| In this paper, we discuss the limit theory of the independent case, the B-value and the NA r.v. series. We divide the paper into three parts: In the first chapter, under the conditionof independence but no identical function, we discuss the convergent rate problem of r.v. series about , and we extend the conclusion by the function series{φn(x), n ≥ 1} satisfying the more dimension function set is non-negative and even function; and there is x0≥0, makes φ(x)/x2 and x3/φ(x) be quasi-increasing function on the interval(x0,+∞)}, which makes some relative results in the paper [1], [2] specialized, in the same time, we also discuss the similar result on the condition of the NA r.v. series; In the second chapter, for , we talk over the holding ofthe convergent rate of continuous independent r.v. series , we have that the successive conversion of continuous independent r.v. series keep their speed under some conditions; And in the third part, we pay our attention to the strong limit problem of the B-value and the NA r.v. series, the result which we obtain spread relative conclusion in the paper [18], [19],[22].And throughout this paper, we state that r.v. is defined on the same probability space (Ω, F,P) if without special explanation.
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