Abstract
We establish strong large deviation results for an arbitrary sequence of randomvectors under some assumptions on the normalized cumulant generating function.In other words, we give asymptotic approximations for a multivariate tail probabilityof the same kind as the one obtained by Bahadur and Rao (Ann Math Stat 31:1015–1027, 1960) for the sample mean (in the one-dimensional case).The proof of our resultsfollows the same lines as in Chaganty and Sethuraman (J Stat Plan Inference, 55:265–280, 1996). We also present three statistical applications to illustrate our results, thefirst one dealing with a vector of independent sample variances, the second one witha Gaussian multiple linear regression model and the third one with the multivariateNadaraya–Watson estimator. Some numerical results are also presented for the firsttwo applications.