Stable Lévy Motion with Values in the Skorokhod Space: Construction and Approximation

In this article, we introduce an infinite-dimensional analogue of the $$\alpha $$α-stable Lévy motion, defined as a Lévy process $$Z=\{Z(t)\}_{t \ge 0}$$Z={Z(t)}t≥0 with values in the space $${\mathbb {D}}$$D of càdlàg functions on [0, 1], equipped with Skorokhod’s $$J_1$$J1 topology. For each $$t \ge 0$$t≥0, Z(t) is an $$\alpha $$α-stable process with sample paths in $${\mathbb {D}}$$D, denoted by $$\{Z(t,s)\}_{s\in [0,1]}$${Z(t,s)}s∈[0,1]. Intuitively, Z(t, s) gives the value of the process Z at time t and location s in space. This process is closely related to the concept of regular variation for random elements in $${\mathbb {D}}$$D introduced in de Haan and Lin (Ann Probab 29:467–483, 2001) and Hult and Lindskog (Stoch Proc Appl 115:249–274, 2005). We give a construction of Z based on a Poisson random measure, and we show that Z has a modification whose sample paths are càdlàg functions on $$[0,\infty )$$[0,∞) with values in $${\mathbb {D}}$$D. Finally, we prove a functional limit theorem which identifies the distribution of this modification as the limit of the partial sum sequence $$\{S_n(t)=\sum _{i=1}^{[nt]}X_i\}_{t\ge 0}$${Sn(t)=∑i=1[nt]Xi}t≥0, suitably normalized and centered, associated with a sequence $$(X_i)_{i\ge 1}$$(Xi)i≥1 of i.i.d. regularly varying elements in $${\mathbb {D}}$$D.