# Random partitions tending to identity

The following is a doubt that I have about the definition of sequence of random partitions tending to the identity. What does the author mean by tending to the identity? It is obvious from the context that the author refers to a sequence of random partitions such that $\lim_{n \to \infty} \Vert \sigma_{n} \Vert = 0$, but tending to identity is not a clear and intuitive definition.

I would really appreciate if someone could help with the definition, please.

By the way:

$\mathbb{L}$ is the space of adapted caglad processes

$\mathbb{D}$ is the space of adapted cadlag processes

$S$ is the space of simple predictable processes

• The basic idea is that if the largest of the stopping times goes to infinity and greatest distance between them goes to zero, then a process sampled at the partition will look more and more like the original process, so the partition "tends to the identity" in the sense that the sampling operation it represents tends to the identity, i. e. sampling "at infinity" recovers the original process (obviously speaking non-rigorously). – joriki Jul 18 '18 at 17:26