Random sequence

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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 X1,...,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".[1]

Axiomatic probability theory deliberately avoids a definition of a random sequence.[2] 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.[3] During the 20th century various technical approaches to defining random sequences were developed and now three distinct paradigms can be identified.

Contents

Early history

Émile Borel was one of the first mathematicians to formally address randomness in 1909.[4] 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.[5]

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.[6] This definition is often called Mises-Church randomness.

Modern approaches

In the mid 1960s, A. N. Kolmogorov and D. W. Loveland independently proposed a more permissive selection rule.[7][8] In their view Church's recursive function definition was too restrictive in that it read the elements in order. 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. This definition is often called Kolmogorov-Loveland 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.

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