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

In this article, we introduce an infinite-dimensional analogue of the α -stable Lévy motion, defined as a Lévy process Z = { Z ( t ) } t ≥ 0 with values in the space D of càdlàg functions on [0, 1], equipped with Skorokhod’s J 1 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 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 , ∞ ) 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 { S n ( t ) = ∑ i = 1 [ n t ] X i } t ≥ 0 , suitably normalized and centered, associated with a sequence ( X i ) i ≥ 1 of i.i.d. regularly varying elements in D .