Random Real Numbers

Weihrauch, Klaus

In 1966 Martin-Löf [ML66] has defined random sequences of symbols as a recursion theoretical concept. A sequence of symbols is non-random, iff it is an element of the intersection of a computable sequence of recursively enumerable open sets the measures of which converge to zero with 2⁻ⁿ, a computable null set for short. We use the same concept to introduce random real numbers. The central definition is that of a computable sequence of recursively enumerable open subsets of real numbers. We give several arguments which show that our definition is natural. We prove that, as in the case of random sequences, there is a universal randomness test. Finally we prove that a real number is random, iff its natural positional representation with basis Q (Q ≥ 2) is a random sequence. This shows that our definition of random real numbers is equivalent to that of Calude and Jürgensen [Cal94, CJ94] by means of random positional representations.

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Weihrauch, Klaus: Random Real Numbers. Hagen 1997. FernUniversität in Hagen.

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12 Monate

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