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• Stronger than Martin-Löf randomness

• Weaker than Martin-Löf randomness

• See also

• References

• Further reading

Richard von Mises formalized the notion of a test for randomness in order to define a random sequence as one that passed all tests for randomness. He defined a "collective" (kollektiv) to be an infinite binary string x 1 : ∞ ∞ {\displaystyle x_{1:\infty }} defined such that

• There exists a limit lim n 1 n ∑ ∑ i = 1 n x i = p ∈ ∈ ( 0 , 1 ) {\displaystyle \lim _{n}{\frac {1}{n}}\sum _{i=1}^{n}x_{i}=p\in (0,1)} .

• For any "admissible" rule, such that it picks out an infinite subsequence ( x m i ) i {\displaystyle (x_{m_{i}})_{i}} from the string, we still have lim n 1 n ∑ ∑ i = 1 n x m i = p {\displaystyle \lim _{n}{\frac {1}{n}}\sum _{i=1}^{n}x_{m_{i}}=p} . He called this principle "impossibility of a gambling system".

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