Sample path large deviations for levy processes and random walks with regularly varying increments

Let X be a Levy process with regularly varying Levy measure ν. We obtain sample-path large deviations for scaled processes. X_n(t) X(nt)/n and obtain a similar result for random walks with regularly varying increments. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but a full large deviation principle does not hold.