Operator Geometric Stable Laws.pdf

Operator Geometric Stable Laws Tomasz J. Kozubowski1, Mark M. Meerschaert2, Anna K. Panorska3 University of Nevada, Reno and Hans-Peter Scheffler University of Dortmund Operator geometric stable laws are the weak limits of operator normed and centered geometric random sums of independent, identically distributed random vectors. They generalize operator stable laws and geometric stable laws. In this work we characterize operator geometric stable distributions, their divisibility and domains of attraction, and their application to finance. Operator geometric stable laws are useful for modeling finan- cial portfolios where the cumulative price change vectors are sums of a random number of small random shocks with heavy tails, and each component has a different tail index. AMS 2000 subject classifications: 60E07, 60F05, 60G50, 62H05, 62P05 Key words and phrases: currency exchange rates; domains of attraction; geometric stable law; heavy tails; infinite divisibility; Linnik distribution; operator stable law; ran- domized sum; skew Laplace law; stability; stable distribution. 1. Introduction We introduce a new class of multivariate distributions called operator geometric stable, generalizing the geometric stable and operator stable laws. Our motivation comes from 1The research of this author was partially supported by NSF grant DMS-0139927. 2The research of this author was partially supported by NSF grants DES-9980484 and DMS-0139927. 3The research of this author was partially supported by NSF grants ATM-0231781 and DMS-0139927 1 2a problem in finance, where a portfolio of stocks or other financial instruments changes price over time, resulting in a time series of random vectors. The daily price change vectors are each accumulations of a random number of random shocks. Price shocks are typically heavy tailed with a tail parameter that is different for each stock [39]. Operator stable models can handle the variations in tail behavior [33] while geometric stable models [11,