  Two different ways of trimming the sample path of a stochastic process in
D[0, 1]: global ("trim as you go") trimming and record time ("lookback")
trimming are analysed to find conditions for the corresponding operators to be
continuous with respect to the (strong) J1-topology. A key condition is that
there should be no ties among the largest ordered jumps of the limit process.
As an application of the theory, via the continuous mapping theorem we prove
limit theorems for trimmed Levy processes, using the functional convergence of
the underlying process to a stable process. The results are applied to a
reinsurance ruin time problem.
