An Easy Criterion for Randomness

Abstract

We propose a single valued criterion for randomness of a binary sequence $x_{1}x_{2}\cdots x_{n}\in \{0,1\}^{n}$ defined by

$$ \Sigma^{n}(x_{1}x_{2}\cdots x_{n})=\sum_{\xi\in\{0,1\}^{+}}|x_{1}x_{2} \cdots x_{n}|_{\xi}^{2}, $$

where $\{0,1\}^{+}=\cup _{k=1}^{\infty }\{0,1\}^{k}$ is the set of nonempty finite sequences over {0,1} and for ξ∈{0,1}^k,

$$ |x_{1}x_{2}\cdots x_{n}|_{\xi}=\#\{i; 1\le i\le n-k+1, x_{i}x_{i+1} \cdots x_{i+k-1}=\xi\}. $$

We prove that

$$ \lim_{n\to\infty}n^{-2}\Sigma^{n}(X_{1}X_{2}\cdots X_{n})=3/2 $$

holds with probability 1 if X ₁ X ₂⋯X _n is an i.i.d. process with P(X _i =0)=P(X _i =1)=1/2. Moreover, if a sample path x ₁ x ₂⋯ satisfies this almost all condition, then it is a normal number in the sense of E. Borel, but this converse is not true. We also propose a method to generate infinite sequences x ₁ x ₂⋯ satisfying this almost all condition, which are found out to be reasonable pseudorandom numbers from the point view of the block frequencies.