When there is the need to model some form of dependence in a sample of observations, stochastic processes arise quite naturally. Dependence is a generic term which must be specified case by case in statistics and probability, but the most common situations include time and/or spatial dependence. In our context, the most interesting form of dependence is time dependence. Still, time dependency can be modelled in a variety of forms as we will discuss in this chapter.
3.1 Definition and First Properties
We assume to have a probability space . A real-valued, one-dimensional, stochastic process is a family of random variables defined on taking values in . The set may be any abstract set, but we will restrict our attention to particular cases. For each , the random variable is a measurable map . For a given fixed value of , say , the map , seen as a function of ...