## A triangle inequality for covariances of binary FKG random variables.(English)Zbl 0841.60011

For binary random variables $$\sigma_1, \dots, \sigma_n$$ whose joint distribution is denoted by $$\mu$$, the FKG condition reads $$\mu(\alpha \vee \beta) \mu(\alpha \wedge \beta) \geq \mu(\alpha)\mu(\beta)$$ where the $$i$$-th coordinate of the configuration $$(\alpha \vee \beta)$$ is the minimum of the $$i$$-th coordinates of $$\alpha$$ and $$\beta$$ and similarly the $$i$$-th coordinate of the configuration $$(\alpha \wedge \beta)$$ is the maximum of the $$i$$-th coordinates of $$\alpha$$ and $$\beta$$. The authors show that for random variables satisfying the FKG condition the following inequality holds $\text{Var}(\sigma_j) \text{Cov}(\sigma_i, \sigma_k) \geq \text{Cov}(\sigma_i, \sigma_j) \text{Cov}(\sigma_j,\sigma_k).$

### MSC:

 60E15 Inequalities; stochastic orderings 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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