Summary
Consider the rectangles of the first k(n) lines of length n in the right-upper integer-lattice, and suppose that its points are labelled randomly by the numbers 1, 2,..., m. The time i is called coincidence if the points (i, 1), (i, 2),..., (i, k(n)) are labelled identically. Asymptotic properties of the longest run of coincidences are discussed under different conditions on k(n).
The results are related to a problem of P. Révész: If the points of an n ×n integer-lattice are coloured red and white randomly, what is the largest area of rectangles with red points only.
A conjecture is formulated, indicating some peculiar number-theoretic characteristics of some limit-relations in this area.
Article PDF
Similar content being viewed by others
References
Erdös, P., Révész, P.: On the length of the longest head-run. In: “Topics in Information Theory”, ed.: Csíszár, Colloquia Mathematica Societatis Janos Bolyai, Budapest, 1975, pp. 219–228
Petrov, V.V.: On the probabilities of large deviations for sums of independent random variables. Teor. Verojatnost i Primenen 10 (1965) pp. 310–322, Theor. Probab. Appl. 10, 287–298 (1965)
Révész, R.: A Characterization of the Asymptotic Properties of a Stochastics Process by four Classes of Deterministic Curves. [To be published]
Shannon, C.: Communication theory of secrecy systems. Bell System Techn. Journal 28, 656–715 (1949)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nemetz, T., Kusolitsch, N. On the longest run of coincidences. Z. Wahrscheinlichkeitstheorie verw Gebiete 61, 59–73 (1982). https://doi.org/10.1007/BF00537225
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00537225