In the field of plasma physics, suprathermal ions are encountered e.g. in astrophysical jets, the solar wind, as well as fusion plasmas, where they originate from neutral beam injection or fusion reactions. One aspect of shared interest with astrophysical settings is their transport by different forms of plasma turbulence that may generally exhibit non-diffusive character. Therefore, fast ion cross-field transport by electrostatic turbulence is investigated on the TORoidal Plasma EXperiment (TORPEX), in the simple magnetized torus geometry. Lithium ions in the 30-70 eV range are injected toroidally into hydrogen plasmas, with electron temperatures of typically below 5 eV. Previous studies identified sub- to superdiffusive regimes of transport, depending on the fast ion energy and related gyro- and drift-averaging effects. The first part of this thesis investigates the phenomenon of local intermittency, as quantified by the skewness of fast ion time-series. A comprehensive data-set of fast ion time-series is presented to establish observations of intermittency across all non-diffusive transport regimes. Through the development of an experiment-based particle tracing algorithm, the physical picture of a small, but meandering fast ion beam is clarified and the generation of local intermittency demonstrated through synthetic time-series. Furthermore, an analytical model is presented to predict the skewness of such time-series solely based on their time-average value and two basic beam parameters. Its application to the experimental data shows very good agreement between the predicted and the measured skewness. These combined findings demonstrate conclusively how intermittency is generated across all transport regimes in our system, or by any type of meandering beam. The second part of this thesis advances the statistical description of non-diffusive fast ion transport through Fractional Diffusion Equations (FDEs). To improve upon physical shortcomings of models that assign distributions with infinite variance to the random jumps of particles, we utilize tempered Lévy distributions that exponentially truncate jumps beyond a chosen scale. The FDE and propagator of Truncated Asymmetrical Fractional Lévy Motion (TAFLM) are derived by judiciously adapting earlier path-integral methods. Being generally non-Gaussian at early times, the propagators converge arbitrarily slowly to Gaussians in the long-time limit, while preserving their overall non-diffusive behaviour. Very good agreement with fluid-tracer results from the Global Braginskii Solver is found in the most strongly spread, quasi-diffusive regime. Here, the finite domain size of the turbulent plasma structures can now take effect through the truncation scale. Further use of these statistical methods can be discussed across the wider field of bounded, non-diffusive transport.