On the L2-rates of histograms and frequency polygons from sampled data

In this paper, we study some asymptotic properties for nonparametric density estimators based upon $n$ discretized observations $X_{t_{1}}$,...,$X_{t_{n}}$ of a weakly stationary continuous-time process. More precisely, we establish that histograms and frequency polygons can attain the same optimal $L_{2}$-rates as in the i.i.d. case. Moreover, if sample paths are sufficiently irregular, a ``parametric'' rate is even possible by using a suitable ``high frequency'' sampling design.