Exact Expressions for Some Randomness Tests

The notion of randomness which v. MISESattempted to formalise first has received a long time only moderate attention in contemporary probability theory. It was hard to find a convincing distinction between random and nonrandom elements of a probability space and, for most probabilists, it has been up to now not clear how much would one be happier finding it. Though for a statistician nothing seems ,to be more interesting than the question about randomness. Given an element w of the event space Q as the outcome of an experiment, and a distribution P he wants to find out how justified it is to suppose that the underlying distribution to the experiment was P; i.e. that w is random w.r.t. P. However, his model is slightly different because in the typica,l cases he has an access to a large number of independently repeated experiments pn = P X P X . . • x P and what he wishes to decide on the basis of w= (WI' ... , wn) is only the question about P, the product structure taken for granted. The decisions can then be made on the basis of central limit theorems, and it is, roughly said, the investigation of the conditions of such decisions to which most of mathematical statistics is devoted. There are some highly interesting statistical situations where the product-space framework is not applicable: e,g. prediction problems or testing of pseudo-random sequences. After its revival in the sixties by the work of KOLMOGOROV and MARTIN-LoF(continued by LEVIN, CHAITIN,SCHNORR) the modern theory of randomness approaches now to a satisfiable form and its solutions to these problems are of convincing simplicity and generality. Unfortunately, to understand them one has to learn some computability theory, and if later one tries to apply them one notes with some disappointment the large gap between theoretical and practical computability. The present paper does not bridge this gap, either. It gives some more exact relations between complexity and randomness and one can only hope that when the theory using general computability will be more perfect then the chances to find its practical extension increase. In Section 1 we give the necessary definitions, in Section 2 some known results on MARTIN-Lcm's tests. In Section 3 apriori probability and its known relation to randomness is described. Section 4 is devoted to various definitions of complexity and their estimates. In Section 5 we give some exact expressions for MARTIN-Lcm'stest in terms of different types of complexities. The possibility of such expressions shows once again the technical flexibility of the complexity apparatus. Finally, in Section 6, we follow up the connections of LEVIN's uniform tests to the previous theory and introduce a somewhat modified uniform test and a simple one having the conservation property.