142 RANDOM NUMBERS 3.5 (Multiple domain web hosting) 3.5. WHAT IS A
142 RANDOM NUMBERS 3.5 3.5. WHAT IS A RANDOM SEQUENCE? A. Introductory remarks. We have seen in this chapter how to generate sequences (1) of real numbers in the range 0 2 U, < 1, and we have called them random sequences even though they are completely deterministic in character. To justify this terminology, we claimed that the numbers behave as if they are truly random. Such a statement may be satisfactory for practical purposes (at the present time), but it sidesteps a very important philosophical and theoretical question: Precisely what do we mean by random behavior ? A quantitative definition is needed. It is undesirable to talk about concepts that we do not really understand, especially since many apparently paradoxical statements can be made about random numbers. The mathematical theory of probability and statistics carefully sidesteps the question; it refrains from making absolute statements, and instead expresses everything in terms of how much probability is to be attached to statements involving random sequences of events. The axioms of probability theory are set up so that abstract probabilities can be computed readily, but nothing is said about what probability really signifies, or how this concept can be applied meaningfully to the actual world. In the book Probability, Statistics, and Truth (New York: Macmillan, 1957), R. von Mises discusses this situation in detail, and presents the view that a proper definition of probability depends on obtaining a proper definition of a random sequence. Let us paraphrase here some statements made by two of the many authors who have commented on the subject. D. H. Lehmer (1951): A random sequence is a vague notion embodying the idea of a sequence in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests, traditional with statisticians and depending somewhat on the uses to which the sequence is to be put. J. IV. Franklin (1962): The sequence (1) is random if it has every property that is shared by all infinite sequences of independent samples of random variables from the uniform distribution. Franklin s statement essentially generalizes Lehmer s to say that the se- quence must satisfy all statistical tests. His definition is not completely precise, and we will see later that a reasonable interpretation of his statement leads us to conclude that there is no such thing as a random sequence! So let us begin with Lehmer s less restrictive statement and attempt to make it precise. What we really want is a relatively short list of mathematical properties, each of which is satisfied by our intuitive notion of a random sequence; furthermore, the list is to be complete enough so that we are willing to agree that any sequence satisfying these properties is random. In this section, we will develop what seems to be