Summary
Many of the classical submartingale inequalities, including Doob's maximal inequality and upcrossing inequality, are valid for sequences S j such that the (S j+1 -S j's are associated (positive mean) random variables, and for more general “demisubmartingales”. The demisubmartingale maximal inequality is used to prove weak convergence to the two-parameter Wiener process of the partial sum processes constructed from a stationary two-parameter sequence of associated random variables X ijwith \(\mathop \sum \limits_i \mathop \sum \limits_j {\text{Cov}}\left( {X_{{\text{oo}}} ,X_{ij} } \right) < \infty \).
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Alfred P. Sloan Research Fellow. Research supported in part by NSF Grants MCS 77-20683 and MCS 80-19384
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Newman, C.M., Wright, A.L. Associated random variables and martingale inequalities. Z. Wahrscheinlichkeitstheorie verw Gebiete 59, 361–371 (1982). https://doi.org/10.1007/BF00532227
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DOI: https://doi.org/10.1007/BF00532227