We present a model of one-dimensional symmetric and asymmetric random walks. The model is applied to an experiment studying fluid transport in a rapidly rotating annulus. In the model, random walkers alternate between flights (steps of constant velocity) and sticking (pauses between flights). Flight time and sticking time probability distribution functions (PDFs) have power law decays: P(t) ~ t^(-u) and t^(-v) for flights and sticking, respectively. We calculate the dependence of the variance exponent d (sigma^2 ~ t^d) on the PDF exponents u and v. For a broad distribution of flight times (u < 3), the motion is superdiffusive (1 < d < 2), and the PDF has a divergent second moment, i.e., it is a Levy distribution. For a broad distribution of sticking times (v < 3), either superdiffusion or subdiffusion (d < 1) can occur, with qualitative differences between symmetric and asymmetric random walks. For narrow PDFs (u > 3, v > 3), normal diffusion (d=1) is recovered. Predictions of the model are related to experimental observations of transport in a rotating annulus. The Eulerian velocity field is chaotic, yet it is still possible to distinguish between well-defined sticking events (particles trapped in vortices) and flights (particles making long excursions in a jet). The distribution of flight lengths is well described by a power law with a divergent second moment (Levy distribution). The observed transport is strongly asymmetric and is well described by the proposed model.