Generalized gamma approximation with rates for urns, walks and trees
Generalized gamma approximation with rates for urns, walks and trees
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广义伽玛近似与瓮、步道和树木的比率
DOI：
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发表时间：
2013
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通讯作者：
Nathan Ross
中科院分区：
文献类型：
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作者：
E. Pekoz;Adrian Röllin;Nathan Ross
我们研究了一类新的时间非齐次波利亚型瓮方案，并给出了给定颜色的适当缩放数量的球的分布的最佳收敛速率，以达到具有整数参数的几乎全类广义伽马分布，该类包括瑞利分布、半正态分布和伽马分布。我们的主要工具是斯坦因方法，结合将广义伽玛极限分布表征为与更新理论的平衡分布变换相关的分布变换的不动点。我们在随机游走路径和树的递归构造中识别这些瓮模型的特殊情况，产生简单随机游走路径的本地时间和高度统计的收敛率，以及均匀随机二叉树和平面树的随机子树的大小。
We study a new class of time inhomogeneous Polya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein’s method combined with characterizing the generalized gamma limiting distributions as fixed points of distributional transformations related to the equilibrium distributional transformation from renewal theory. We identify special cases of these urn models in recursive constructions of random walk paths and trees, yielding rates of convergence for local time and height statistics of simple random walk paths, as well as for the size of random subtrees of uniformly random binary and plane trees.