We prove that the position of a tagged particle in a stationary simple exclusion process satisfies a law of large numbers. For this purpose, we show the extremality of an invariant measure for the process "seen" from the tagged particle, and we use the ergodicity properties of the initial process.
"A Limit Theorem for the Position of a Tagged Particle in a Simple Exclusion Process." Ann. Probab. 15 (1) 375 - 381, January, 1987. https://doi.org/10.1214/aop/1176992275