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Renz (Ann. Probab. 1996) has established a rate of convergence $1/\sqrt{n}$in the central limit theorem for martingales with some restrictive conditions.In the present paper a modification of the methods, developed by Bolthausen(Ann. Probab. 1982) and Grama and Haeusler (Stochastic Process. Appl. 2000), isapplied for obtaining the same convergence rate for a class of more generalmartingales. An application to linear processes is discussed.

A Berry-Esseen bound of order $ 1/\sqrt{n} $ for martingales