The definition of random sequences

The Definition of Random Sequences

Kolmogorov has defined tile conditional complexity of an object y when the object x is already given to us as the minimal length of a binary program which by means of x computes y on a certain asymptotically optimal machine. On the basis of this definition he has proposed to consider those elements of a given large finite population to be random whose complexity is maximal. Almost all elements ...

Von Mises' Definition of Random Sequences Reconsidered

FURTHER RESULTS OF CONVERGENCE OF UNCERTAIN RANDOM SEQUENCES

Convergence is an issue being widely concerned about. Thus, in this paper, we mainly put forward two types of concepts of convergence in mean and convergence in distribution for the sequence of uncertain random variables. Then some of theorems are proved to show the relations among the three convergence concepts that are convergence in mean, convergence in measure and convergence in distributio...

Definition of necessary length of random sequences at execution of the empirical tests (criterion x2)

This article is devoted to the problem solution, arising from the necessity to test a sequence of random numbers. The application of Pearson’s criterion imposes restriction from below on the number of elements of the random sequence. In the article, the set of the classical empirical tests is considered and the expressions for calculation of necessary length of the random sequence are given.

Nonrandom Sequences between Random Sequences

Let us say that an infinite binary sequence q lies above an infinite binary sequence p if q can be obtained from p by replacing selected 0’s in p by 1’s. We show that above any infinite binary Martin-Löf random sequence p there exists an infinite binary nonrandom sequence q above which there exists an infinite binary random sequence r. This result is of interest especially in connection with th...