalized $U$-statistics and subgraph counting in the binomial random-connection model |
| topic | Probability 60F05, 60G50, 05C80 |
| url | https://arxiv.org/abs/2505.12338 |