The Doob inequality and strong law of large numbers for multidimensional arrays in general Banach spaces

We establish the Doob inequality for martingale difference arrays and provide a sufficient condition so that the strong law of large numbers would hold for an arbitrary array of random elements without imposing any geometric condition on the Banach space. Some corollaries are derived from the main results, they are more general than some well-known ones.

[1]  John B. Walsh,et al.  Stochastic integrals in the plane , 1975 .

[2]  A. Rosalsky,et al.  On Almost Sure and Mean Convergence of Normed Double Sums of Banach Space Valued Random Elements , 2007 .

[3]  Strong laws for blockwise martingale difference arrays in Banach spaces , 2010 .

[4]  Marcinkiewicz-zygmund weak laws of large numbers for unconditional random elements in Banach spaces , 1981 .

[5]  Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces , 1994 .

[6]  Nguyen Van Huan,et al.  On the strong law of large numbers and Lp-convergence for double arrays of random elements in p-uniformly smooth Banach spaces , 2009 .

[7]  On the strong law of large numbers for d-dimensional arrays of random variables , 2007 .

[8]  A. Gut.  Probability: A Graduate Course , 2005 .

[9]  N. Quang,et al.  A Hájek-Rényi-Type Maximal Inequality and Strong Laws of Large Numbers for Multidimensional Arrays , 2010 .

[10]  Hyun-Chull Kim.  The H´ ajeck-R` enyi inequality for weighted sums of negatively orthant dependent random variables , 2006 .

[11]  Strong limit theorems for quasi-orthogonal random fields , 1989 .

[12]  R. Smythe.  Strong Laws of Large Numbers for $r$-Dimensional Arrays of Random Variables , 1973 .

[13]  Qi-Man Shao,et al.  A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables , 2000 .

[14]  Ferenc Móricz,et al.  Strong Laws for Blockwise ℳ-Dependent Random Fields , 2008 .

[15]  A General Approach to Strong Laws of Large Numbers for Fields of Random Variables , 2008 .

[16]  A characterization of p-uniformly smooth Banach spaces and weak laws of large numbers for d-dimensional adapted arrays , 2010 .