Factorial moments for random mappings by means of indicator variables

1993 ◽  
Vol 30 (1) ◽  
pp. 167-174 ◽  
Author(s):  
Brian J. English
Keyword(s):  
Factorial Moments ◽  
Number Of Components ◽  
Random Mapping ◽  
Indicator Variables ◽  
Random Mappings

A simple identity for the incomplete factorial of sums of zero-one variables is exploited to provide the factorial moments of the number of components and the number of cyclical elements of the random mapping (T, {pi}) considered by Ross (1981).

Factorial moments for random mappings by means of indicator variables

1993 ◽  
Vol 30 (01) ◽  
pp. 167-174
Author(s):  
Brian J. English
Keyword(s):  
Factorial Moments ◽  
Number Of Components ◽  
Random Mapping ◽  
Indicator Variables ◽  
Random Mappings

A simple identity for the incomplete factorial of sums of zero-one variables is exploited to provide the factorial moments of the number of components and the number of cyclical elements of the random mapping (T, {pi }) considered by Ross (1981).

Functionals of random mappings: exact and asymptotic results

1991 ◽  
Vol 23 (3) ◽  
pp. 437-455 ◽  
Author(s):  
P. J. Donnelly ◽  
W. J. Ewens ◽  
S. Padmadisastra
Keyword(s):  
Frequency Spectrum ◽  
Exact Results ◽  
Asymptotic Results ◽  
Number Of Components ◽  
Random Mapping ◽  
Random Mappings ◽  
Simulation Results ◽  
Central Tool

A random mapping partitions the set {1, 2, ···, m} into components, where i and j are in the same component if some functional iterate of i equals some functional iterate of j. We consider various functionals of these partitions and of samples from it, including the number of components of ‘small' size and of size O(m) as m → ∞the size of the largest component, the number of components, and various symmetric functionals of the normalized component sizes. In many cases exact results, while available, are uniformative, and we consider various approximations. Numerical and simulation results are also presented. A central tool for many calculations is the ‘frequency spectrum', both exact and asymptotic.

Functionals of random mappings: exact and asymptotic results

1991 ◽  
Vol 23 (03) ◽  
pp. 437-455 ◽  
Author(s):  
P. J. Donnelly ◽  
W. J. Ewens ◽  
S. Padmadisastra
Keyword(s):  
Frequency Spectrum ◽  
Exact Results ◽  
Asymptotic Results ◽  
Number Of Components ◽  
Random Mapping ◽  
Random Mappings ◽  
Simulation Results ◽  
Central Tool

A random mapping partitions the set {1, 2, ···,m} into components, whereiandjare in the same component if some functional iterate ofiequals some functional iterate ofj. We consider various functionals of these partitions and of samples from it, including the number of components of ‘small' size and of sizeO(m) asm→ ∞the size of the largest component, the number of components, and various symmetric functionals of the normalized component sizes. In many cases exact results, while available, are uniformative, and we consider various approximations. Numerical and simulation results are also presented. A central tool for many calculations is the ‘frequency spectrum', both exact and asymptotic.

scholarly journals Predecessors and Successors in Random Mappings with Exchangeable In-Degrees

2013 ◽  
Vol 50 (3) ◽  
pp. 721-740 ◽  
Author(s):  
Jennie C. Hansen ◽  
Jerzy Jaworski
Keyword(s):  
Asymptotic Behaviour ◽  
Preferential Attachment ◽  
Degree Sequence ◽  
Discrete Distributions ◽  
Threshold Behaviour ◽  
Mapping Model ◽  
Random Mapping ◽  
Exact Distributions ◽  
Random Mappings ◽  
Expected Values

In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, TnD̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂1,D̂2,…,D̂n). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂1,D̂2,…,D̂n. As an application of these results, we consider two special examples of TnD̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

The distribution and moments of the number of components of a random function

1990 ◽  
Vol 27 (01) ◽  
pp. 202-207 ◽  
Author(s):  
Joseph Kupka
Keyword(s):  
Probability Distribution ◽  
Random Function ◽  
Simple Formula ◽  
Computer Calculation ◽  
Factorial Moments ◽  
Number Of Components ◽  
Mean And Variance ◽  
The Mean

A relatively simple formula is presented for the probability distribution of the number K of components of a random function. This formula facilitates the (computer) calculation of the factorial moments of K and yields new expressions for the mean and variance of K.

Random mappings with an attracting center: Lagrangian distributions and a regression function

1990 ◽  
Vol 27 (03) ◽  
pp. 622-636 ◽  
Author(s):  
Sven Berg ◽  
Ljuben Mutafchiev
Keyword(s):  
Regression Function ◽  
Central Component ◽  
Conditional Distributions ◽  
Mapping Model ◽  
Random Mapping ◽  
Random Mappings ◽  
The Relationship

For a random mapping model with a single attracting center (Stepanov (1971)) we study the relationship between the sizes of the central tree, the adjacent points, and the free points. Joint, marginal and conditional distributions are shown to be of well-known Lagrangian type. Exact and asymptotic moment properties are investigated with the aid of Riordan's (1968) Abel identities. In particular, the relationship between the size of the central tree and that of the central component is discussed in terms of a regression function.

On a random mapping (T, Pj)

1984 ◽  
Vol 21 (1) ◽  
pp. 186-191 ◽  
Author(s):  
Jerzy Jaworski
Keyword(s):  
Fundamental Lemma ◽  
Number Of Components ◽  
Random Mapping ◽  
Finite Set

A random mapping (T,Pj) of a finite set V into itself is studied. We give a new proof of the fundamental lemma of [6]. Our method leads to the derivation of several results which cannot be deduced from [6]. In particular we determine the distribution of the number of components, cyclical points and ancestors of a given point.

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