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May, 1985 Asymptotical Growth of a Class of Random Trees
B. Pittel
Ann. Probab. 13(2): 414-427 (May, 1985). DOI: 10.1214/aop/1176993000


We study three rules for the development of a sequence of finite subtrees $\{t_n\}$ of an infinite $m$-ary tree $t$. Independent realizations $\{\omega(n)\}$ of a stationary ergodic process $\{\omega\}$ on $m$ letters are used to trace out paths in $t$. In the first rule, $t_n$ is formed by adding a node to $t_{n - 1}$ at the first location where the path defined by $\omega (n)$ leaves $t_{n - 1}$. The second and third rules are similar, but more complicated. For each rule, the height $L_n$ of the added node is shown to grow, in probability, as $\ln n$ divided by $h$ the entropy per symbol of the generic process. A typical retrieval time has the same behavior. On the other hand, $\lim \inf_nL_n/\ln n = \sigma_1, \lim \sup_n L_n/\ln n = \sigma_2$ a.s., where the constants $\sigma_1, \sigma_2$, are, in general, different, depend on the rule in use, and $\sigma_1 < 1/h < \sigma_2$. It is proven along the way that the height of $t_n$ grows as $\sigma_2\ln n$ with probability one.


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B. Pittel. "Asymptotical Growth of a Class of Random Trees." Ann. Probab. 13 (2) 414 - 427, May, 1985.


Published: May, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0563.60010
MathSciNet: MR781414
Digital Object Identifier: 10.1214/aop/1176993000

Primary: 60C05
Secondary: 28D20 , 60F15 , 68C25

Keywords: asymptotic growth , ergodic process , lengths of the paths , Random trees , strong , weak convergence

Rights: Copyright © 1985 Institute of Mathematical Statistics


Vol.13 • No. 2 • May, 1985
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