scholarly journals Derangement Polynomials and Excedances of Type $B$

10.37236/81 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
William Y. C. Chen ◽  
Robert L. Tang ◽  
Alina F. Y. Zhao
Keyword(s):  
Normal Distribution ◽  
Generating Function ◽  
Type B ◽  
Eulerian Polynomials ◽  
Basic Properties ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal

Based on the notion of excedances of type $B$ introduced by Brenti, we give a type $B$ analogue of the derangement polynomials. The connection between the derangement polynomials and Eulerian polynomials naturally extends to the type $B$ case. Using this relation, we derive some basic properties of the derangement polynomials of type $B$, including the generating function formula, the Sturm sequence property, and the asymptotic normal distribution. We also show that the derangement polynomials are almost symmetric in the sense that the coefficients possess the spiral property.

scholarly journals On the Variety of Shapes on the Fringe of a Random Recursive Tree

2010 ◽  
Vol 47 (01) ◽  
pp. 191-200 ◽  
Author(s):  
Qunqiang Feng ◽  
Hosam M. Mahmoud
Keyword(s):  
Normal Distribution ◽  
Contraction Method ◽  
Tree Shape ◽  
Recursive Tree ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal ◽  
The Given ◽  
Fixed Tree

We consider a variety of subtrees of various shapes lying on the fringe of a recursive tree. We prove that (under suitable normalization) the number of isomorphic images of a given fixed tree shape on the fringe of the recursive tree is asymptotically Gaussian. The parameters of the asymptotic normal distribution involve the shape functional of the given tree. The proof uses the contraction method.

Central limit theorem for germination-growth models in ℝd with non-Poisson locations

2001 ◽  
Vol 33 (04) ◽  
pp. 751-755
Author(s):  
S. N. Chiu ◽  
M. P. Quine
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Central Limit ◽  
Growth Models ◽  
Germination Time ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal ◽  
Dependent Point ◽  
Germinated Seeds

Seeds are randomly scattered in ℝ d according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.

Central limit theorem for germination-growth models in ℝd with non-Poisson locations

2001 ◽  
Vol 33 (4) ◽  
pp. 751-755 ◽  
Author(s):  
S. N. Chiu ◽  
M. P. Quine
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Central Limit ◽  
Growth Models ◽  
Germination Time ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal ◽  
Dependent Point ◽  
Germinated Seeds

Seeds are randomly scattered in ℝd according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.

scholarly journals On the asymptotic behaviour of the jackknife for stochastic processes

1983 ◽  
Vol 27 (3) ◽  
pp. 329-337
Author(s):  
P.N. Kokic ◽  
N.C. Weber
Keyword(s):  
Asymptotic Behaviour ◽  
Stochastic Processes ◽  
Normal Distribution ◽  
Renewal Processes ◽  
Underlying Process ◽  
Limiting Behaviour ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal ◽  
Jackknife Estimator

The limiting behaviour of the J∞ jackknife estimator for parameters associated with stochastic processes is shown to depend on the nature of the underlying process through the asymptotic behaviour of the estimator being jackknifed. In particular, the jackknifed versions of certain estimators associated with renewal processes are shown to have an asymptotic normal distribution.

scholarly journals On colored set partitions of type B n

2014 ◽  
Vol 12 (9) ◽  
Author(s):  
David Wang
Keyword(s):  
Generating Function ◽  
Asymptotic Expression ◽  
Type B ◽  
Set Partitions ◽  
Asymptotic Normal

AbstractGeneralizing Reiner’s notion of set partitions of type B n, we define colored B n-partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored B n-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored B n-partition. We find an asymptotic expression of the total number of colored B n-partitions up to an error of O(n −1/2log7/2 n], and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored B n-partitions.

scholarly journals On the Variety of Shapes on the Fringe of a Random Recursive Tree

2010 ◽  
Vol 47 (1) ◽  
pp. 191-200 ◽  
Author(s):  
Qunqiang Feng ◽  
Hosam M. Mahmoud
Keyword(s):  
Normal Distribution ◽  
Contraction Method ◽  
Tree Shape ◽  
Recursive Tree ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal ◽  
The Given ◽  
Fixed Tree

We consider a variety of subtrees of various shapes lying on the fringe of a recursive tree. We prove that (under suitable normalization) the number of isomorphic images of a given fixed tree shape on the fringe of the recursive tree is asymptotically Gaussian. The parameters of the asymptotic normal distribution involve the shape functional of the given tree. The proof uses the contraction method.

scholarly journals Decision-Making for the Lifetime Performance Index

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Basim S. O. Alsaedi ◽  
M. M. Abd El-Raouf ◽  
E. H. Hafez ◽  
Zahra Almaspoor ◽  
Osama Abdulaziz Alamri ◽  
...  
Keyword(s):  
Normal Distribution ◽  
Performance Index ◽  
Supplier Selection ◽  
Selection Criterion ◽  
Real Data ◽  
Censored Samples ◽  
Lifetime Performance ◽  
Asymptotic Normal Distribution ◽  
Lower Specification Limit ◽  
Asymptotic Normal

The purpose of this research is to develop a maximum likelihood estimator (MLE) for lifetime performance index CL for the parameter of mixture Rayleigh-Half Normal distribution (RHN) under progressively type-II right-censored samples under the constraint of knowing the lower specification limit (L). Additionally, we suggest an asymptotic normal distribution for the MLE for CL in order to construct a mechanism for evaluating products’ lifespan efficiency. We have specified all the steps to carry out the test. Additionally, not only does hypothesis testing successfully assess the lifetime performance of items, but it also functions as a supplier selection criterion for the consumer. Finally, we have added two real data examples as illustration examples. These two applications are provided to demonstrate how the results can be applied.

DRAWING MULTISETS OF BALLS FROM TENABLE BALANCED LINEAR URNS

2013 ◽  
Vol 27 (2) ◽  
pp. 147-162 ◽  
Author(s):  
Hosam M. Mahmoud
Keyword(s):  
Normal Distribution ◽  
Case Analysis ◽  
Linear Case ◽  
Urn Models ◽  
Average Case Analysis ◽  
Average Case ◽  
Probability Structure ◽  
Nondegenerate Case ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal

We investigate the evolution of an urn of balls of two colors, where one chooses a pair of balls and observes rules of ball addition according to the outcome. A nonsquare ball addition matrix of the form $\left( \matrix{a & b \cr c & d \cr e & f}\right)$ corresponds to such a scheme, in contrast to pólya urn models that possess a square ball addition matrix. We look into the case of constant row sum (the so-called balanced urns) and identify a linear case therein. Two cases arise in linear urns: the nondegenerate and the degenerate. Via martingales, in the nondegenerate case one gets an asymptotic normal distribution for the number of balls of any color. In the degenerate case, a simpler probability structure underlies the process. We mention in passing a heuristic for the average-case analysis for the general case of constant row sum.

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