In this paper, we consider the following linear process Xn = ∞∑ i=−∞ ciξn−i, n ∈ Z, and establish the central limit theorem of the randomly indexed partial sums Sνn := X1 + · · · + Xνn , where {ci; i ∈ Z} is a sequence of real numbers, {ξn; n ∈ Z} is a stationary m-dependent sequence and {νn; n ≥ 1} is a sequence of positive integer valued random variables. In addition, in order to show the main result, we prove the central limit theorems for randomly indexed m-dependent random variables, which improve some known results