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Discrete-Time Risk-Sensitive Filters with Non-Gaussian Initial Conditions and Their Ergodic Properties |
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| Authors: | Subhrakanti Dey Charalambos D. Charalambous |
| Affiliation: | 1. Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, VIC 3052, Australia. Subhrakanti Dey: was born in Calcutta, India in 1968. He obtained his Bachelor of Technology and Master of Technology Degree from the Dept. of Electronics and Electrical Communication Engineering, Indian Institute of Technology, Kharagpur, India, in 1991 and 1993, respectively. He obtained his Doctor of Philosophy degree from the Dept. of Systems Engineering, Research School of Informations Sciences and Engineering in the Australian National University, Canberra, Australia in 1996. He is currently appointed as a Lecturer with the Department of Electrical and Electronic Engineering in the University of Melbourne since February 2000. During September 1995-September 1997 and September 1998-February 2000, he was appointed as a postdoctoral Research Fellow in the Department of Systems Engineering, Australian National University. During September 1997-September 1998, he was a post-doctoral Research Associate at the Institute for Systems Research, University of Maryland, College Park, U.S.A. His current research interests include signal processing for telecommunications, wireless communications and networks, performance analysis of communication networks, stochastic and adaptive estimation and control and statistical and adaptive signal processing. He is currently an academic staff member of the ARC Special Research Centre for Ultra-Broadband Informamation Networks and Co-operative Research Centre for Sensor, Signal and Information Processing in the Department of Electrical and Electronic Engineering, University of Melbourne.;2. Department of Electrical Engineering, McGill University, Montreal, Canada H3A 2A7. Charalambos D. Charalambous: was born in Cyprus, on July 4, 1962. He received a Ph.D. degree from Old Dominion University, Norfolk, VA, in December 1992. Currently, he is an Associate Professor with University of Ottawa, School of Information Technology and Engineering, CA, and Adjunct Professor with McGill University, ECE Department, and an associate member of the Centre for Intelligent Machines. From 1995 to 1999, he was a visiting assistant professor with McGill University, ECE Department;3. from 1993 to 1995, he was a Post-Doctoral Research Fellow with Idaho State University, Engineering Department, Idaho. His research interests incluede stochastic processes and their applications in estimation, decision and control, robust control, information theory, and wireless communication at the physical and network level, and mathematical finance. |
| Abstract: | In this paper, we study asymptotic stability properties of risk-sensitive filters with respect to their initial conditions. In particular, we consider a linear time-invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risk-sensitive filter asymptotically converges to a suboptimal filter initialized with an incorrect covariance matrix for the initial state vector in the mean square sense provided the incorrect initializing value for the covariance matrix results in a risk-sensitive filter that is asymptotically stable, that is, results in a solution for a Riccati equation that is asymptotically stabilizing. For non-Gaussian initial conditions, we derive the expression for the risk-sensitive filter in terms of a finite number of parameters. Under a boundedness assumption satisfied by the fourth order absolute moment of the initial state variable and a slow growth condition satisfied by a certain Radon-Nikodym derivative, we show that a suboptimal risk-sensitive filter initialized with Gaussian initial conditions asymptotically approaches the optimal risk-sensitive filter for non-Gaussian initial conditions in the mean square sense. Some examples are also given to substantiate our claims. |
| Keywords: | Risk-sensitive estimantion asymptotic stability non-Gaussian optimal filtering |