Let (xt, t ∈ S) be a family of observations of a phenomenon which may be physical, economic, biological, etc. To model the mechanism that generates the xt, we may suppose them to be realizations of random variables (Xt, t ∈ S) that are, in general, correlated. The overall phenomenon is described by (Xt, t ∈ T) where t is generally interpreted as a time: (Xt, t ∈ T) is said to be a stochastic process or a random function.
If T is denumerable, it concerns a discrete-time process, and if T is an interval in , it concerns a continuous-time process. If the set S of observation times is random, we say that we observe a point process (this notion will be elaborated subsequently).
– Discrete-time processes:
1) The daily electricity consumption of Paris.
2) The monthly number of vehicle registrations in France.
3) The annual production of gasoline.
4) The evolution of a population: growth, the extinction of surnames, the propagation of epidemics.
5) The evolution of sunspots over the past two centuries.
6) The series of outcomes for a sportsman.
– Continuous-time processes:
1) The trajectory of a particle immersed in a fluid, where it is subjected to successive collisions with the molecules of the fluid.
2) The reading from an electrocardiogram.