Can intermittent long-range jumps of a random walker compensate for lethargy?

We study the dynamics of a lazy random walker who is inactive for extended times and tries to make up for her laziness with very large jumps. She remains in a condition of rest for a time tau derived from a waiting-time distribution. psi (tau) alpha 1/tau(mu W), with mu(W) < 2, thereby making jumps only from time to time from a position x to a position x' of a one-dimensional path. However, when the random walker jumps, she moves by quantities l = vertical bar x - x'vertical bar derived randomly from a distribution pi(l) alpha 1/l(mu xi), with mu(xi) > 1. The most convenient choice to make up for the random walker laziness would be to select mu(xi) < 3, which in the ordinary case mu(W) > 2 would produce Levy flights with scaling delta = 1/(mu(xi) - 1) and consequently super-diffusion. According to the Sparre Andersen theorem, the distribution density of the first times to go from x(A) to x(B) > x(A) has the inverse power law form f (t) alpha 1/t(mu)FPT with mu(FPT) = mu(SA) = 1.5. We find the surprising result that there exists a region of the phase space (mu(xi), mu(W)) with mu(W) < 2, where mu(FPT) > mu(SA) and the lazy walker compensates for her laziness. There also exists an extended region breaking the Sparre Andersen theorem, where the lazy runner cannot compensate for her laziness. We make conjectures concerning the possible relevance of this mathematical prediction, supported by numerical calculations, for the problem of animal foraging.