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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 $$J_1$$ J 1 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 $$\{S_n(t)=\sum _{i=1}^{[nt]}X_i\}_{t\ge 0}$$ { S n ( t ) = ∑ i = 1 [ n t ] X i } t ≥ 0 , suitably normalized and centered, associated with a sequence $$(X_i)_{i\ge 1}$$ ( X i ) i ≥ 1 of i.i.d. regularly varying elements in $${\mathbb {D}}$$ D .

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