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.
Probability axioms|Axiomatic probability theory deliberately avoids a definition of a random sequence. 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|Bourbaki school considered the statement let us consider a random sequence an abuse of language. During the 20th century various technical approaches to defining random sequences were developed and now three distinct paradigms can be identified.
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.
Probability axioms|Axiomatic probability theory deliberately avoids a definition of a random sequence. 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|Bourbaki school considered the statement let us consider a random sequence an abuse of language. During the 20th century various technical approaches to defining random sequences were developed...