Home Images of a finite set under iterations of two random dependent mappings
Article
Licensed
Unlicensed Requires Authentication

Images of a finite set under iterations of two random dependent mappings

  • Aleksandr A. Serov EMAIL logo
Published/Copyright: July 15, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Discrete Mathematics and Applications
From the journal Discrete Mathematics and Applications Volume 26 Issue 3

Abstract

Let N be a set of N elements and F1,G1,F2,G2,… be a sequence of independent pairs of random dependent mappings N→N such that Fk and Gk are random equiprobable mappings and P{Fk(x)=Gk(x)}=α for all x∈N and k = 1, 2, … For a subset S0⊂N,|S0|=n, we consider a sequences of its images Sk=Fk(…F2(F1(S0))…), Tk=Gk(…G2(G1(S0))…), k = 1, 2 …, and a sequences of their unions Sk∪Tk and intersections Sk∩Tk, k = 1, 2 … We obtain two-sided inequalities for M|Sk∪Tk| and M|Sk∩Tk| such that upper and lower bounds are asymptotically equivalent if N,n,k→∞, nk=o(N) and α=O1N.

Keywords: random mappings of finite sets; joint distributions; iterations of random mappings; Markov chain

Note: Originally published in Diskretnaya Matematika (2015) 27, No_4, 133–140 (in Russian).

Funding source: Russian Science Foundation

Award Identifier / Grant number: 14-50-00005

Funding statement: This work was supported by the Russian Science Foundation under grant no. 14-50-00005

References

[1] Kolchin V.F., Random mappings, Optimization Software Inc. Publications Division, N.Y., 1986, 207 pp.Search in Google Scholar

[2] Kolchin V. F., Sevastyanov B. A., Chistyakov V. P., Random allocations, V. H. Winston & Sons, Washington, 1978, 262 pp.Search in Google Scholar

[3] Gantmacher F.R., The Theory of Matrices, 1, 2, Chelsea Publishing Company. NewYork, 1959, vol. 1: x+374 pp., vol. 2: x+277 pp.Search in Google Scholar

[4] Zubkov A. M., Serov A. A., “Images of subset of finite set under iterations of random mappings”, Discrete Math. Appl., 25:3 (2015), 179–185.10.1515/dma-2015-0017Search in Google Scholar

[5] Stepanov V. E., “On the distribution of the number of vertices in strata of a random tree”, Theory Probab. Appl., 14:1 (1969), 64–77.10.1137/1114007Search in Google Scholar

[6] Dalal A., Schmutz E. “Compositions of random functions on a finite set”, Electr. J. Comb., 9:R26 (2002).10.37236/1642Search in Google Scholar

[7] Flajolet P., Odlyzko A. M., “Random mapping statistics”, Lect. Notes Comput. Sci., 434, 1990, 329–354.10.1007/3-540-46885-4_34Search in Google Scholar

[8] Goh W. M. Y., Hitczenko P., Schmutz E., “Iterating random functions on a finite set”, 2014, 7 pp., arXiv: arXiv:math/0207276v2.Search in Google Scholar

[9] Kingman J. F. C., “The coalescent”, Stoch.Proc. Appl., 13 (1982), 235–248.10.1016/0304-4149(82)90011-4Search in Google Scholar

Received: 2015-10-30
Published Online: 2016-7-15
Published in Print: 2016-7-1

© 2016 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Research Article
  2. Finite automata and numbers
  3. Research Article
  4. On coincidences of tuples in a binary tree with random labels of vertices
  5. Research Article
  6. On elementary word functions obtained by bounded prefix concatenation
  7. Research Article
  8. Functions without short implicents. Part II: Construction
  9. Research Article
  10. Images of a finite set under iterations of two random dependent mappings
  11. Research Article
  12. Extinction of decomposable branching processes
https://doi.org/10.1515/dma-2016-0015
Keywords for this article
random mappings of finite sets; joint distributions; iterations of random mappings; Markov chain

Articles in the same Issue

  1. Research Article
  2. Finite automata and numbers
  3. Research Article
  4. On coincidences of tuples in a binary tree with random labels of vertices
  5. Research Article
  6. On elementary word functions obtained by bounded prefix concatenation
  7. Research Article
  8. Functions without short implicents. Part II: Construction
  9. Research Article
  10. Images of a finite set under iterations of two random dependent mappings
  11. Research Article
  12. Extinction of decomposable branching processes
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 27.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/dma-2016-0015/html
Scroll to top button