On asymptotic properties of high moments of compound Poisson distribution
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866909379499917312 |
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| author | Khorunzhiy, O. |
| author_facet | Khorunzhiy, O. |
| contents | We study asymptotic behavior of the moments $M_k(λ)$ of the sum $X_1+\dots+X_{N_λ}$, where $N_λ$ follows the Poisson probability distribution with mean value $λ$ and $\{X_j\}$ is a family of i.i.d. random variables also independent from $N_λ$. We obtain an explicit expression for the leading term of $M_k(λ)$ as $k\to\infty$ and study it in dependence of the asymptotic behavior of $λ= λ_k$.
In application, we establish a concentration property of maximal vertex degree of large weighted random graphs. Another application is related with a variable that arises in the studies of high moments of large random matrices. Finally, regarding three particular cases of probability distribution of $X_j$, we comment on the asymptotic behavior of certain combinatorial polynomials, including the Bell polynomials of even partitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_00379 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On asymptotic properties of high moments of compound Poisson distribution Khorunzhiy, O. Probability Combinatorics 05C80, 11C99, 60E05, 60F99, 60G50 We study asymptotic behavior of the moments $M_k(λ)$ of the sum $X_1+\dots+X_{N_λ}$, where $N_λ$ follows the Poisson probability distribution with mean value $λ$ and $\{X_j\}$ is a family of i.i.d. random variables also independent from $N_λ$. We obtain an explicit expression for the leading term of $M_k(λ)$ as $k\to\infty$ and study it in dependence of the asymptotic behavior of $λ= λ_k$. In application, we establish a concentration property of maximal vertex degree of large weighted random graphs. Another application is related with a variable that arises in the studies of high moments of large random matrices. Finally, regarding three particular cases of probability distribution of $X_j$, we comment on the asymptotic behavior of certain combinatorial polynomials, including the Bell polynomials of even partitions. |
| title | On asymptotic properties of high moments of compound Poisson distribution |
| topic | Probability Combinatorics 05C80, 11C99, 60E05, 60F99, 60G50 |
| url | https://arxiv.org/abs/2007.00379 |