The frequency interpretation in probability (Q1970521)

The aim of the present paper is to provide a definite meaning to the notion of randomness and to the related notion of freguency interpretation. The author shows that there exist sequences of data values that model ``all'' relevant properties associated with a sequence of random variables, ``all'' referring to all properties that hold with probability 1 and may be described by an algorithm (in the sense of the computability theory). The key notion for the new concept of randomness is the so-called generic sequence, defined as follows. Let \(X_1,X_2,\dots,X_n,\dots\) be a sequence of random variables on the probability space \(\{S,P\}\), where \(S\) is a sample space and \(P\) is a probability on \(S\). Let \(x_1,x_2,\dots,x_n,\dots\) be a sample sequence, i.e. \(x_i=X_i(s)\) for some \(s\in S\). In order to define \(x_2,x_2,\dots,x_n,\dots\) as a generic sequence there is considered a program \(\Pi\) that behaves as follows: suppose \(a_1,a_2,\dots,a_n,\dots\) is a sequence of real numbers, (some of) whose terms are inputs to the program \(\Pi\). The program \(\Pi\) first outputs a natural number \(n_1\) indicating that \(a_{n_1}\) should be the input, then computes a natural number \(n_2\) indicating that \(a_{n_2}\) is the next input, and so on. \(\{n_j\}_{j\in\mathbb{N}}\) is not supposed to be an increasing or even non-constant sequence (although this is the relevant case). Another hypothesis is that sometime, after \(a_{n_j}\), the program computes and outputs the value \(f_j(a_{n_1}, \dots, a_{n_j})\), where \(f_j\) is a function on some domain in \(\mathbb{R}^j\). Finally, the following equality is supposed to hold: \[ P \Bigl( \limsup_{j\to \infty} f_j(X_{n_1},\dots,X_{n_j}) =\alpha\Bigr) =1,\tag{E1} \] where \(\alpha\) is a real number. It is not assumed that infinitely many \(f_j\) are computed for all sequences \(\{a_i\}_{i\in\mathbb{N}}\), but this is true for almost all sample sequences. Also, the computed \(n_j\) and \(f_j\) depend on the sample sequence. Now, \(\{x_n\}_{n \in\mathbb{N}}\) is defined as a generic sequence for the random variable sequence \(\{X_n\}_{n\in\mathbb{N}}\) if for all algorithms (programs) described and satisfying (E1), the following equality holds: \[ \limsup_{j \to\infty} f_j(x_{n_1}, \dots,x_{n_j}) =\alpha. \tag{E2} \] From (E1) and the fact that there are only countably many algorithms it follows that the sample points \(s\in S\) for which \(X_1(s),X_2(s), \dots, X_n(s),\dots\) is a generic sequence have the probability 1. The new concept of randomness is not only proved to be consistent with the classical theory of probability but also very useful to its applications.