This Tract has now been out of print for a number of years. Since there still seems to be some demand for it, the Syndics of the Cambridge University Press have judged it desirable to publish a new edition.

However, owing to the vigorous development of Mathematical Probability Theory since 1937, any attempt to bring the book up to date would have meant rewriting it completely, a task that would have been utterly beyond my possibilities under present conditions. Thus I have had to restrict myself in the main to a number of minor corrections, otherwise leaving the work—including the Bibliography—where it was in 1937.

Besides the minor corrections, most of which are concerned with questions of terminology, there are, in fact, only two major alterations. In the first place, a serious error in the statement and proof of Theorem 11 has been put right. Further, the contents of Chapter IV, § 4, which are fundamental for the theory of asymptotic expansions, etc., developed in Chapter VII, have been revised and simplified. This permits a new formulation of the important Lemma 4, on which the proofs of Theorems 24-26 are based. Finally a brief list of recent works on the subject in the English language has been added.

When this Tract was first published in 1937, an important part of it was Chapter VII, containing Liapounoff's classical inequality for the remainder in the Central Limit Theorem, as well as the theory of the related asymptotic expansions. For the Third Edition, this chapter has been partly rewritten, and now brings a proof of the sharper inequality due to Berry and Esseen. Moreover, several minor changes have been made, and the terminology has been somewhat modernized.

The Mathematical Theory of Probability has lately become of growing importance owing to the great variety of its applications, and also to its purely mathematical interest. The subject of this tract is the development of the purely mathematical side of the theory, without any reference to the applications. The axiomatic foundations of the theory have been chosen in agreement with the theory given by A. Kolmogoroff in his work Grundbegriffe der Wahrscheinlichkeitsrechnung, to which I am greatly indebted. In accordance with this theory, the subject has been treated as a branch of the theory of completely additive set functions. The method principally used has been that of characteristic functions (or Fourier-Stieltjes transforms).

The limitation of space has made it necessary to restrict the programme somewhat severely. Thus in the first place it has proved necessary to consider exclusively probability distributions in spaces of a finite number of dimensions. With respect to the advanced part of the theory, I have found it convenient to confine myself almost entirely to problems connected with the so-called Central Limit Theorem for sums of independent variables, and with some of its generalizations and modifications in various directions. This limitation permits a certain uniformity of method, but obviously a great number of important and interesting problems will remain unmentioned.

My most sincere thanks are due to my friends W. Feller, O. Lundberg and H. Wold for valuable help with the preparation of this work.

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.