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Sharp asymptotics of large deviations in \(\mathbb{R}^ d\). (English) Zbl 0831.60042

Summary: Given a sequence \(\{X_i\}\), \(i= 1, 2, 3,\dots\), of i.i.d. random variables taking values in \(\mathbb{R}^d\), \(d\geq 2\), let \(S_n= \sum^n_{i= 1} X_i\). For \(\Gamma\) a Borel set in \(\mathbb{R}^d\) having smooth boundary, with \(a= \inf_{x\in \Gamma} I(x)\), the minimal value of the large deviation rate function \(I(x)\) over \(\Gamma\), we find, under suitable hypotheses, asymptotic results as \(n\to \infty\), of the form \[ P(S_n\in n\Gamma)= n^\gamma e^{- na}(d_0+ o(1)), \] where the constant \(\gamma\) depends sensitively on the geometry of \(\Gamma\) and the dimension \(d\), and takes values \(- \infty< \gamma\leq (d- 2)/2\). For fixed \(a= \inf_{x\in \Gamma} I(x)\), we construct examples having any specific \(\gamma\) in this range.

MSC:

60F10 Large deviations
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