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An extension of a theorem of Mikosch. (English) Zbl 1352.60048
Summary: Let $$0 < \alpha \leq 2$$. Let $$\mathbb{N}^d$$ be the $$d$$-dimensional lattice equipped with the coordinate-wise partial order $$\leq$$, where $$d\geq 1$$ is a fixed integer. For $$\mathbf{n} = (n_1, \dots, n_d) \in \mathbb{N}^d$$, define $$|\mathbf{n}| = \prod_{i = 1}^d n_i$$. Let $$\{X, X_{\mathbf{n}}: \mathbf{n} \in \mathbb{N}^d \}$$ be a field of independent and identically distributed real-valued random variables. Set $$S_{\mathbf{n}} = \sum_{\mathbf{k} \leq \mathbf{n}} X_{\mathbf{k}}$$, $$\mathbf{n} \in \mathbb{N}^d$$ and write $$\log x = \log_e(e \vee x)$$, $$x \geq 0$$. This note is devoted to an extension of a strong limit theorem of T. Mikosch [“The law of the iterated logarithm for independent random variables outside the domain of partial attraction of the normal law”, Vestn. Leningr. Univ. Mat. Mekh. Astron. 3, 35–39 (1984)]. By applying an idea of D. Li and P. Chen [“A characterization of Chover-type law of iterated logarithm”, SpringerPlus 3, Article ID 386, 7 p. (2014; doi:10.1186/2193-1801-3-386)] and the classical Marcinkiewicz-Zygmund strong law of large numbers for random fields, we obtain necessary and sufficient conditions for $\limsup_{\mathbf{n}} |S_{\mathbf{n}}|^{(\log |\mathbf{n}|)^{- 1}}:=\lim_{m \to \infty} \sup_{|\mathbf{n}|\geq m} |S_{\mathbf{n}}|^{(\log |\mathbf{n}|)^{- 1}} = e^{1 / \alpha} \;\;\text{almost\;surely}.$

##### MSC:
 60F15 Strong limit theorems 60G60 Random fields 60G50 Sums of independent random variables; random walks
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##### References:
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