Abstract
We give a short analytic proof of local large deviations for i.i.d. random variables in the domain of a multivariate α-stable law, α∈(0,1)∪(1,2]. Our method simultaneously covers lattice and nonlattice distributions (and mixtures thereof), bypassing aperiodicity considerations. The proof applies also to the dynamical setting.
Highlights
Local large deviation results for i.i.d. random variables in the domain of a stable law have been recently obtained by Caravenna and Doney [7, Theorem 1.1] and refined by Berger [5, Theorem 2.3]
We refer to such results as stable local large deviations
We provide a new proof of the stable LLD in Theorem 1.1
Summary
Local large deviation results for i.i.d. random variables in the domain of a stable law have been recently obtained by Caravenna and Doney [7, Theorem 1.1] and refined by Berger [5, Theorem 2.3]. A major advantage of this approach is that it generalises naturally to the dynamical setting This is the third main aim of this paper where in Theorem 3.2 we establish the. Our result covers distributions that are jointly lattice and nonlattice avoiding the consideration of numerous different cases that arise in the corresponding local limit theorems (see the discussion in [8]). We provide an analytic proof of Theorem 1.6 establishing local large deviations for i.i.d. random variables in the domain of a multivariate stable law.
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