Deterministic first-long-block extraction

Ladder times at non-jump levels, the first long block's endpoints, and its stopped translated path converge under J1 convergence of skip-free walks to a good extraction path. The proof uses joint continuity of moving stopped cuts and an exact one-mesh terminal-step comparison.

References

• T. Duquesne and J.-F. Le Gall, Random Trees, Lévy Processes and Spatial Branching Processes, Astérisque 281 (2002), Chapter 2.
• P. Billingsley, Convergence of Probability Measures, 2nd ed., Wiley, 1999, Chapter 3 (the Skorokhod topology).