## Precise asymptotics in the law of the iterated logarithm.(English)Zbl 1044.60024

Summary: Let $$X,X_1,X_2,\dots$$ be i.i.d. random variables with mean 0 and positive, finite variance $$\sigma^2$$, and set $$S_n=X_1+ \cdots+X_n$$, $$n>1$$. Continuing earlier work related to strong laws, we prove the following analogs for the law of the iterated logarithm: $\lim_ {\varepsilon\searrow\sigma\sqrt 2}\sqrt{\varepsilon^2-2\sigma^2} \sum_{n\geq 3} \frac 1n P\bigl(| S_n|\geq \varepsilon\sqrt {n \log \log n}+a_n\bigr)= \sigma\sqrt 2$ whenever $$a_n=O(\sqrt n(\log\log n)^{-\gamma})$$ for some $$\gamma\geq 1/2$$ (assuming slightly more than finite variance), and $\lim_{\varepsilon\searrow 0}\varepsilon^2 \sum_{n\geq 3} \frac{1} {n\log n}P\bigl(| S_n|\geq \varepsilon \sqrt {n\log\log n}\bigr) =\sigma^2.$

### MSC:

 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks 60E15 Inequalities; stochastic orderings
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### References:

 [1] Baum, L. E. and Katz, M. (1965). Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120 108-123. · Zbl 0142.14802 [2] Chen, R. (1978). A remark on the tail probability of a distribution. J. Multivariate Anal. 8 328-333. · Zbl 0376.60033 [3] Cramér, H. (1946). Mathematical Methods of Statistics. Princeton Univ. Press. · Zbl 0063.01014 [4] Davis, J. A. (1968). Convergence rates for the law of the iterated logarithm. Ann. Math. Statist. 39 1479-1485. · Zbl 0174.49902 [5] Erd os, P. (1949). On a theorem of Hsu and Robbins. Ann. Math. Statist. 20 286-291. · Zbl 0033.29001 [6] Erd os, P. (1950). Remark on my paper ”On a theorem of Hsu and Robbins.” Ann. Math. Statist. 21 138. · Zbl 0035.21403 [7] Feller, W. (1968). An Introduction to Probability Theory and Its Applications 1. Wiley, New York. · Zbl 0155.23101 [8] Fuk, D. H. and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Probab. Appl. 16 643-660. · Zbl 0259.60024 [9] Gut, A. (1978). Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 6 469-482. · Zbl 0383.60030 [10] Gut, A. (1980). Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices. Ann. Probab. 8 298-313. · Zbl 0429.60022 [11] Gut, A. and Sp ataru, A. (2000). Precise asymptotics in the Baum-Katz and Davis law of large numbers. J. Math. Anal. Appl. 248 233-246. · Zbl 0961.60039 [12] Heyde, C. C. (1975). A supplement to the strong law of large numbers. J. Appl. Probab. 12 173-175. JSTOR: · Zbl 0305.60008 [13] Hsu, P. L. and Robbins, H. (1947). Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A. 33 25-31. JSTOR: · Zbl 0030.20101 [14] Petrov, V. V. (1995). Limit Theorems of Probability Theory. Oxford Univ. Press. · Zbl 0826.60001 [15] Sp ataru, A. (1999). Precise asymptotics in Spitzer’s law of large numbers. J. Theoret. Probab. 12 811-819. · Zbl 0932.60026 [16] Spitzer, F. (1956). A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc. 82 323-339. JSTOR: · Zbl 0071.13003
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