Consistency results for the kernel density estimate on continuous time stationary and dependent data

The aim of this paper is to study the consistency of the kernel density estimator pertaining to a continuous time stationary process X=(Xt)t≥0, with an underlying density f. More precisely, in a rather general dependency setting, where we use a martingale difference device and a technique based on a sequence of projections on σ-fields, we establish the almost sure pointwise and uniform consistencies with rates of the estimate fT of f built upon the part (Xt)0≤t≤T of the process X.