Limiting Distributions for Path Lengths in Recursive Trees

1991 ◽  
Vol 5 (1) ◽  
pp. 53-59 ◽  
Author(s):  
Hosam M. Mahmoud
Keyword(s):  
Path Length ◽  
Elementary Proof ◽  
Random Variable ◽  
Limiting Distribution ◽  
Limiting Distributions ◽  
Quadratic Mean ◽  
Recursive Tree ◽  
Asymptotically Normal ◽  
Path Lengths ◽  
Recursive Trees

The depth of insertion and the internal path length of recursive trees are studied. Luc Devroye has recently shown that the depth of insertion in recursive trees is asymptotically normal. We give a direct alternative elementary proof of this fact. Furthermore, via the theory of martingales, we show that In, the internal path length of a recursive tree of order n, converges to a limiting distribution. In fact, we show that there exists a random variable I such that (In – n In n)/n→I almost surely and in quadratic mean, as n → α. The method admits, in passing, the calculation of the first two moments of In.

On the size of paged recursive trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750021
Author(s):  
Mehri Javanian
Keyword(s):  
Contraction Method ◽  
Recursive Tree ◽  
Asymptotically Normal ◽  
Recursive Trees ◽  
Normal Formula

A paged recursive tree is constructed as a recursive tree except that it depends on an integer parameter [Formula: see text] representing a page capacity, small subtrees with size [Formula: see text]. We investigate the number of nodes [Formula: see text] (size of the tree) in paged recursive trees built from labels [Formula: see text]. The expectation and variance of [Formula: see text] are derived, and it is also shown that [Formula: see text] is asymptotically normal. [Formula: see text] as [Formula: see text] by applying the contraction method.

On the distribution of distances in recursive trees

1996 ◽  
Vol 33 (03) ◽  
pp. 749-757 ◽  
Author(s):  
Robert P. Dobrow
Keyword(s):  
Exact Distribution ◽  
Recursive Tree ◽  
Asymptotically Normal ◽  
Ancient Manuscripts ◽  
The Law ◽  
Recursive Trees ◽  
Pyramid Schemes ◽  
Bernoulli Distributions ◽  
Family Trees

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree Tn with n labeled nodes is a recursive tree if n = 1, or n > 1 and Tn can be constructed by joining node n to a node of some recursive tree Tn– 1. For arbitrary nodes i < n in a random recursive tree we give the exact distribution of Xi,n , the distance between nodes i and n. We characterize this distribution as the convolution of the law of Xi,j+ 1 and n – i – 1 Bernoulli distributions. We further characterize the law of Xi,j+ 1 as a mixture of sums of Bernoullis. For i = in growing as a function of n, we show that is asymptotically normal in several settings.

scholarly journals On a memory game and preferential attachment graphs

2016 ◽  
Vol 48 (2) ◽  
pp. 585-609 ◽  
Author(s):  
Hüseyin Acan ◽  
Paweł Hitczenko
Keyword(s):  
Preferential Attachment ◽  
Random Variable ◽  
Urn Model ◽  
Limiting Distributions ◽  
Asymptotically Normal ◽  
Generalized Pólya Urn ◽  
Limit Result ◽  
Few Vertices ◽  
Memory Game ◽  
Main Technical Tool

Abstract In their recent paper Velleman and Warrington (2013) analyzed the expected values of some of the parameters in a memory game; namely, the length of the game, the waiting time for the first match, and the number of lucky moves. In this paper we continue this direction of investigation and obtain the limiting distributions of those parameters. More specifically, we prove that when suitably normalized, these quantities converge in distribution to a normal, Rayleigh, and Poisson random variable, respectively. We also make a connection between the memory game and one of the models of preferential attachment graphs. In particular, as a by-product of our methods, we obtain the joint asymptotic normality of the degree counts in the preferential attachment graphs. Furthermore, we obtain simpler proofs (although without rate of convergence) of some of the results of Peköz et al. (2014) on the joint limiting distributions of the degrees of the first few vertices in preferential attachment graphs. In order to prove that the length of the game is asymptotically normal, our main technical tool is a limit result for the joint distribution of the number of balls in a multitype generalized Pólya urn model.

On the distribution of distances in recursive trees

1996 ◽  
Vol 33 (3) ◽  
pp. 749-757 ◽  
Author(s):  
Robert P. Dobrow
Keyword(s):  
Exact Distribution ◽  
Recursive Tree ◽  
Asymptotically Normal ◽  
Ancient Manuscripts ◽  
The Law ◽  
Recursive Trees ◽  
Pyramid Schemes ◽  
Bernoulli Distributions ◽  
Family Trees

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree Tn with n labeled nodes is a recursive tree if n = 1, or n > 1 and Tn can be constructed by joining node n to a node of some recursive tree Tn–1. For arbitrary nodes i < n in a random recursive tree we give the exact distribution of Xi,n, the distance between nodes i and n. We characterize this distribution as the convolution of the law of Xi,j+1 and n – i – 1 Bernoulli distributions. We further characterize the law of Xi,j+1 as a mixture of sums of Bernoullis. For i = in growing as a function of n, we show that is asymptotically normal in several settings.

Total Path Length for Random Recursive Trees

1999 ◽  
Vol 8 (4) ◽  
pp. 317-333 ◽  
Author(s):  
ROBERT P. DOBROW ◽  
JAMES ALLEN FILL
Keyword(s):  
Path Length ◽  
Recurrence Relations ◽  
Rooted Tree ◽  
Random Variable ◽  
Finite Variance ◽  
Density Estimator ◽  
Search Cost ◽  
Total Path Length ◽  
Recursive Trees ◽  
Mutually Independent

Total path length, or search cost, for a rooted tree is defined as the sum of all root-to-node distances. Let Tn be the total path length for a random recursive tree of order n. Mahmoud [10] showed that Wn := (Tn − E[Tn])/n converges almost surely and in L2 to a nondegenerate limiting random variable W. Here we give recurrence relations for the moments of Wn and of W and show that Wn converges to W in Lp for each 0 < p < ∞. We confirm the conjecture that the distribution of W is not normal. We also show that the distribution of W is characterized among all distributions having zero mean and finite variance by the distributional identityformula herewhere [Escr ](x) := − x ln x − (1 minus; x) ln(1 − x) is the binary entropy function, U is a uniform (0, 1) random variable, W* and W have the same distribution, and U, W and W* are mutually independent. Finally, we derive an approximation for the distribution of W using a Pearson curve density estimator.

scholarly journals An almost sure result for path lengths in binary search trees

2003 ◽  
Vol 35 (02) ◽  
pp. 363-376
Author(s):  
F. M. Dekking ◽  
L. E. Meester
Keyword(s):  
Path Length ◽  
Random Variable ◽  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Total Path Length ◽  
Path Lengths

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variableZ. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜mof finite binary trees with at mostmnodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector timesZ.

scholarly journals ON WEIGHTED PATH LENGTHS AND DISTANCES IN INCREASING TREES

2007 ◽  
Vol 21 (3) ◽  
pp. 419-433 ◽  
Author(s):  
M. Kuba ◽  
A. Panholzer
Keyword(s):  
Divisible Distribution ◽  
Limiting Distribution ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Path Lengths ◽  
Recursive Trees ◽  
Insertion Process

We study weighted path lengths (depths) and distances for increasing tree families. For those subclasses of increasing tree families, which can be constructed via an insertion process (e.g., recursive trees, plane-oriented recursive trees, and binary increasing trees), we can determine the limiting distribution that can be characterized as a generalized Dickman's infinitely divisible distribution.

scholarly journals An almost sure result for path lengths in binary search trees

2003 ◽  
Vol 35 (2) ◽  
pp. 363-376 ◽  
Author(s):  
F. M. Dekking ◽  
L. E. Meester
Keyword(s):  
Path Length ◽  
Random Variable ◽  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Total Path Length ◽  
Path Lengths

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variable Z. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜m of finite binary trees with at most m nodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector times Z.

scholarly journals Label-based parameters in increasing trees

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Probability Distribution ◽  
Random Variable ◽  
Growth Behavior ◽  
Limiting Distribution ◽  
Factorial Moments ◽  
International Audience ◽  
Recursive Trees ◽  
Insertion Process

International audience Grown simple families of increasing trees are a subclass of increasing trees, which can be constructed by an insertion process. Three such tree families contained in the grown simple families of increasing trees are of particular interest: $\textit{recursive trees}$, $\textit{plane-oriented recursive trees}$ and $\textit{binary increasing trees}$. Here we present a general approach for the analysis of a number of label-based parameters in a random grown simple increasing tree of size $n$ as, e.g., $\textit{the degree of the node labeled j}$, $\textit{the subtree-size of the node labeled j}$, etc. Further we apply the approach to the random variable $X_{n,j,a}$, which counts the number of size-$a$ branches attached to the node labeled $j$ (= subtrees of size $a$ rooted at the children of the node labeled $j$) in a random grown simple increasing tree of size $n$. We can give closed formulæ for the probability distribution and the factorial moments. Furthermore limiting distribution results for $X_{n,j,a}$ are given dependent on the growth behavior of $j=j(n)$ compared to $n$.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

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