We prove that a continuous-time branching particle system which starts from a discrete stable number of particles and is characterized by scaled Sibuya branching must have evolved from a Poisson field. We provide a local version of this result and derive other local limit theorems. We consider the “dual” subcritical Galton-Watson process constructed starting from a Galton-Watson process with scaled Sibuya branching conditioned by extinction. For the “dual” process, we derive pgf of the law of the total progeny. In a special case, we obtain its pmf and discover a relationship between this law and the distribution of a busy period of the server in the M/M/1 queue. We establish local approximation for the long-time evolution of a Poisson field of “dual” subcritical Galton-Watson processes with exponentially tilted Sibuya branching, and delineate its connection with the logarithmically distributed Yaglom limit. We relate the extinction probability for supercritical Galton-Watson processes with discrete stable branching to the Lambert W function and its generalizations.

Bio: Dr. Vladimir Vinogradov is a Professor of Mathematics at Ohio University in Athens, Ohio. He earned his M.Sc. in Mathematics and Ph.D. in Probability and Statistics from Moscow State University with his dissertation published in his monograph "Refined Large Deviation Limit Theorems". Professor Vinogradov has taught in various post-secondary institutions of Canada, Japan, Russia and U.S.A., and held an NSERC Canada postdoctoral fellowship at Carleton University. His research focuses on various topics of Probability Theory, Stochastic Processes, Mathematical Statistics, Analysis and Financial Mathematics. Professor Vinogradov has published articles in many professional journals and presents frequently at national and international conferences. He has been recipient of a British Columbia – Asia Pacific Scholars' Award, and served on the Ontario Graduate Scholarships Committee as well as NSERC Canada external reviewer and graduate coordinator at the University of Northern British Columbia. Professor Vinogradov is advisor to Actuarial Science and Mathematical Statistics majors.