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Large deviations for weighted sums of stretched exponential random variables. (English) Zbl 1314.60076

Summary: We consider the probability that a weighted sum of \(n\) i.i.d.random variables \(X_j, j = 1,\dots,n\), with stretched exponential tails is larger than its expectation and determine the rate of its decay, under suitable conditions on the weights. We show that the decay is subexponential, and identify the rate function in terms of the tails of \(X_j\) and the weights. Our result generalizes the large deviation principle given by R. Kiesel and U. Stadtmüller [J. Math. Anal. Appl. 251, No. 2, 929–939 (2000; Zbl 0967.60025)] as well as the tail asymptotics for sums of i.i.d.random variables provided by A. V. Nagaev [Theory Probab. Appl. 14, 51–64 (1969), translation from Teor. Veroyatn. Primen. 14, 51–63 (1969; Zbl 0196.21002); Ann. Probab. 7, No. 5, 745–789 (1979; Zbl 0418.60033)]. As an application of our result, motivated by random projections of high-dimensional vectors, we consider the case of random, self-normalized weights that are independent of the sequence \((X_j)\), identify the decay rate for both the quenched and annealed large deviations in this case, and show that they coincide. As another example we consider weights derived from kernel functions that arise in nonparametric regression.

MSC:

60F10 Large deviations
62G08 Nonparametric regression and quantile regression
62G32 Statistics of extreme values; tail inference
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