### Convergence for Martingale Sequences of Random Bounded Linear Operators

## Main Article Content

## Abstract

**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.

*Keywords:*Random operators, product of random operators, convergence of random 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.