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  • Author or Editor: T. Móri x
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Abstract  

For everyk≥1 consider the waiting time until each pattern of lengthk over a fixed alphabet of sizen appears at least once in an infinite sequence of independent, uniformly distributed random letters. Lettingn→∞ we determine the limiting finite dimensional joint distributions of these waiting times after suitable normalization and provide an estimate for the rate of convergence. It will turn out that these waiting times are getting independent.

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Abstract  

A process of evolving random graphs is considered where vertices are added to the graph one by one, and edges connecting the new vertex to the old ones are drawn independently, each with probability depending linearly on the degree of the endpoint. In the paper the asymptotic degree distribution and the order of the maxdegree are determined.

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In a one-parameter model for evolution of random trees strong law of large numbers and central limit theorem are proved for the number of vertices with low degree. The proof is based on elementary martingale theory.

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In a string ofn independent coin tosses we consider the difference between the lengths of the longest blocks of consecutive heads resp. tails. A complete characterization of the a.s. limit properties of this quantity is proved.

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Abstract  

Sufficient conditions of covariance type are presented for weighted averages of random variables with arbitrary dependence structure to converge to 0, both for logarithmic and general weighting. As an application, an a.s. local limit theorem of Csáki, Földes and Révész is revisited and slightly improved.

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