An Easy Criterion for Randomness

Kamae, Teturo; Xue, Yumei

doi:10.1007/s13171-014-0054-3

Cited by 6 publications

(8 citation statements)

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“…Since the function f (x) = x 2 is convex, the value ξ∈{0,1} k |x 1 x 2 · · · x n | 2 ξ for any k = 1, 2, · · · is smaller if the values |x 1 x 2 · · · x n | ξ for ξ ∈ {0, 1} k are less deviated as a whole from the mean value (n − k + 1)/2 k , that is, the sequence x 1 x 2 · · · x n is more random. In fact, it is proved in [3] that lim inf n→∞ Σ(x 1 x 2 · · · x n ) n 2 ≥ 3 2 holds for any x 1 x 2 · · · ∈ {0, 1} ∞ , while lim n→∞ Σ(X 1 X 2 · · · X n ) n 2 = 3 2 holds with probability 1 if X 1 X 2 · · · X n is the i.i.d. process with P (X i = 0) = P (X i = 1) = 1/2.…”

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confidence: 99%

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A characterization of eventual periodicity

Kamae

Kim

2015

Theoretical Computer Science

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Section: Introductionmentioning

confidence: 99%

“…In [3], a criterion of randomness for binary words is introduced. As stated in Definition 1 and 3, let…”

Section: Introductionmentioning

confidence: 99%

A characterization of eventual periodicity

Kamae

Kim

2015

Theoretical Computer Science

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“…In addition, they satisfy some long-range recurrence properties which are not necessarily satisfied by normal numbers. In fact, these properties together characterize the -randomness [1]. The other extreme, the eventual periodicity of…”

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“…Forgetting the arithmetical point of view, we focus on the uniformity of the block frequency in the finite words, which is also an important factor of the randomness. From this point of view, the ideal finite words are equidistributed ones, that is, x 1 x 2 • • • x n ∈ A n is said to be equidistributed [1] if for any k = 1, 2, . .…”

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Entropy estimate by a randomness criterion

Kamae¹

2016

Ergod. Th. Dynam. Sys.

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We propose a new criterion for randomness of a word $x_{1}x_{2}\cdots x_{n}\in \mathbb{A}^{n}$ over a finite alphabet $\mathbb{A}$ defined by $$\begin{eqnarray}\unicode[STIX]{x1D6EF}^{n}(x_{1}x_{2}\cdots x_{n})=\mathop{\sum }_{\unicode[STIX]{x1D709}\in \mathbb{A}^{+}}\unicode[STIX]{x1D713}(|x_{1}x_{2}\cdots x_{n}|_{\unicode[STIX]{x1D709}}),\end{eqnarray}$$ where $\mathbb{A}^{+}=\bigcup _{k=1}^{\infty }\mathbb{A}^{k}$ is the set of non-empty finite words over $\mathbb{A}$, for $\unicode[STIX]{x1D709}\in \mathbb{A}^{k}$, $$\begin{eqnarray}|x_{1}x_{2}\cdots x_{n}|_{\unicode[STIX]{x1D709}}=\#\{i;~1\leq i\leq n-k+1,~x_{i}x_{i+1}\cdots x_{i+k-1}=\unicode[STIX]{x1D709}\},\end{eqnarray}$$ and for $t\geq 0$, $\unicode[STIX]{x1D713}(0)=0$ and $\unicode[STIX]{x1D713}(t)=t\log t~(t>0)$. This value represents how random the word $x_{1}x_{2}\cdots x_{n}$ is from the viewpoint of the block frequency. In fact, we define a randomness criterion as $$\begin{eqnarray}Q(x_{1}x_{2}\cdots x_{n})=(1/2)(n\log n)^{2}/\unicode[STIX]{x1D6EF}^{n}(x_{1}x_{2}\cdots x_{n}).\end{eqnarray}$$ Then, $$\begin{eqnarray}\lim _{n\rightarrow \infty }(1/n)Q(X_{1}X_{2}\cdots X_{n})=h(X)\end{eqnarray}$$ holds with probability 1 if $X_{1}X_{2}\cdots \,$ is an ergodic, stationary process over $\mathbb{A}$ either with a finite energy or $h(X)=0$, where $h(X)$ is the entropy of the process. Another criterion for randomness using $t^{2}$ instead of $t\log t$ has already been proposed in Kamae and Xue [An easy criterion for randomness. Sankhya A77(1) (2015), 126–152]. In comparison, our new criterion provides a better fit with the entropy. We also claim that our criterion not only represents the entropy asymptotically but also gives a good representation of the randomness of fixed finite words.

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