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Convergence for martingale sequences of random bounded linear operators

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In this paper, we study the convergence for martingale sequences of random bounded linear operators. The condition for the existence of such a infinite product of random bounded linear operators is established. | VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 62-69 Convergence for Martingale Sequences of Random Bounded Linear Operators Tran Manh Cuong1,*, Ta Cong Son1, Le Thi Oanh2 1 Faculty of Mathematics, Mechanics and Informatics, VNU Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam 2 Department of Mathematics, Hong Duc University, 565 Quang Trung, Dong Ve, Thanh Hoa, Viet Nam Received 03 December 2018 Revised 20 December 2018; Accepted 20 December 2018 Abstract: In this paper, we study the convergence for martingale sequences of random bounded linear operators. The condition for the existence of such a infinite product of random bounded linear operators is established. AMS Subject classification 2000: 60H05, 60B11, 60G57, 60K37, 37L55. Keywords and phrases: Random bounded linear operators, products of random bounded linear operators, martingales of random bounded linear operators, convergence of random bounded linear operators. 1. Introduction Let ( , , P ) be a complete probability space and X, Y be separable Banach spaces. A mapping : X LY0 ( ) is said to be a random operator, where LY0 ( ) stands for the space of Y valued random variables and is equipped with the topology of convergence in probability. If a random : X LY0 ( ) is linear and continuous then it is called a random linear operator. The set of all random linear operators A : X LY0 ( ) is denoted by L ( , X, Y). The random operator theory is one of the branches of the theory of random processes and functions; its creation is a natural step in the development of random analysis. Research in theory of random operators has been carried out in many directions such as random fixed points of random operators, random operator equations, random linear operators (see [1]-[4]). _ Corresponding author Email: cuongtm@vnu.edu.vn https//doi.org/ 10.25073/2588-1124/vnumap.4305 62 T.M. Cuong et al. / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4