## 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>