Exact Asymptotics in log log Laws for Random Fields

A. Spătaru
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A. Spătaru: Institute of Mathematical Statistics and Applied Mathematics

Journal of Theoretical Probability, 2004, vol. 17, issue 4, 943-965

Abstract: 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]

Keywords: Multidimensional indices; tail probabilities of sums of i.i.d. random variables; law of the iterated logarithm; Dirichlet divisor problem (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/s10959-004-0584-z

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