A note on characterizing tightness of random sets of càdlàg paths

We record two natural tightness criteria for random sets of càdlàg paths, within the state space introduced by Etheridge et al. in [4]. This state space is the equivalent, for càdlàg paths, of the (continuous path) state space introduced for the Brownian web by Fontes et al. in [6].

We discuss some implications of these criteria, in particular for cases of coalescing random walkers with heavy tailed jumps, in which tightness fails.