Functional estimation in continuous or discretized time

In many applications, observations take the form of curves, such as electricity consumption, variations in stock market prices over the course of a session, electrocardiograms...

Such phenomena can be modeled by considering families $\left(X_t,\mathrm{\hspace{0.17em}}t\in ℝ\right)$ of time-indexed random variables.

If X _t have the same distribution μ, then we can define the empirical distribution function associated with the observation of (X _t ,0 ≤ t≤T) by posing :