Gamma, Gaussian and Poisson approximations for random sums using size-biased and generalized zero-biased couplings
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866912108541640704 |
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| author | Daly, Fraser |
| author_facet | Daly, Fraser |
| contents | Let $Y=X_1+\cdots+X_N$ be a sum of a random number of exchangeable random variables, where the random variable $N$ is independent of the $X_j$, and the $X_j$ are from the generalized multinomial model introduced by Tallis (1962). This relaxes the classical assumption that the $X_j$ are independent. We use zero-biased coupling and its generalizations to give explicit error bounds in the approximation of $Y$ by a Gaussian random variable in Wasserstein distance when either the random variables $X_j$ are centred or $N$ has a Poisson distribution. We further establish an explicit bound for the approximation of $Y$ by a gamma distribution in stop-loss distance for the special case where $N$ is Poisson. Finally, we briefly comment on analogous Poisson approximation results that make use of size-biased couplings. The special case of independent $X_j$ is given special attention throughout. As well as establishing results which extend beyond the independent setting, our bounds are shown to be competitive with known results in the independent case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_13815 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Gamma, Gaussian and Poisson approximations for random sums using size-biased and generalized zero-biased couplings Daly, Fraser Probability 62E17 (Primary) 60E10, 60E15, 60F05 (Secondary) Let $Y=X_1+\cdots+X_N$ be a sum of a random number of exchangeable random variables, where the random variable $N$ is independent of the $X_j$, and the $X_j$ are from the generalized multinomial model introduced by Tallis (1962). This relaxes the classical assumption that the $X_j$ are independent. We use zero-biased coupling and its generalizations to give explicit error bounds in the approximation of $Y$ by a Gaussian random variable in Wasserstein distance when either the random variables $X_j$ are centred or $N$ has a Poisson distribution. We further establish an explicit bound for the approximation of $Y$ by a gamma distribution in stop-loss distance for the special case where $N$ is Poisson. Finally, we briefly comment on analogous Poisson approximation results that make use of size-biased couplings. The special case of independent $X_j$ is given special attention throughout. As well as establishing results which extend beyond the independent setting, our bounds are shown to be competitive with known results in the independent case. |
| title | Gamma, Gaussian and Poisson approximations for random sums using size-biased and generalized zero-biased couplings |
| topic | Probability 62E17 (Primary) 60E10, 60E15, 60F05 (Secondary) |
| url | https://arxiv.org/abs/2011.13815 |