Mathematical Statistics and Stochastic Processes

Chapter 9

Introduction to Statistics for Stochastic Processes

9.1. Modeling a family of observations

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).

EXAMPLE 9.1.–

– 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.

   3) ...