A subexponential version of Cramer's theorem
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913948879552512 |
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| author | Ferré, Grégoire |
| author_facet | Ferré, Grégoire |
| contents | We consider the large deviations associated with the empirical mean of independent and identically distributed random variables under a subexponential moment condition. We show that non-trivial deviations are observable at a subexponential scale in the number of variables, and we provide the associated rate function, which is non-convex and is not derived from a Legendre-Fenchel transform. The proof adapts the one of Cramer's theorem to the case where the fluctuation is generated by a single variable. In particular, we develop a new tilting strategy for the lower bound, which leads us to introduce a condition on the second derivative of the moment generating function. Our results are illustrated by a couple of simple examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_05791 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A subexponential version of Cramer's theorem Ferré, Grégoire Probability Statistical Mechanics We consider the large deviations associated with the empirical mean of independent and identically distributed random variables under a subexponential moment condition. We show that non-trivial deviations are observable at a subexponential scale in the number of variables, and we provide the associated rate function, which is non-convex and is not derived from a Legendre-Fenchel transform. The proof adapts the one of Cramer's theorem to the case where the fluctuation is generated by a single variable. In particular, we develop a new tilting strategy for the lower bound, which leads us to introduce a condition on the second derivative of the moment generating function. Our results are illustrated by a couple of simple examples. |
| title | A subexponential version of Cramer's theorem |
| topic | Probability Statistical Mechanics |
| url | https://arxiv.org/abs/2206.05791 |