DEFINITIONS OF RANDOMNESS

Randomness is a fundamental but elusive concept in mathematics and physics. Even for the elementary case of a random binary sequence, a generally accepted and operational definition is lacking. However, when ensembles are used for the foundation of probability theory, randomness has to be identified with uncorrelatedness, a neglected notion that yet solves many puzzles surrounding randomness. In any single sequence, a definite amount of correlation is unavoidable. At best, the remaining correlations are of high order or cover a large distance. This is the case for properly defined well-tempered pseudorandom sequences. Reliable algorithms for the generation of pseudorandom bits at rates of 1 GHz result.