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.

On a random mapping (T, Pj )

1984 ◽  
Vol 21 (01) ◽  
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.

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.

Epidemic processes on digraphs of random mappings

1999 ◽  
Vol 36 (3) ◽  
pp. 780-798 ◽  
Author(s):  
Jerzy Jaworski
Keyword(s):  
Probability Distributions ◽  
Exact Probability ◽  
Threshold Functions ◽  
Random Mapping ◽  
Finite Set ◽  
Random Mappings

A random mapping (Tn;q) of a finite set V, V = {1,2,…,n}, into itself assigns independently to each i ∊ V its unique image j ∊ V with probability q if i = j and with probability P = (1-q)/(n−1) if i ≠ j. Three versions of epidemic processes on a random digraph GT representing (Tn;q) are studied. The exact probability distributions of the total number of infected elements as well as the threshold functions for these epidemic processes are determined.

Epidemic processes on digraphs of random mappings

1999 ◽  
Vol 36 (03) ◽  
pp. 780-798 ◽  
Author(s):  
Jerzy Jaworski
Keyword(s):  
Probability Distributions ◽  
Exact Probability ◽  
Threshold Functions ◽  
Random Mapping ◽  
Finite Set ◽  
Random Mappings

A random mapping (T n ;q) of a finite set V, V = {1,2,…,n}, into itself assigns independently to each i ∊ V its unique image j ∊ V with probability q if i = j and with probability P = (1-q)/(n−1) if i ≠ j. Three versions of epidemic processes on a random digraph G T representing (T n ;q) are studied. The exact probability distributions of the total number of infected elements as well as the threshold functions for these epidemic processes are determined.

Generalized allocation scheme with cell occupancies from a fixed finite set

2020 ◽  
Vol 30 (5) ◽  
pp. 347-352
Author(s):  
Aleksandr N. Timashev
Keyword(s):  
Explicit Form ◽  
Asymptotic Estimates ◽  
Allocation Scheme ◽  
Number Of Components ◽  
Finite Set ◽  
Probabilistic Character ◽  
Normal Law ◽  
Generalized Allocation Scheme ◽  
Positive Integers ◽  
Number Of Particles

AbstractWe consider a generalized scheme of allocation of n particles (elements) over unordered cells (components) under the condition that the number of particles in each cell belongs to a fixed finite set A of positive integers. A new asymptotic estimates for the total number In(A) of variants of allocations of n particles are obtained under some conditions on the set A; these estimates have an explicit form (up to equivalence). Some examples of combinatorial-probabilistic character are given to illustrate by particular cases the notions introduced and results obtained. For previously known theorems on the convergence to the normal law of the total number of components and numbers of components with given cardinalities the norming parameters are obtained in the explicit form without using roots of algebraic or transcendent equations.

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 (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.

Sign in / Sign up

Export Citation Format

Share Document