A new proof is given of the complete convergence theorem for the $d$-dimensional basic contact process provided that the infection parameter is larger than the critical value in the one-dimensional case. This proof is much more elementary than the known one since it does not depend on exponential estimates and does not use the subadditive ergodic theory in the extension from one to more dimensions.
Roberto Henrique Schonmann. "A New Proof of the Complete Convergence Theorem for Contact Processes in Several Dimensions with Large Infection Parameter." Ann. Probab. 15 (1) 382 - 387, January, 1987. https://doi.org/10.1214/aop/1176992276