scholarly journals Convergence Rates for Generalized Descents

10.37236/723 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
John Pike
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Explicit Formula ◽  
Central Limit ◽  
Convergence Rates ◽  
Stein’S Method ◽  
Permutation Statistics ◽  
Mean And Variance ◽  
The Mean

d-descents are permutation statistics that generalize the notions of descents and inversions. It is known that the distribution of d-descents of permutations of length n satisfies a central limit theorem as n goes to infinity. We provide an explicit formula for the mean and variance of these statistics and obtain bounds on the rate of convergence using Stein's method.

scholarly journals A General Central Limit Theorem for Shape Parameters of $m$-ary Tries and PATRICIA Tries

10.37236/3763 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael Fuchs ◽  
Chung-Kuei Lee
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Slight Modification ◽  
Wiener Index ◽  
Central Limit Theorems ◽  
Shape Parameters ◽  
Mean And Variance ◽  
The Mean

Tries and PATRICIA tries are fundamental data structures in computer science with numerous applications. In a recent paper, a general framework for obtaining the mean and variance of additive shape parameters of tries and PATRICIA tries under the Bernoulli model was proposed. In this note, we show that a slight modification of this framework yields a central limit theorem for shape parameters, too. This central limit theorem contains many of the previous central limit theorems from the literature and it can be used to prove recent conjectures and derive new results. As an example, we will consider a refinement of the size of tries and PATRICIA tries, namely, the number of nodes of fixed outdegree and obtain (univariate and bivariate) central limit theorems. Moreover, trivariate central limit theorems for size, internal path length and internal Wiener index of tries and PATRICIA tries are derived as well.

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