13.1. The law of large numbers and the central limit theorem

At the beginning of the course, we already said that the mathematical laws of the theory of probability were obtained by abstraction of real statistical regularities inherent in mass random phenomena. The presence of these regularities is connected precisely with the mass character of phenomena, that is, with a large number of homogeneous experiments performed or with a large number of emerging random actions, generating in their totality a random variable, subject to a well-defined law. The property of stability of mass random phenomena has been known to mankind since ancient times. In whatever area it manifests itself, its essence boils down to the following: the specific features of each individual random phenomenon have almost no effect on the average result of the masses and such phenomena; random deviations from the average, unavoidable in each individual phenomenon, in the mass are mutually canceled, leveled, aligned. It is this stability of the averages that represents the physical content of the “law of large numbers” understood in the broad sense of the word: with a very large number of random phenomena, their average result practically ceases to be random and can be predicted with a large degree of certainty.

In the narrow sense of the word, the "law of large numbers" in the theory of probability is understood as a series of mathematical theorems, in each of which, for certain conditions, the fact of approximation of the average characteristics of a large number of experiments to certain definite constants is established.

AT 2.3 we have already formulated the simplest of these theorems — the theorem of J. Bernoulli. She argues that with a large number of experiments, the frequency of the event approaches (more precisely, it converges in probability) to the probability of this event. With other, more general forms of the law of large numbers, we will introduce in this chapter. All of them establish the fact and conditions of convergence in probability of one or another random variable to constant, non-random variable.

The law of large numbers plays an important role in practical applications of probability theory. The property of random variables, under certain conditions, to behave practically as non-random allows us to confidently operate with these quantities, to predict the results of mass random phenomena with almost complete certainty.

The possibilities of such predictions in the field of mass random phenomena are further extended by the presence of another group of limit theorems, concerning not limit values ​​of random variables, but limit distribution laws. This is a group of theorems, known as the "central limit theorem". We have already said that when summing up a sufficiently large number of random variables, the law of the distribution of the sum unboundedly approaches the normal one if certain conditions are met. These conditions, which can be mathematically formulated in various ways — in a more or less general form — essentially boil down to the requirement that the effect on the sum of individual addends be evenly small, that is, that the sum does not include members that clearly dominate the population. the rest in their effect on the dispersion of the sum. Different forms of the central limit theorem differ among themselves by the conditions for which this limit property of the sum of random variables is established.

Different forms of the law of large numbers, together with various forms of the central limit theorem, form the totality of the so-called limit theorems of probability theory. Limit theorems make it possible not only to carry out scientific predictions in the field of random phenomena, but also to evaluate the accuracy of these predictions.

In this chapter we consider only some of the most simple forms of limit theorems. First, we will consider theorems related to the group of the “law of large numbers”, then the theorems related to the group of the “central limit theorem”.