SUB-GAUSSIAN TAIL BOUNDS FOR THE WIDTH AND HEIGHT OF CONDITIONED GALTON-WATSON TREES

• 成果类型: Article
• 署名作者: Addario-Berry, Louigi; Devroye, Luc; Janson, Svante
• 署名单位: McGill University; McGill University; Uppsala University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP758
• 发表日期: 2013
• 页码: 1072-1087
• 关键词: behavior

摘要

We study the height and width of a Galton-Watson tree with offspring distribution xi satisfying E xi = 1, 0 < Var xi < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 <= k <= n.