2018/09/06 by Raluca M. Balan, Balan, Raluca M., Becem Saidani +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1809.02103
openalex publication_date 2018/09/06 · openalex created_date 2022/08/03 · openalex updated_date 2026/07/28
In this article, we introduce an infinite-dimensional analogue of the\n\α-stable L 'evy motion, defined as a L 'evy process Z= Z(t) t \≥\n0 with values in the space mathbbD of c `adl `ag functions on [0,1],\nequipped with Skorokhod's J1 topology. For each t \≥ 0, Z(t) is an\n\α-stable process with sample paths in mathbbD, denoted by\n Z(t,s) s\∈ [0,1]. Intuitively, Z(t,s) gives the value of the process\nZ at time t and location s in space. This process is closely related to\nthe concept of regular variation for random elements in mathbbD introduced\nin de Haan and Lin (2001) and Hult and Lindskog (2005). We give a construction\nof Z based on a Poisson random measure, and we show that Z has a\nmodification whose sample paths are c `adl `ag functions on [0,\∞) with\nvalues in mathbbD. Finally, we prove a functional limit theorem which\nidentifies the distribution of this modification as the limit of the partial\nsum sequence Sn(t)=\∑i=1[nt]Xi t\≥ 0, suitably normalized\nand centered, associated to a sequence (Xi)i\≥ 1 of i.i.d. regularly\nvarying elements in mathbbD.\n