Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-09T01:32:22.990Z Has data issue: false hasContentIssue false
  • English
  • Français

Arbitrary Threshold Widths for Monotone, Symmetric Properties

Part of: Distribution theory - Probability Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Raphaël Rossignol
Raphaël Rossignol*
Affiliation:
Université de Neuchâtel
*
∗ Postal address: Institut de Mathématiques, Faculté des Sciences, Université de Neuchâtel, 11 rue Emile Argand, case postal 158, 2009 Neuchâtel, Switzerland. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that there exist symmetric properties in the discrete n-cube whose threshold widths range asymptotically between 1/√n and 1/logn. These properties are built using a combination of failure sets arising in reliability theory. This combination of sets is simply called a product. Some general results on the threshold width of the product of two sets A and B in terms of the threshold locations and widths of A and B are provided.

Keywords

Threshold width0-1 lawparallel-series systemseries-parallel systemk-out-of-n system

MSC classification

Primary: 60F20: Zero-one laws
Secondary: 60E15: Inequalities; stochastic orderings 60K10: Applications (reliability, demand theory, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 1 , March 2007 , pp. 164 - 180
Copyright
Copyright © Applied Probability Trust 2007 

References

Barlow, R. and Proschan, F. (1965). Mathematical Theory of Reliability. John Wiley, New York.Google Scholar
Bobkov, S. (1997). An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Prob. 25, 206–214.CrossRefGoogle Scholar
Bobkov, S. and G{ötze, F.} (1999). Discrete isoperimetric and Poincaré-type inequalities. Prob. Theory Relat. Fields 114, 245–277.Google Scholar
Bollobás, B. (1985). Random Graphs. Academic Press, London.Google Scholar
Bollobás, B. et al. (2001). The scaling window of the 2-{SAT} transition. Random Structures Algorithms 18, 201–256.Google Scholar
Boucheron, S., Lugosi, G. and Massart, P. (2003). Concentration inequalities using the entropy method. Ann. Prob. 31, 1583–1614.CrossRefGoogle Scholar
Bourgain, J. et al. (1992). The influence of variables in product spaces. Israel J. Math. 77, 55–64.Google Scholar
Compton, K. (1989). 0–1 laws in logic and combinatorics. In Algorithms and Order, ed. Rival, I., Kluwer, Dordrecht, pp. 353–383.Google Scholar
Coupier, D., Desolneux, A. and Ycart, B. (2005). Image denoising by statistical area thresholding. J. Math. Imag. Vision 22, 183–197.Google Scholar
Creignou, N. and Daudé, H. (1999). Satisfiability threshold for random {XOR-CNF} formulas. Discrete Appl. Math. 96/97, 41–53.CrossRefGoogle Scholar
Friedgut, E. (1999). Sharp threshold of graph properties and the k-sat problem. J. Amer. Math. Soc. 12, 1017–1054.Google Scholar
Friedgut, E. and Kalai, G. (1996). Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124, 2993–3002.CrossRefGoogle Scholar
Grimmett, G. (1989). Percolation. Springer, New York.CrossRefGoogle Scholar
Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–30.CrossRefGoogle Scholar
Kołowrocki, K. (1994). Limit reliability functions of some series-parallel and parallel-series systems. Appl. Math. Comput. 62, 129–151.Google Scholar
Kołowrocki, K. (1994). On limiting forms of the reliability functions sequence of the parallel-series system. Appl. Math. Comput. Sci. 4, 575–590.Google Scholar
Kołowrocki, K. (2001). Asymptotic approach to reliability evaluation of a rope transportation system. Reliab. Eng. System Safety 71, 57–64.CrossRefGoogle Scholar
Kołowrocki, K. (2004). Reliability of Large Systems. Elsevier, Amsterdam.Google Scholar
Kontoleon, J. (1980). Reliability determination of a r-successive-out-of-n:f system. IEEE Trans. Reliab. 29, 437.CrossRefGoogle Scholar
Ledoux, M. (1996). On {Talagrand's} deviation inequalities for product measures. ESAIM Prob. Statist. 1, 63–87.Google Scholar
Margulis, G. (1974). Probabilistic characteristics of graphs with large connectivity. Prob. Pedarachi Inf. 10, 101–108 (in Russian).Google Scholar
Paroissin, C. and Ycart, B. (2003). Zero-one law for the non-availability of multistate repairable systems. Internat. J. Reliab. 10, 311–322.Google Scholar
Paroissin, C. and Ycart, B. (2004). Central limit theorem for hitting times of functionals of Markov Jump processes. ESAIM Prob. Statist. 8, 66–75.Google Scholar
Rossignol, R. (2005). Largeur de seuil dans les lois du zéro-un. , Université Paris 5. Available at http://www.math-info.univ-paris5.fr/∼rost/these_pdf.pdf.Google Scholar
Rossignol, R. (2006). Threshold for monotone symmetric properties through a logarithmic Sobolev inequality. Ann. Prob. 34, 1707–1725.Google Scholar
Talagrand, M. (1993). Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and {Margulis'} graph connectivity theorem. Geom. Funct. Anal. 3, 295–314.Google Scholar
Tillich, J.-P. and Zémor, G. (2000). Discrete isoperimetric inequalities and the probability of a decoding error. Combin. Prob. Comput. 9, 465–479.Google Scholar