On necessary and sufficient conditions for the local large deviation principle
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908990892408832 |
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| author | Borovkov, Konstantin |
| author_facet | Borovkov, Konstantin |
| contents | One says that the local large deviation principle (LLDP) is satisfied for a family of random vectors $\{ζ_T\}_{T\ge 0}$ in $\mathbb R^d,$ $d\ge 1,$ if there exists a function $D:\mathbb R^d\to [0,\infty],$ $D\not \equiv \infty,$ such that, for any $α\in \mathbb R^d$, \[ \lim_{T\to \infty}T^{-1}\ln \mathbf{P} (|ζ_T -α|<\varepsilon_T)= - D(α)\] for $\varepsilon_T\to 0$ slowly enough. In this paper, we establish necessary and sufficient conditions for the LLDP that are very close to each other. Namely, if the LLDP is satisfied then, for $M_T\to\infty$ slowly enough as $T\to\infty$, there exists the limit \[ A(μ):= \lim_{T\to\infty}T^{-1}\ln \mathbf{E} (e^{T\langle μ, ζ_T\rangle}; |ζ_T|\le M_T)\in (-\infty, \infty],\quad μ\in \mathbb R^d,\] which is equal to the Legendre--Fenchel transform $\mathcal L_D$ of the rate function $D$. Conversely, if the above limit $A(\cdot )$ exists and is an essentially smooth function, then the LLDP is satisfied with the rate function $D$ equal to $\mathcal L_A.$ This "relaxed version" of the Gärtner--Ellis theorem's main condition does not involve the restrictive integrability assumptions from the latter and is most adequate to the nature of the local large deviation problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22257 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On necessary and sufficient conditions for the local large deviation principle Borovkov, Konstantin Probability 60F10 (Primary) 26B25 (Secondary) One says that the local large deviation principle (LLDP) is satisfied for a family of random vectors $\{ζ_T\}_{T\ge 0}$ in $\mathbb R^d,$ $d\ge 1,$ if there exists a function $D:\mathbb R^d\to [0,\infty],$ $D\not \equiv \infty,$ such that, for any $α\in \mathbb R^d$, \[ \lim_{T\to \infty}T^{-1}\ln \mathbf{P} (|ζ_T -α|<\varepsilon_T)= - D(α)\] for $\varepsilon_T\to 0$ slowly enough. In this paper, we establish necessary and sufficient conditions for the LLDP that are very close to each other. Namely, if the LLDP is satisfied then, for $M_T\to\infty$ slowly enough as $T\to\infty$, there exists the limit \[ A(μ):= \lim_{T\to\infty}T^{-1}\ln \mathbf{E} (e^{T\langle μ, ζ_T\rangle}; |ζ_T|\le M_T)\in (-\infty, \infty],\quad μ\in \mathbb R^d,\] which is equal to the Legendre--Fenchel transform $\mathcal L_D$ of the rate function $D$. Conversely, if the above limit $A(\cdot )$ exists and is an essentially smooth function, then the LLDP is satisfied with the rate function $D$ equal to $\mathcal L_A.$ This "relaxed version" of the Gärtner--Ellis theorem's main condition does not involve the restrictive integrability assumptions from the latter and is most adequate to the nature of the local large deviation problem. |
| title | On necessary and sufficient conditions for the local large deviation principle |
| topic | Probability 60F10 (Primary) 26B25 (Secondary) |
| url | https://arxiv.org/abs/2604.22257 |