Abstract
This chapter focuses on stochastic processes by considering a down-to-earth class of such processes, those whose random variables have finite second moments. When the term “L2 theory” is used in connection with stochastic processes, it refers to the properties of an L2 process that can be deduced from its covariance function. One advantage of concentrating on the L2 theory of stochastic processes is that questions concerning measurability and continuity of the sample functions can be avoided. Brownian motion has been used as a model for the movement of a microscopic particle undergoing molecular bombardment in a liquid. The Karhunen-Loève expansion assumes a special form when the process is Gaussian. The Karhunen-Loève expansion is a series of independent random variables. The Kaiman filter is a more realistic model for estimation problems; stationarity is not assumed, and only a finite number of observations are necessary to construct any particular estimate.
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