In the most varied fields of practical and scientific experience, cases occur where certain observations or trials may be repeated a large number of times under similar circumstances. Our attention is then directed to a certain quantity, which may assume different numerical values at successive observations. In many cases each observation yields not only one, but a certain number of quantities, say k, so that generally we may say that the result of each observation is a definite point X in a space of k dimensions (k ≥ 1), while the result of the whole series of observations is a sequence of points: X1, X2, ….

Thus if we make a series of throws with a given number of dice, we may observe the sum of the points obtained at each throw. We are then concerned with a variable quantity, which may assume every integral value between m and 6m (both limits inclusive), where m is the number of dice. On the other hand, in a series of measurements of the state of some physical system, or of the size of certain organs in a number of individuals belonging to the same biological species, each observation furnishes a certain number of numerical values, i.e. a definite point X in a space R of a fixed number of dimensions.

In certain cases, the observed characteristic is only indirectly expressed as a number.

1. For a distribution in a one-dimensional space, the only possible discontinuities arise from discrete points which, in terms of the mechanical interpretation used in Chapter II, are bearers of positive quantities of mass. As soon as the number of dimensions exceeds unity, the question of the discontinuities becomes, however, more complicated. Thus in a k-dimensional space, the whole mass may be concentrated to a sub-space of less than k dimensions (line, surface, …), though there is no single point that carries a positive quantity of mass.

Given a random variable X = (ξ1 …,ξk) in the k-dimensional space Rk, we denote as in Chapter II the corresponding pr.f. by P (S) and the d.f. by F (x1 …, xk). Just as in the case k = 1, there can at most be a finite number of points A such that P (A) > a > 0, and hence at most an enumerable set of points B such that P (B) > 0. We shall call this set the point spectrum of the distribution.

According to II, § 3, every component ξi of X is itself a random variable, and the corresponding (one-dimensional) distribution is found by projecting the original distribution on the axis of ξi.

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.