Stochastic Processes. General Properties. Trajectories, Finite‐dimensional Distributions
Yuliya Stepanovna Mishura, Georgiy M. Shevchenko · 2017
A stochastic process is a function of two variables, one of them being a time variable and the other one a sample point (elementary event). This chapter considers some examples of random processes and draws their trajectories. There are two main approaches to characterizing a stochastic process: by the properties of its trajectories and by some number-valued characteristics, for example by finite-dimensional distributions of the values of the process. These approaches are closely related; however, any of them has its own specifics. For the stochastic process with the values in a metric separable space, consistency conditions for the families of sets and are fulfilled simultaneously. If some stochastic process is defined, then statistician knows in particular its finite-dimensional distributions. They can say that a family of finite-dimensional distributions corresponds to a stochastic process.