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Mathematical Statistics and Stochastic Processes by Denis Bosq

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Chapter 9

Introduction to Statistics for Stochastic Processes

9.1. Modeling a family of observations

Let (xt, tS) 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, tS) that are, in general, correlated. The overall phenomenon is described by (Xt, tT) where t is generally interpreted as a time: (Xt, tT) 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 images, 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) ...

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