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

Consider now the formation of Zn 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 Xr might e.g. be the gains of a certain player during a series of games, every game requiring the time δ, so that Zn is the total gain realized after n games, or after the time nδ.)

The sum Zn then arises after the time nδ, and the d.f. of Zn 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 Zn may then tend to a definite limit, which will depend on the continuous time parameterτ.