Let fn be a non-parametric kernel density estimator based on a kernel function K. and a sequence of independent and identically distributed random variables taking values in R. The goal of this article is to prove moderate deviations and large deviations for the statistic sup x∈R |fn(x) - fn(-x)|.
We first give a functional moderate deviation principle for random processes with stationary and independent increments under the Ledoux's condition.Then we apply the result to laws of the iterated logarithm.
