Regularly varying random variables play an important role in probability theory, being used as models for heavy-tailed observations (observations which may assume extreme values with high probability). With the rapid advancement of technology, data is no longer observed at fixed moments of time, but continuously over a fixed interval in time or space (which we may identify with the interval [0, 1]). If this measurement is expected to exhibit a sudden drop or increase over this fixed interval, then an appropriate model for it could be a random element in an infinite dimensional space, such as the Skorohod space D = D([0, 1]) of cadlag functions on [0, 1] (i.e. right-continuous functions with left limits).
In this talk, we examine the macroscopic limit (as time gets large) of the partial sum sequences associated to i.i.d. regularly varying elements with values in D. This limit is an interesting object in itself, which we call a “Dvalued stable Levy motion”. Our methods were inspired by the construction of the classical stable Levy motion (in dimension d), and of its approximation by partial sums of i.i.d. regularly varying vectors, as presented in the monograph Resnick (2007). We extend these two results to the infinite-dimensional setting, using the concept of regular variation for random elements in D introduced by de Haan and Lin (2001), and developed further by Hult and Lindskog (2005). The approximation result is an extension to functional convergence of a limit theorem obtained by Roueff and Soulier (2015).
This talk is based on joint work with Becem Saidani.