Correlation inequalities and monotonicity properties of the Ruelle operator

In this paper, we provide sufficient conditions for the validity of the FKG Inequality, on Thermodynamic Formalism setting, for a class of eigenmeasures of the dual of the Ruelle operator. We use this correlation inequality to study the maximal eigenvalue problem for the Ruelle operator associated to low regular potentials. As an application, we obtain explicit upper bounds for the main eigenvalue (consequently, for the pressure) of the Ruelle operator associated to Ising models with a power law decay interaction energy.

Negative Dependence Through the FKG Inequality

We investigate random variables arising in occupancy problems, and show the variables to be negatively associated, that is, negatively dependent in a strong sense. Our proofs are based on the FKG correlation inequality, and they suggest a useful, general technique for proving negative dependence among random variables. We also show that in the special case of two binary random variables, the notions of negative correlation and negative association coincide.

LRU is better than FIFO under the independent reference model

Consider a two-level storage system operating with the least recently used (LRU) or the first-in, first-out (FIFO) replacement strategy. Accesses to the main storage are described by the independent reference model (IRM). Using the FKG inequality, we prove that the miss ratio for LRU is smaller than or equal to the miss ratio for FIFO.

LRU is better than FIFO under the independent reference model

Consider a two-level storage system operating with the least recently used (LRU) or the first-in, first-out (FIFO) replacement strategy. Accesses to the main storage are described by the independent reference model (IRM). Using the FKG inequality, we prove that the miss ratio for LRU is smaller than or equal to the miss ratio for FIFO.

Multivariate monotone likelihood ratio and uniform conditional stochastic order

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn. Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.

Multivariate monotone likelihood ratio and uniform conditional stochastic order

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn . Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.