We consider a particle moving in one dimension, its velocity being a reversible diffusion process, with constant diffusion coefficient, of which the invariant measure behaves like (1 + |v|) −β for some β > 0. We prove that, under a suitable rescaling, the position process resembles a Brownian motion if β ≥ 5, a stable process if β ∈ [1, 5) and an integrated symmetric Bessel process if β ∈ (0, 1). The critical cases β = 1 and β = 5 require special rescalings. We recover some results of [24, 10, 19] and [1] with an alternative approach.