Uniform convergence of reversed martingales
The author studies the uniform convergence of reversed martingales and submartingales. Theorem 3.1 is the analogue of the well-known Neveu's lemma for reversed martingales and submartingales; it gives a series of necessary and sufficient conditions for the almost sure convergence and \(L^ 1\)-convergence of supremum of a countable family of reversed martingales and submartingales. Theorem 4.1 and Theorem 4.2, which are completed by Theorem 4.3 and Theorem 4.4, are extensions of Theorem 3.1; then a necessary and sufficient condition for the convergence of family of reversed martingales indexed by a separable topological space converging to a degenerated limiting process is given. The condition is expressed by means of regular convergence (in Hardy's sense) of corresponding mean. Furthermore, it will be shown that the given condition for the uniform convergence of reversed martingales is equivalent to condition of eventually totally boundedness in mean established for the uniform law of large numbers by J. Hoffmann- Jørgensen. An important application of the results is given in Example 4.1 and it concerns the uniform law of large numbers; finally the author gives reference where further applications of their results in statistics are given.
convergence of reversed martingales and submartingales
scientific article; zbMATH DE number 757602