## Precise asymptotics in the Baum-Katz and Davis laws of large numbers.(English)Zbl 0961.60039

Let $$X_1,X_2,\dots$$ be a sequence of i.i.d. random variables with mean $$0$$ and partial sums $$\{S_n, n\geq 1\}$$. The law of large numbers is concerned with the probabilities $$P(|S_n|>\varepsilon n)$$ as $$n\to\infty$$. In order to study convergence rates, the classical Hsu-Robbins-Spitzer-Erdős-Baum-Katz results establish necessary and sufficient conditions on the moments in order for sums like $$\sum^\infty_{n=1} P(|S_n|> \varepsilon n)$$ to converge. The present paper is devoted to another aspect, namely the limiting behaviour of the sums, in terms of $$\varepsilon$$, as $$\varepsilon\to 0$$. More precisely, the authors prove limit theorems for $$\sum^\infty_{n=1} n^{r/p- 2}P(|S_n|> \varepsilon n^{1/p})$$, properly normalized, as $$\varepsilon\searrow 0$$.

### MSC:

 60F05 Central limit and other weak theorems

### Keywords:

convergence rates; limiting behaviour of the sums
Full Text:

### References:

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