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Block-distribution in random strings. (English) Zbl 0778.60023
A notion of \(k\)-discrepancy for infinite sequences of independent random variables taking the values 0 and 1 with probabilities \(p\) and \(q\) is introduced: \[ D^ k_ N(x_ 1,\ldots,x_ N)=\max_{A\in\{0,1\}^ k}{1\over\sqrt{p^ k\mu_ k(A)}} \left|{\#\{1\leq n\leq N-k\mid x_ nx_{n+1}\cdots x_{n+k}=A\}\over N}-\mu_ k(A)\right|, \] where \(p\leq q\), \(A=a_ 1a_ 2\cdots a_ k\) and \(\mu_ k\) is the \(k\)-fold product measure generated by \(\mu(\{0\})=p\) and \(\mu(\{1\})=q\). For an increasing sequence of integers \(k(N)\) a sequence of 0’s and 1’s \(x_ 1,x_ 2,\ldots\) is called \(k(N)\)-uniformly distributed if \[ \lim_{N\to\infty}D^{k(N)}_ N(x_ 1,\ldots,x_ N)=0. \] It is proved that almost all sequences in \(\{0,1\}^{\mathbf N}\) are \(k(N)\)- uniformly distributed, if and only if \(\log_{1/p}n-\log_{1/p}\log n- k(n)\to\infty\). Otherwise the set of \(k(N)\)-uniformly distributed sequences has measure 0 which is an immediate consequence of Kolmogoroff’s 0-1-law. This generalizes a result due to P. Flajolet, P. Kirschenhofer and R. F. Tichy [Probab. Theory Relat. Fields 80, No. 1, 139-150 (1988; Zbl 0638.68058)], who considered the case \(p=q=1/2\) and proved that in this case almost all sequences are \(k(N)\)-uniformly distributed, if \(\log_ 2n-\log_ 2\log_ 2n- k(n)\to\infty\). K. Grill [A note on randomness, Stat. Probab. Lett. (to appear)] proved that almost no sequences are \(k(N)\)-uniformly distributed in the opposite case. M. Goldstern [J. Number Theory (to appear)] gave a general argument that almost all sequences are \(k(N)\)-uniformly distributed for all \(k(N)\) satisfying the above condition. The proof of the theorem uses a generalization of L. J. Guibas’ and A. M. Odlyzko’s theory of correlation polynomials of strings [J. Comb. Theory, Ser. A 30, 183-208 (1981; Zbl 0454.68109)].
Reviewer: P.J.Grabner (Graz)

MSC:
60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
60F10 Large deviations
68R05 Combinatorics in computer science
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References:
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