Algorithmic tests and randomness with respect to a class of measures

This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the Martin-Löf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability p of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli ∗LIAFA, CNRS & Université Paris Diderot, Paris 7, Case 7014, 75205 Paris Cedex 13, France, e-mail: Laurent dot Bienvenu at liafa dot jussieu dot fr †Department of Computer Science, Boston University, 111 Cummington st., Room 138, Boston, MA 02215, e-mail: gacs at bu dot edu ‡LORIA & INRIA Nancy Grand Est – B248, 615, rue du Jardin Botanique, BP 239, 54506 Vandœuvre-lès-Nancy, France, e-mail: Mathieu dot Hoyrup at loria dot fr §Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., Toronto, Ontario, Canada, M5S 2E4, e-mail: crojas at math dot utoronto dot ca ¶LIF, Universitè Aix – Marseille, CNRS, 39, rue Joliot-Curie, 13453 Marseille cedex 13, France, on leave from IITP RAS, Bolshoy Karetny, 19, Moscow. Supported by NAFIT ANR-08EMER-008-01, RFBR 0901-00709-a grants. e-mail: sasha dot shen at gmail dot com. ha l-0 06 44 78 5, v er si on 1 25 N ov 2 01 1 Author manuscript, published in "Proceedings of the Steklov Institute of Mathematics 274, 1 (2011) 34-89" DOI : 10.1134/S0081543811060058