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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.07695 |
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| _version_ | 1866916268521553920 |
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| author | Talarczyk, Anna Wiśniewolski, Maciej |
| author_facet | Talarczyk, Anna Wiśniewolski, Maciej |
| contents | Recognizing the regime of positive definiteness for a strictly logarithmic covariance kernel, we prove that the small deviations of a related Gaussian multiplicative chaos (GMC) $M_γ$ are for each natural dimension $d$ always of lognormal type, i.e. the upper and lower limits as $t\to \infty$ of $$
-\ln\Big(\mathbb{P}(M_γ(B(0,r))\le δ\Big)/(\ln δ)^2 $$
are finite and bounded away from zero. We then place the small deviations in the context of Laplace transforms of $M_γ$ and discuss the explicit bounds on the associated constants. We also provide some new representations of the Laplace transform of GMC related to a strictly logarithmic covariance kernel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07695 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On small deviations of Gaussian multiplicative chaos with a strictly logarithmic covariance on Euclidean ball Talarczyk, Anna Wiśniewolski, Maciej Probability 60G15, 60G60 Recognizing the regime of positive definiteness for a strictly logarithmic covariance kernel, we prove that the small deviations of a related Gaussian multiplicative chaos (GMC) $M_γ$ are for each natural dimension $d$ always of lognormal type, i.e. the upper and lower limits as $t\to \infty$ of $$ -\ln\Big(\mathbb{P}(M_γ(B(0,r))\le δ\Big)/(\ln δ)^2 $$ are finite and bounded away from zero. We then place the small deviations in the context of Laplace transforms of $M_γ$ and discuss the explicit bounds on the associated constants. We also provide some new representations of the Laplace transform of GMC related to a strictly logarithmic covariance kernel. |
| title | On small deviations of Gaussian multiplicative chaos with a strictly logarithmic covariance on Euclidean ball |
| topic | Probability 60G15, 60G60 |
| url | https://arxiv.org/abs/2401.07695 |