Convergence for Martingale Sequences of Random Bounded Linear Operators - Cuong, Tran Manh; Son, Ta Cong; Oanh, Le Thi (VNU Journal of Science: Mathematics - Physics; Vol 34 No 4DO - 10.25073/2588-1124/vnumap.4305, 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. References [1] D.H. Thang, T.N. Anh, On random equations and applications to random fixed point theorems, Random Oper.Stoch.Equ. 18(2010), 199-212. [2] D.H. Thang, T.C. Son, On the convergence of the product of independent random operators, Stochas.Int. J. Prob. Stochas. Process. 88(2016), 927-945. [3] D.H. Thang and N. Thinh, Random bounded operators and their extension, Kyushu J.Math. 58 (2004), 257-276. [4] D.H. Thang, N. Thinh, Generalized random linear operators on a Hilbert space, Stochas. Int. J. Prob. Stochas. Process. 85(2013), 1040-1059. [5] Dung, L.V., Son, T. C. and Tien, N. D., L1 bounds for some martingale central limit theorems, Lithuanian Mathematical Journal, 54 (1), 48–60 (2014). [6] T.C.Son and D.H. Thang, The Brunk-Prokhorov strong law of large numbers for fields of martingale differences taking values in a Banach space, Statistics & Probability Letters (2013) 83: 1901-1910. [7] T.C. Son and D.H. Thang, On the convergence of series of martingale differences with multidimensional indices, Journal of the Korean Mathematical Society, 52(5) (2015): 1023-1036. [8] T.C.Son, D.H.Thang and L.V.Dung, Rate of complete convergence for maximums of moving average sums of martingale difference fields in Banach spaces, Statist.Probab.Lett. 82(2012), 1978-1985. [9] Y. Kifer, Ergodic Theory of Random transformation, Birkhauser, (1986) [10] S.D.Chatterji, Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scand, 22 (1986) 21-41.