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Inequalities with applications to percolation and reliability

Inequalities with applications to percolation and reliability <jats:p>A probability measure <jats:italic>μ</jats:italic> on ℝ<jats:sup>n</jats:sup><jats:sub>+</jats:sub> is defined to be strongly new better than used (SNBU) if <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0021900200029326_inline1" /> for all increasing subsets <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0021900200029326_inline2" />. For <jats:italic>n</jats:italic> = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if <jats:italic>μ</jats:italic><jats:sub>1</jats:sub>, <jats:italic>μ</jats:italic><jats:sub>2</jats:sub>, ···, <jats:italic>μ<jats:sub>n</jats:sub></jats:italic> are NBU probability measures on ℝ<jats:sub>+</jats:sub>, then the product-measure <jats:italic>μ</jats:italic> = <jats:italic>μ</jats:italic> × <jats:italic>μ</jats:italic><jats:sub>2</jats:sub> × ··· × <jats:italic>μ<jats:sub>n</jats:sub></jats:italic> on ℝ<jats:sup>n</jats:sup><jats:sub>+</jats:sub> is SNBU. A discrete analog (i.e., with N instead of ℝ<jats:sub>+</jats:sub>) also holds.</jats:p><jats:p>Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.</jats:p> http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Probability CrossRef

Inequalities with applications to percolation and reliability

Journal of Applied Probability , Volume 22 (3): 556-569 – Sep 1, 1985

Inequalities with applications to percolation and reliability


Abstract

<jats:p>A probability measure <jats:italic>μ</jats:italic> on ℝ<jats:sup>n</jats:sup><jats:sub>+</jats:sub> is defined to be strongly new better than used (SNBU) if <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0021900200029326_inline1" /> for all increasing subsets <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0021900200029326_inline2" />. For <jats:italic>n</jats:italic> = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if <jats:italic>μ</jats:italic><jats:sub>1</jats:sub>, <jats:italic>μ</jats:italic><jats:sub>2</jats:sub>, ···, <jats:italic>μ<jats:sub>n</jats:sub></jats:italic> are NBU probability measures on ℝ<jats:sub>+</jats:sub>, then the product-measure <jats:italic>μ</jats:italic> = <jats:italic>μ</jats:italic> × <jats:italic>μ</jats:italic><jats:sub>2</jats:sub> × ··· × <jats:italic>μ<jats:sub>n</jats:sub></jats:italic> on ℝ<jats:sup>n</jats:sup><jats:sub>+</jats:sub> is SNBU. A discrete analog (i.e., with N instead of ℝ<jats:sub>+</jats:sub>) also holds.</jats:p><jats:p>Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.</jats:p>

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Publisher
CrossRef
ISSN
0021-9002
DOI
10.2307/3213860
Publisher site
See Article on Publisher Site

Abstract

<jats:p>A probability measure <jats:italic>μ</jats:italic> on ℝ<jats:sup>n</jats:sup><jats:sub>+</jats:sub> is defined to be strongly new better than used (SNBU) if <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0021900200029326_inline1" /> for all increasing subsets <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0021900200029326_inline2" />. For <jats:italic>n</jats:italic> = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if <jats:italic>μ</jats:italic><jats:sub>1</jats:sub>, <jats:italic>μ</jats:italic><jats:sub>2</jats:sub>, ···, <jats:italic>μ<jats:sub>n</jats:sub></jats:italic> are NBU probability measures on ℝ<jats:sub>+</jats:sub>, then the product-measure <jats:italic>μ</jats:italic> = <jats:italic>μ</jats:italic> × <jats:italic>μ</jats:italic><jats:sub>2</jats:sub> × ··· × <jats:italic>μ<jats:sub>n</jats:sub></jats:italic> on ℝ<jats:sup>n</jats:sup><jats:sub>+</jats:sub> is SNBU. A discrete analog (i.e., with N instead of ℝ<jats:sub>+</jats:sub>) also holds.</jats:p><jats:p>Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.</jats:p>

Journal

Journal of Applied ProbabilityCrossRef

Published: Sep 1, 1985

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