Stable Levy motion with values in the Skorokhod space: construction and approximatio

In this article, we introduce an infinite-dimensional analogue of the α-stable Levy motion, defined as a Levy process Z = {Z(t)}t≥0 with values in the space D of cadlag functions on [0, 1], equipped with Skorokhod’s J1 topology. For each t ≥ 0, Z(t) is an α-stable process with sample paths in D, denoted by {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 D . We give a construction of Z based on a Poisson random measure, and we show that Z has a modification whose sample paths are cadlag functions on [0, ∞) with values in D. Finally, we prove a functional limit theorem which identifies the distribution of this modification as the limit of the partial sum sequence {Sn(t) = P[nt] i=1 Xi}t≥0, suitably normalized and centered, associated to a sequence (Xi)i≥1 of i.i.d. regularly varying elements in D.