Title:

Some problems in the mathematical theory of probability

The group of problems with whioh we are concerned may be reduced to the consideration of a system whose state changes with time. It is assumed only that a knowledge of the past history of the system and of its present state detenninss a probability distribution for its state at any instant in the future. This generalizes the situation in classical mechanics, where the future development of the system is completely determined by its present state. The process in time is a stochastic process. The most usual vase is where the conditional probability distribution for future states, given the present state, is unaffected by any additional knowledge of the past history of the system. The process is then a Markov process. We now make some further simplifying assumptions. 1) We regard "time" as a discrete sequence of instants. 2) We assume that the system possesses only a finite or enumeraably infinite number of possible states. 3) We assume that the probability distribution for states at time m+n, given the state at time m, is independent of m. The process is now called a (temporally homogeneous) Markov chain. Its properties depend on those of the stoohastic matrix, [p_{ij}], whose elements are the onestep transition probabilities from the i^{th} to the j^{th} state of the system.
