Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes

► The subordinated arithmetic Brownian motion with tempered stable and inverse tempered stable subordinators is examined. ► We compare the main statistical properties of considered systems. ► We propose the parameter's estimation procedures. ► We analyze the ensemble averaged MSD as a tool of recognition between subdiffusive and diffusive models. ► We examine real environmental data by using the considered processes.