Asymptotic Connection Between Sum and Extreme Order Statistics in a Random Sequence.
ABSTRACT:
We consider independent identically distributed random variables (vectors), whose
distribution belongs to the domain of attraction of a stable law. Ordering the variables
(vectors) by increase of their modules (norms) we investigate the asymptotic properties of a
ratio of the members of the variation series. Using the characteristic functions of the ratio
we analyze the total contribution of the first k maximal modulus of summands to the sum. In
case when the variables have infinite moments of all orders we show that for distributions
which have a certain regularity at infinity, the sum is completely dominated by its k largest
terms. This case is different as any linear setting by constants of sequence of sums leads
either to the degenerated limiting laws or to their absence. However, as it follows from our
results that the limiting distributions can be derived by using non-linear setting.
About the Speaker
Dr. Ilham Akhundov received his MS and Ph.D. from
Saint-Petersburg State University under the
supervision of Prof. Velery Nevzorov. His research interests include limit
theorems for order statistics based on
cumulative sums, extremes and heavy tailed phenomena, first passage times and
maximum sums. He was a Fulbright
Fellow at Texas A&M University
from September 1999 to September 2000. During his tenure in Mathematics
Departments of Azerbaijan Economy University and the
Fatih University in Turkey he taught
various courses in Mathematics and Statistics
for undergraduate and graduate students. He joined the McMaster University
Department of Mathematics and Statistics as a
part-time Instructor in December 2001.
