A joint generalization of Harris-Spitzer's and Szatzschneider's one-dimensional collision models is given and for the path of a given particle an approximation is obtained.
Peter Major. Domokos Szasz. "On the Effect of Collisions on the Motion of an Atom in $R^1$." Ann. Probab. 8 (6) 1068 - 1078, December, 1980. https://doi.org/10.1214/aop/1176994568