Asymptotical growth of a class of random trees. (English) Zbl 0563.60010

Pittel, B.

Ann. Probab. 13, 414-427 (1985).

We study three rules for the development of a sequence of finite subtrees \(\{t_ n\}\) of an infinite m-ary tree t. Independent realizations \(\{\) w(n)\(\}\) of a stationary ergodic process \(\{\) \(w\}\) 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\({}_ nL_ n/\ln n=\sigma_ 2\) 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 proved along the way that the height of \(t_ n\) grows as \(\sigma_ 2\ln n\) with probability one.

MSC:

60C05	Combinatorial probability
60F15	Strong limit theorems
28D20	Entropy and other invariants
68Q25	Analysis of algorithms and problem complexity

Keywords:

random trees; lengths of the paths; ergodic process; asymptotic growth; strong, weak convergence