On Adam Jakubowski's approach to proving asymptotic results for regularly varying sequences

Speaker: Thomas Mikosch, University of Copenhagen, Denmark
Email: mikosch@math.ku.dk

Co-author: Olivier Wintenberger, Paris Dauphine, France

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In recent work [1, 4], an idea of Adam Jakubowski [2, 3] was used to prove in- finite stable limit theory and precise large deviation results for sums of strictly stationary regularly varying sequences. The idea of Jakubowski consists of approximating tail probabilities of distributions for such sums with increasing index by the corresponding quantities for sums with fixed index. This idea can also be made to work for Laplace functionals of point processes, the distribution function of maxima and the characteristic functions of partial sums of stationary sequences. In each of these situations, extremal dependence manifests in the appearance of suitable cluster indices (extremal index for maxima, cluster index for sums,...). The proposed method can be easily understood and has the potential to function as heuristics for proving limit results for weakly dependent heavy-tailed sequences.

[1] K. Bartkiewicz, A. Jakubowski, T. Mikosch and O. Wintenberger. Infinite variance stable limits for sums of dependent random variables. Probab. Theory Rel. Fields 150, 337- 372, 2011
[2] A. Jakubowski. Minimal conditions in p-stable limit theorems. Stoch. Proc. Appl. 44, 291 - 327, 1993.
[3] A. Jakubowski. Minimal conditions in p-stable limit theorems - II. Stoch. Proc. Appl. 68, 1 - 20, 1997.
[4] T. Mikosch and O. Wintenberger. Precise large deviations for dependent regularly varying sequences. Probab. Theory Rel. Fields, to appear, 2012.

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