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random sequence
The concept of a random sequence is essential in probability theory and statistics. The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words "let ''X''1,...,''Xn'' be independent random variables...". Yet as D. H. Lehmer stated in 1951: "A random sequence is a vague notion... in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians".〔"What is meant by the word Random" in ''Mathematics and common sense'' by Philip J. Davis 2006 ISBN 1-56881-270-1 pages 180-182〕
Axiomatic probability theory ''deliberately'' avoids a definition of a random sequence.〔''Inevitable Randomness in Discrete Mathematics'' by József Beck 2009 ISBN 0-8218-4756-2 page 44〕 Traditional probability theory does not state if a specific sequence is random, but generally proceeds to discuss the properties of random variables and stochastic sequences assuming some definition of randomness. The Bourbaki school considered the statement "let us consider a random sequence" an abuse of language.〔''Algorithms: main ideas and applications'' by Vladimir Andreevich Uspenskiĭ, Alekseĭ, Lʹvovich Semenov 1993 Springer ISBN 0-7923-2210-X page 166〕 During the 20th century various technical approaches to defining random sequences were developed and now three distinct paradigms can be identified.
==Early history==
Émile Borel was one of the first mathematicians to formally address randomness in 1909.〔E. Borel, ''Les probabilites denombrables et leurs applications arithmetique'' Rend. Circ. Mat. Palermo 27 (1909) 247-271〕 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. 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. 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.〔Laurant Bienvenu "Kolmogorov Loveland Stochastocity" in STACS 2007: 24th Annual Symposium on Theoretical Aspects of Computer Science by Wolfgang Thomas ISBN 3-540-70917-7 page 260〕
The sub-sequence selection criterion imposed by von Mises is important, because although 0101010101... is not biased, by selecting the odd positions, we get 000000... which is not random. 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. Church was a pioneer in the field of computable functions, and the definition he made relied on the Church Turing Thesis for computability.〔Alonzo Church, "On the Concept of Random Sequence," Bull. Amer. Math. Soc., 46 (1940), 254–260〕 This definition is often called ''Mises-Church randomness''.
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