Multidimensional strong large deviation results
Résumé
We establish strong large deviation results for an arbitrary sequence of random
vectors under some assumptions on the normalized cumulant generating function.
In other words, we give asymptotic approximations for a multivariate tail probability
of 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 results
follows 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, the
first one dealing with a vector of independent sample variances, the second one with
a Gaussian multiple linear regression model and the third one with the multivariate
Nadaraya–Watson estimator. Some numerical results are also presented for the first
two applications.