Title: On the long range dependence property of fractional counting processes

Abstract

The fractional generalizations of the Poisson process has drawn the attention of many researchers since the last decade. Recent works on fractional extensions of the Poisson process, commonly known as the fractional Poisson processes, lead to some interesting connections between the areas of fractional calculus, stochastic subordination and renewal theory. The state probabilities of such processes are governed by the systems of fractional differential equations which display a slowly decreasing memory. It seems a characteristic feature of all real systems. Here, we discuss some recently introduced generalized counting processes and their fractional variants. Various fractional counting processes such as the fractional Poisson process and its mixed variants exhibit the longrange dependence property. It is proved by establishing an asymptotic result for the covariance of inverse stable subordinator.

Biography

Dr. Kuldeep Kumar Kataria received his BSc (Hons) degree in Mathematics from St. Stephen’s College, University of Delhi. He received his MSc degree in Mathematics from IIT Kanpur. In 2018, he received his PhD in Mathematics from IIT Bombay. Later, he joined IISc, Bangalore as a NBHM Post-Doctoral Fellow. He is currently working as an Assistant Professor in the Department of Mathematics at IIT Bhilai. The research interest of Dr. Kataria lies in the area of fractional stochastic processes and subordinated (time-changed) versions of certain counting processes. He deals with stable subordinators and space-time fractional versions of the Poisson process. In his PhD thesis, he has studied the applications of Adomian Decomposition Method to certain fractional stochastic processes. So far, Dr. Kataria has published 19 research articles in international journals of repute like Journal of Theoretical Probability, ALEA. Latin American Journal of Probability and Mathematical Statistics, Journal of Mathematical Analysis and Applications, Comptes Rendus Math´ematique, Statistics and Probability Letters, Stochastic Analysis and Applications, etc. Also, he has published several expository articles in reputed mathematical magazine like American Mathematical Monthly, Mathematics Magazine, etc. For his research contributions he has been honoured with the Award of Excellence in Thesis Work for the year 2016- 2018 by Hon’ble Prime Minister of India at the 56th Convocation of the Institute (IIT Bombay).