Professor Uwe Einmahl brought the following fact to my attention: The limit law established in Theorem 1.1 of [1] was obtained by Robbins and Siegmund [3], in the context of Brownian motions rather than random walk. Actually, the result of Robbins and Siegmund provides an asymptotic order that is higher than the one described in Theorem 1.1 of [1].
As also pointed out by Professor Einmahl, the work of Robbins and Siegmund appears (to some degree) as a response to conjectures made by Darling and Erdös [2].
References
Chen, X.: The limit law of the iterated logarithm. J. Theor. Probab. 28, 721–725 (2015)
Darling, D.A., Erdös, P.: A limit theorem for the maximum of normalized sums of independent random variables. Duke Math. J. 23, 143–155 (1956)
Robbins, H., Siegmund, D.: On the law of the iterated logarithm for maxima and minima. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, University of California, Berkeley, CA, 1970/1971, Vol. III: Probability theory, pp. 51–70. Univ. California Press, Berkeley, CA (1972)
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Chen, X. Acknowledgment of Priority. J Theor Probab 30, 700 (2017). https://doi.org/10.1007/s10959-015-0649-1
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DOI: https://doi.org/10.1007/s10959-015-0649-1