Today, the theory of random processes represents a large field of
mathematics with many different branches, and the task of choosing
topics for a brief introduction to this theory is far from being simple.
This introduction to the theory of random processes uses mathematical
models that are simple, but have some importance for applications. We
consider different processes, whose development in time depends on some
random factors. The fundamental problem can be briefly circumscribed in
the following way: given some relatively simple characteristics of a
process, compute the probability of another event which may be very
complicated; or estimate a random variable which is related to the
behaviour of the process. The models that we consider are chosen in such
a way that it is possible to discuss the different methods of the theory
of random processes by referring to these models. The book starts with a
treatment of homogeneous Markov processes with a countable number of
states. The main topic is the ergodic theorem, the method of
Kolmogorov's differential equations (Secs. 1-4) and the Brownian motion
process, the connecting link being the transition from Kolmogorov's
differential-difference equations for random walk to a limit diffusion
equation (Sec. 5).