Chapter VIII - A class of stochastic processes

Summary

1. In the preceding Chapters, we have been concerned with distributions of sums of the type Z_n = X₁ + … + X_n, where the X_r are independent random variables. Z_n is then a variable depending on a discontinuous parameter n, and the passage from Z_n to Z_n+1 means that Z_n receives the additive contribution X_n+1, so that we have Z_n+1 = Z_n + X_n+1 where Z_n and X_n+1 are independent.

Consider now the formation of Z_n by successive addition of the mutually independent contributions X₁, X₂, …, and let us assume that each addition of a new contribution takes a finite time δ. (In a concrete interpretation the X_r might e.g. be the gains of a certain player during a series of games, every game requiring the time δ, so that Z_n is the total gain realized after n games, or after the time nδ.)

The sum Z_n then arises after the time nδ, and the d.f. of Z_n is thus the d.f. of the sum that has been formed during the time interval (0, nδ). Suppose now that we allow δ to tend to zero and n to tend to infinity, in such a way that nδ tends to a finite limit _τ. It is conceivable that the distribution of Z_n may then tend to a definite limit, which will depend on the continuous time parameter_τ.