We prove functional limit theorems for stochastic processes which have clusters of large values which, when summed and suitably normalised in time and space, get collapsed in a jump of the limiting process observed at the same time point. In order to keep track of the clustering information, which gets lost in the usual Skorohod topologies in the space of c\`adl`ag functions, we introduce a new space specially designed for that purpose which generalises the already more general spaces introduced by Whitt. Our main applications are to hyperbolic and non-uniformly expanding dynamical systems with heavy tailed observable functions maximised at dynamically linked maximal sets (such as periodic points). We also study limits of extremal processes and record times point processes for observables not necessarily heavy tailed. The applications studied include hyperbolic systems such as Anosov diffeomorphisms, but also non-uniformly expanding maps such as maps with critical points of Benedicks-Carleson type or indifferent fixed points such as Manneville-Pomeau maps. The main tool is a limit theorem for point processes with decorations derived from a bi-infinite sequence which we call the piling process, designed to keep the clustering information. This piling process is based on the tail process defined in the regularly varying context rather than the general stationary stochastic processes we consider here.
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