# Abstract:

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.

# Keywords:

strong law of large numbers, the Doob inequality, martingale difference array, Banach space

# Classification:

60E15, 60F15, 60G42, 60B12

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