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Large deviation local limit theorems for arbitrary sequences of random variables. (English) Zbl 0559.60030

Consider a sequence \(\{T_ n\}_{n=1,2,..}\) of real-valued random variables with an analytic moment-generating function \(\phi_ n(z)\), nonvanishing for \(| Real (z)| <a\), \(a>0\). Set \(\psi_ n(z)=n^{-1}\log \phi_ n(z)\) and \(\gamma_ n(u)=\sup_{-a<s<a}\{us-\psi_ n(s)\}\) for real u. Under certain conditions on \(\psi_ n\), the main results provide precise large deviation local limit theorems, in case of \(T_ n/n\) possessing a density function (say) \(k_ n\) as well as for lattice-valued \(T_ n\). In the density case the asymptotic expansion is \[ (1)\quad k_ n(m_ n)=\{n/(2\pi\psi''_ n(\tau_ n))\}^{1/2}\exp \{-n \gamma_ n(m_ n)\}\{1+O(n^{-1})\}, \] where \(\{m_ n\}\) is a given real sequence and \(\tau_ n\) is defined by \(\psi'_ n(\tau_ n)=m_ n\). A result similar to (1) also applies in the lattice case. Applications to the Wilcoxon signed-rank test and Kendall’s distribution-free test for independence are discussed in detail.
Reviewer: J.Steinebach

MSC:

60F10 Large deviations
62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
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