A Physical Approach to some Limiting Stochastic Operations

Analytical operations like convergence for sequences of random variables, differentiation and integration for random functions are usually denned in close analogy with ordinary sequences and functions. In the opinion of the authors, for a proper understanding of these operations from a physical point of view, it is worthwhile examining systematically the connection between the concepts of correlation and convergence. In addition, the correspondence between a discrete sequence of random variables and a random function is discussed.

The paper deals with these extensions and also with the relationship between stationarity and ergodicity on the one hand and correlation on the other, using the concept of ' realisation '. The connection with previous papers of one of the authors (R) on stochastic integration is pointed out.

I n effect, the paper amounts to a critical discussion and extension of the results contained in the chapter on ' Limiting stochastic operations ' in Bartlett's book.