Von Mises' Definition of Random Sequences Reconsidered

We review briefly the attempts to define random sequences (?0). These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence (??1-3 and 5); the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests (?4). Acknowledgement. We thank Johan van Benthem for his deconstruction of an earlier version of this manuscript, and also Roger Cooke and Mike Keane for helpful conversations.

[1]  A. Kechris.  Measure and category in effective descriptive set theory , 1973 .

[2]  A. Lichtenberg,et al.  Regular and Stochastic Motion , 1982 .

[3]  A. N. Kolmogorov.  Combinatorial foundations of information theory and the calculus of probabilities , 1983 .

[4]  C. Caramanis.  What is ergodic theory , 1963 .

[5]  P. Martin-Löf.  Complexity oscillations in infinite binary sequences , 1971 .

[6]  Péter Gács,et al.  On the relation between descriptional complexity and algorithmic probability , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[7]  Gregory J. Chaitin,et al.  Algorithmic Information Theory , 1987, IBM J. Res. Dev..

[8]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[9]  S. Kakutani.  On Equivalence of Infinite Product Measures , 1948 .

[10]  I. Good,et al.  Mathematical Theory of Probability and Statistics , 1966 .

[11]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[12]  T. Kamae.  Subsequences of normal sequences , 1973 .

[13]  Joseph Ford,et al.  How random is a coin toss , 1983 .

[14]  A. Church.  On the concept of a random sequence , 1940 .

[15]  Marc Snir,et al.  Probabilities over rich languages, testing and randomness , 1982, Journal of Symbolic Logic.

[16]  Gregory J. Chaitin,et al.  A recent technical report , 1974, SIGA.

[17]  Claus-Peter Schnorr,et al.  Process complexity and effective random tests , 1973 .

[18]  E. Bishop,et al.  Constructive measure theory , 1972 .

[19]  R. Mises.  Grundlagen der Wahrscheinlichkeitsrechnung , 1919 .

[20]  G. Sacks.  Measure-theoretic uniformity in recursion theory and set theory , 1969 .