Ž . Journal of Mathematical Analysis and Applications 246, 465479 2000 doi:10.1006jmaa.2000.6799, available online at http:www.idealibrary.com on Exponential Mixture Representation of Geometric Stable Densities Boris P. Belinskiy Uni  ersity of Tennessee, Chattanooga, Tennessee 37403 and Tomasz J. Kozubowski Uni  ersity of Tennessee, Chattanooga, Tennessee 37403; and Uni  ersity of California, Santa Barbara, Santa Barbara, California 93106 E-mail: tkozubow@pstat.ucsb.edu Submitted by Ulrich Stadtmueller Received November 9, 1998 1. INTRODUCTION AND STATEMENT OF RESULTS Let 1    t  1   t  t  i  t , t  R , 1 Ž. Ž. Ž.  ,  where  exp i sign t 2, if   1, Ž. Ž .  2  t  2 Ž. Ž.  ,   1  i sign t log t , if   1, Ž.   Ž . Ž . be the characteristic function ch.f. of a geometric stable GS random Ž .  Ž  variable r.v. ; see 7 . The parameter   0, 2 is the index of stability   Ž . determining the tail of the distribution,  , where   min 1, 2  1 , is the skeweness parameter, while   R and   0 control the location and Ž . scale, respectively. We shall write GS ,  ,  for the GS distribution  Ž.Ž. given by 1  2 and denote the corresponding probability density by Ž. Ž p  . Without loss of generality we shall assume that   1 if  , ,  ,  Ž 1   . 1  Ž . . Y  GS 1,  ,  then  Y  GS ,  ,  whenever   1.   465 0022-247X00 $35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved.