ABSTRACT
The simplest random processes are those in which the value of the variable at a given time is independent of its value at any other time. The joint probabilities of finding values of the variable at different times are then a product over the first-order or single-fold probability densities:
… …= × ×P x t x t x t P x t P x t P x t( ; , ; , ; ) ( ; ) ( ; ) ( ; ).n n n n1 1 2 2 1 1 2 2 (3.1)
Thus such processes are completely described by these single-fold probability densities. It is not difficult to find examples of this type of process when time is discrete; for example, the toss of a coin at stated times yielding a sequence of heads and tails. However, in the case of most physical systems, the value of a variable at one time is usually related to its value at one or more previous times. One intensively studied example is that of the motion of a particle that is undergoing random changes in direction due to collisions: the position of the particle at any given time is evidently dependent on its previous trajectory. We have seen in the last chapter that in the case of random variables this kind of relationship is expressed in the form of correlation between functions of the variable at different times.
