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Exact Asymptotics in log log Laws for Random Fields

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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

$${\mathop {\lim }\limits_{\varepsilon \searrow \sigma \sqrt {2r}} } \sqrt {\varepsilon ^2 - 2r\sigma ^2 } \sum\limits_n {\frac{1}{{|\,n\,|}}P(|S_n \geqslant \varepsilon \sqrt {|\,n\,|\log \log |\,n\,|} ) = \frac{{\sigma \sqrt {2r} }}{{r!}},}$$

whenever EX=0,EX 2=σ2 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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  1. Romanian Academy, Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie, no. 13, 76100, Bucharest 5, Romania

    A. Spătaru

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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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