Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914643657621504 |
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| author | Rednoß, Benedikt Thäle, Christoph |
| author_facet | Rednoß, Benedikt Thäle, Christoph |
| contents | This paper deals with sequences of random variables $X_n$ only taking values in $\{0,\ldots,n\}$. The probability generating functions of such random variables are polynomials of degree $n$. Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for $X_n$ is established in a unified way. In the real rooted case the result is classical and only involves the variances of $X_n$, while in the cyclotomic case the fourth cumulants or moments of $X_n$ appear in addition. The proofs are elementary and based on the Stein-Tikhomirov method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09418 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions Rednoß, Benedikt Thäle, Christoph Probability Combinatorics 05A15, 60F05 This paper deals with sequences of random variables $X_n$ only taking values in $\{0,\ldots,n\}$. The probability generating functions of such random variables are polynomials of degree $n$. Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for $X_n$ is established in a unified way. In the real rooted case the result is classical and only involves the variances of $X_n$, while in the cyclotomic case the fourth cumulants or moments of $X_n$ appear in addition. The proofs are elementary and based on the Stein-Tikhomirov method. |
| title | Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions |
| topic | Probability Combinatorics 05A15, 60F05 |
| url | https://arxiv.org/abs/2401.09418 |