This paper is mainly concerned with Riesz spaces valued measures and processes. The author first studied the lattice properties of processes of vector measures valued in order complete Riesz spaces and obtained the Riesz decomposition theorem for o-amarts (order asymptotic martingales) of vector measures. Next he investigated the properties of processes of Banach lattice valued random variables, especially, o-amarts of random variables, whose notion is stable under the lattice operation in comparison with the other notions of amarts (uniform amarts, strong amarts and weak amarts) and shares most of the properties of the others under suitable conditions. He proved a decomposition theorem of o-amarts of random variables and also gave a characterization theorem of these analogous to the others. Moreover he gave necessary and sufficient conditions on the Banach lattice and the ''right'' boundedness conditions on the processes of random variables to ensure the weak convergence, the strong convergence and the order convergence of these processes, which is the aim of this paper.