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The Sigma function, which is the sum of the squares of the number of occurrences of every factor, is a criterion of randomness, measuring specially the uniformity of the block distribution. An infinite word whose prefixes attain asymptotically the smallest possible value of it is called Sigma-random. We prove that the Champernowne word is Sigma-random. We also consider less complex words which have...
We propose a single valued criterion for randomness of a binary sequence x 1 x 2 ⋯ x n ∈ { 0 , 1 } n defined by Σ n ( x 1 x 2 ⋯ x n ) = ∑ ξ ∈ { 0 , 1 } + | x 1 x 2 ⋯ x n | ξ 2 , $$ \Sigma^{n}(x_{1}x_{2}\cdots x_{n})=\sum_{\xi\in\{0,1\}^{+}}|x_{1}x_{2}...
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