Power laws from randomly sampled continuous-time random walks

It has been shown by Reed that random-sampling a Wiener process x(t) at times T chosen out of an exponential distribution gives rise to power laws in the distribution P(x(T))∼x(T)-β. We show, both theoretically and numerically, that this power-law behaviour also follows by random-sampling Lévy flights (as continuous-time random walks), having Fourier distribution w^(k)=e-|k|α, with the exponent β=α.