  Time-changed stochastic processes have attracted great attention and wide
interests due to their extensive applications, especially in financial time
series, biology and physics. This paper pays attention to a special stochastic
process, tempered fractional Langevin motion, which is non-Markovian and
undergoes ballistic diffusion for long times. The corresponding time-changed
Langevin system with inverse $\beta$-stable subordinator is discussed in
detail, including its diffusion type, moments, Klein-Kramers equation, and the
correlation structure. Interestingly, this subordination could result in both
subdiffusion and superdiffusion, depending on the value of $\beta$. The
difference between the subordinated tempered fractional Langevin equation and
the subordinated Langevin equation with external biasing force is studied for a
deeper understanding of subordinator. The time-changed tempered fractional
Brownian motion by inverse $\beta$-stable subordinator is also considered, as
well as the correlation structure of its increments. Some properties of the
statistical quantities of the time-changed process are discussed, displaying
striking differences compared with the original process.
