On martingale tail sums for the path length in random trees

Abstract

For a martingale (Xn) converging almost surely to a random variable X, the sequence (Xn − X) is called martingale tail sum. Recently, Neininger [Random Structures Algorithms, 46 (2015), 346-361] proved a central limit theorem for the martingale tail sum of Régnier's martingale for the path length in random binary search trees. Grübel and Kabluchko [to appear in Annals of Applied Probability, (2016)] gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the law of the iterated logarithm for a family of trees containing binary search trees, recursive trees and plane-oriented recursive trees.

General Edgeworth expansions with applications to profiles of random trees

“…Proof. For β = 0 and γ = 2, the relevant argument is given in the proof of Proposition 6 in [40]. Fix β ∈ (β − , β + ) and γ ∈ (1, 2] such that β ∈ Ω 1 γ ∩ Ω 2 γ .…”

A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk

“…mode of convergence. Recently, CLTs for tail martingales associated with random trees (and related to the derivative of the Biggins martingale at 0) were proved in [14], [25], and [28].…”

“…There have already been several works that investigated how fast W n (1) approaches W ∞ (1) in various senses; see [20] and [21] for the rate of almost-sure convergence, and [4] for the rate of L p -convergence. Laws of the iterated logarithm for martingales related to path length of random trees were obtained in [28]. We also refer the reader to [15] for general CLTs and laws of the iterated logarithm for martingales not necessarily related to branching processes.…”

Dependence and phase changes in random m‐ary search trees

“…It should be mentioned that there is a large literature on K n when m = 2 because it is identical to the comparison cost used by quicksort. Many fine results were obtained; see, for example, the recent papers [3,12,17,20,37,41] and the references therein for more information.…”

“…with conditions on independence and distributions as in (41). Again Lemma 4.2 and its proof apply to T med and imply that the restriction of T med to M 2 3 (0, Id 2 ) has a unique fixed point L(X med , Λ med ).…”