This approach started with the work of Richard von Mises and Alonzo Church.
The frequency / measure-theoretic approach.Three basic paradigms for dealing with random sequences have now emerged: Kolmogorov's definition of a random string was that it is random if has no description shorter than itself via a universal Turing machine. His original definition involved measure theory, but it was later shown that it can be expressed in terms of Kolmogorov complexity. In 1966 Per Martin-Löf introduced a new notion which is now generally considered the most satisfactory notion of algorithmic randomness. But this method was considered too weak by Alexander Shen who showed that there is a Kolmogorov–Loveland stochastic sequence which does not conform to the general notion of randomness. This definition is often called Kolmogorov–Loveland stochasticity. Instead they proposed a rule based on a partially computable process which having read any N elements of the sequence, decides if it wants to select another element which has not been read yet. In their view Church's recursive function definition was too restrictive in that it read the elements in order. Loveland independently proposed a more permissive selection rule. This definition is often called Mises–Church randomness.ĭuring the 20th century various technical approaches to defining random sequences were developed and now three distinct paradigms can be identified. Church was a pioneer in the field of computable functions, and the definition he made relied on the Church Turing Thesis for computability. Von Mises never totally formalized his definition of a proper selection rule for sub-sequences, but in 1940 Alonzo Church defined it as any recursive function which having read the first N elements of the sequence decides if it wants to select element number N + 1. is not biased, by selecting the odd positions, we get 000000. The sub-sequence selection criterion imposed by von Mises is important, because although 0101010101. the frequency of zeros goes to 1/2 and every sub-sequence we can select from it by a "proper" method of selection is also not biased. Using the concept of the impossibility of a gambling system, von Mises defined an infinite sequence of zeros and ones as random if it is not biased by having the frequency stability property i.e. In 1919 Richard von Mises gave the first definition of algorithmic randomness, which was inspired by the law of large numbers, although he used the term collective rather than random sequence.
Émile Borel was one of the first mathematicians to formally address randomness in 1909.
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