Non-Gaussian fluctuations of randomly trapped random walks

In this paper we consider the one-dimensional, biased, randomly trapped random walk when the trapping times have infinite variance. We prove sufficient conditions for the suitably scaled walk to converge to a transformation of a stable L'{e}vy process. As our main motivation, we apply subsequential versions of our results to biased walks on subcritical Galton-Watson trees conditioned to survive. This confirms the correct order of the fluctuations of the walk around its speed for values of the bias that yield a non-Gaussian regime.