Decomposition and convergence for tree martingales

In this paper, the authors firstly construct a graph-theoretic decomposition of an index set for tree martingales, and based on this decomposition, they give a locally finite tree martingale’s notion and a tree martingale decomposition theorem. Secondly, they establish some relations between the locally finite tree martingales and the multiparameter martingales, and furthermore the convergence of tree martingales is shown by using the multiparameter martingale theory of Cairoli–Walsh. Finally, with some mild conditions, two inequalities for tree martingales are obtained by using their decomposition theorem and multiparameter martingale theory.