## Abstract

Let \(\{ X,X_k ,k \in {\mathbb{N}}^r \}\) be i.i.d. random variables, and set *S*
_{
n
}=∑_{
k
}≤ *n*
*X*
_{
k
}. We exhibit a method able to provide exact loglog rates. The typical result is that

whenever *EX*=0,*EX*
^{2}=σ^{2} and *E*[*X*
^{2}(log^{+} | *X* |)^{r-1}] < ∞. To get this and other related precise asymptotics, we derive some general estimates concerning the Dirichlet divisor problem, of interest in their own right.

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## REFERENCES

Chen, R. (1978). A remark on the tail probability of a distribution.

*J*.*Multivariate Anal*.**8**, 328-333.Chow, Y. S., and Lai, T. L. (1978). Paley-type inequalities and convergence rates related to the law of large numbers and extended renewal theory.

*Z*.*Wahrschein-lichkeitstheorie verw*.*Gebiete***45**, 1-19.Deng, D. (1995). Convergence rates for probabilities of moderate deviation for sums of random variables indexed by Z

^{d+}.*Acta Math*.*Hung*.**67**, 131-149.Erdős, P. (1949). On a theorem of Hsu and Robbins.

*Ann*.*Math*.*Stat*.**20**, 286-291.Erdős, P. (1950). Remark on my paper “On a theorem of Hsu and Robbins ”.

*Ann*.*Math*.*Stat*.**21**, 138.Gafurov, M. U. (1983).

*On the Estimate of the Rate of Convergence in the Law of the Iterated Logarithm*, Lecture Notes in Math., Vol.**1021**, Springer-Verlag, New York, pp. 137-144.Gafurov, M. U., and Sira ¢zdinov, S. H. (1979). Some generalizations of Erdős-Katz results related to strong laws of large numbers and their applications (in Russian).

*Kybernetica***15**, 272-292.Galambos, J., Indlekofer, K. H., and Kátai, I. (1987). A renewal theorem for random walks in multidimensional time.

*Trans*.*Amer*.*Math*.*Soc*.**300**, 759-769.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.Gut, A. (1980). Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices.

*Ann*.*Probab*.**8**, 298-313.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.Gut, A., and Sp ¢ataru, A. (2000). Precise asymptotics in the law of the iterated logarithm.

*Ann*.*Probab*.**28**, 1870-1883.Gut, A., and Sp ¢ataru, A. (2003). Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables.

*J*.*Multivariate Anal*.**86**, 398-422.Hardy, G. H., and Wright, E. M. (1979).

*An Introduction to the Theory of Numbers*, 5th ed., Oxford University Press.Heyde, C. C. (1975). A supplement to the strong law of large numbers.

*J*.*Appl*.*Probab*.**12**, 173-175.Hsu, P. L., and Robbins, H. (1947). Complete convergence and the law of large numbers.

*Proc*.*Nat*.*Acad*.*Sci*.**33**, 25-31.Hüsler, J. (1978). A remark on the strong law of large numbers for random variables on partially ordered sets. University of Bern, Unpublished manuscript.

Kend ¢zaev, R. H. (1987). On the rate of convergence in the law of the iterated logarithm for independent random variables indexed by a sector (in Russian).

*Probability Models and Mathematical Statistics*, Fan, Tashkent.Klesov, O. (1982). A remark to the strong law of large numbers (in Russian).

*Theor*.*Probab*.*Math*. Stat.**26**, 69-76.Klesov, O. (1984). On the asymptotics of a series from probability (in Russian).

*Proc*.*Acad*.*Sci*.*Ukr*.**8**, 15-18.Lagodowski, Z. A., and Rychlik, Z. (1984). Some remarks on the strong law of large numbers.

*Bull*.*Acad*.*Sci*.*Polon*.,*Ser*.*Math*.*Astronom*.*Phys*.**32**, 129-134.Lagodowski, Z. A., and Rychlik, Z. (1986). Convergence rates in the strong law of large numbers for random elds.

*Probab*.*Math*.*Stat*.**7**, 149-158.Li, D., Wang, X., and Rao, M. B. (1992). Some results on convergence rates for probabilities of moderate deviations for sums of random variables.

*Int*.*J*.*Math*.*Math*.*Sci*.**15**, 481-498.Maejima, M., and Mori, T. (1984). Some renewal theorems for random walks in multidimensional time.

*Math*.*Proc*.*Camb*.*Philos*.*Soc*.**95**, 149-154.Nagaev, A. V. (1975). A note on the law of large numbers in ℝd, d ⩾ 1. Summary of the 3rd USSR-Japan Symposium on Probability and Mathematical Statistics. Tashkent.

Ney, P., and Wagner, S. (1972). The renewal theorem for a random walk in two-dimensional time.

*Studia Math*.**XLIV**, 71-85.Petrov, V. V. (1975).

*Sums of Independent Random Variables*. Springer-Verlag, New York.Pruss, A. R. (1997). A two-sided estimate in the Hsu-Robbins-Erdős law of large numbers.

*Stochastic Process*.*Appl*.**70**, 173-180.Slivka, J., and Severe, N. C. (1970). On the strong law of large numbers.

*Proc*.*Amer*.*Math*.*Soc*.**24**, 729-734.Smythe, R. (1973). Strong laws of large numbers for

*r*-dimensional arrays of random variables.*Ann*.*Probab*.**1**, 164-170.Sp ¢ataru, A. (1999). Precise asymptotics in Spitzer 's law of large numbers.

*J*.*Theor*.*Probab*.**12**, 811-819.Spitzer, F. (1956). A combinatorial lemma and its applications to probability theory.

*Trans*.*Amer*.*Math*.*Soc*.**82**, 323-339.Titchmarsh, E. C. (1986).

*The Theory of the Riemann Zeta-function*, 4th ed. Oxford University Press, Oxford.Wu, C. F. (1973). A note on the convergence rate of the strong law of large numbers.

*Bull*.*Inst*.*Math*.*Acad*.*Sinica***1**, 121-124.

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Spătaru, A. Exact Asymptotics in log log Laws for Random Fields.
*Journal of Theoretical Probability* **17**, 943–965 (2004). https://doi.org/10.1007/s10959-004-0584-z

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DOI: https://doi.org/10.1007/s10959-004-0584-z