期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2015
卷号：77
期号：1
页码：126-152
DOI：10.1007/s13171-014-0054-3
语种：English
出版社：Indian Statistical Institute
摘要：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.
关键词：Randomness criterion ; pseudorandom number ; normal number
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