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In this article it is shown that the brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete n-vertex ordered graph trees whose search-depth functions converge to the brownian excursion as n→∞. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks...
In this research, strong convergence properties of the tail probability for weighted sums of negatively orthant dependent random variables are discussed. Some sharp theorems for weighted sums of arrays of rowwise negatively orthant dependent random variables are established. These results not only extend the corresponding ones of Cai [4], Wang et al. [19] and Shen [13], but also improve them, respectively.
In this paper the authors study the convergence properties for arrays of rowwise pairwise negatively quadrant dependent random variables. The results extend and improve the corresponding theorems of T. C. Hu, R. L. Taylor: On the strong law for arrays and for the bootstrap mean and variance, Int. J. Math. Math. Sci 20 (1997), 375–382.
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