×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chover, J., A law of the iterated logarithm for stable summands, Proc. Amer. Math. Soc., 17, 441-443, (1966) · Zbl 0144.40503
[2] Gut, A., Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, Ann. Probab., 6, 469-482, (1978) · Zbl 0383.60030
[3] Hartman, P.; Wintner, A., On the law of the iterated logarithm, Amer. J. Math., 63, 169-176, (1941) · JFM 67.0460.03
[4] Klesov, O. I., A law of the iterated logarithm for stable summands, Teor. Verojatnost. i Mat. Statist., 24, 60-63, (1981), 152 (in Russian). English translation in Theory Probab. Math. Stat. 24, 67-70 (1982) · Zbl 0471.60036
[5] Kolmogoroff, A., Sur la loi FORTE des grands nombres, C. R. Acad. Sci. Paris Ser. Math., 191, 910-912, (1930) · JFM 56.0445.06
[6] Li, D.; Chen, P., A characterization of chover-type law of iterated logarithm, Springer Plus, 3, 386, (2014)
[7] Li, D.; Zhang, S., A limit theorem related to the hartman-wintner-Strassen LIL and the chover LIL, Statist. Probab. Lett., 109, 16-21, (2016) · Zbl 1329.60067
[8] Li, D.; Zhang, S., Chover-klesov-type law of iterated logarithm for sums of random fields, Far East J. Theor. Stat., 52, 253-268, (2016) · Zbl 1359.60043
[9] Marcinkiewicz, J.; Zygmund, A., Sur LES fonctions ind√©pendantes, Fund. Math., 29, 60-90, (1937) · JFM 63.0946.02
[10] Mikosch, T., 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), (in Russian) · Zbl 0547.60037
[11] Smythe, R. T., Strong laws of large numbers for \(r\)-dimensional arrays of random variables, Ann. Probab., 1, 164-170, (1973) · Zbl 0258.60026
[12] Strassen, V., A converse to the law of the iterated logarithm, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 4, 265-268, (1966) · Zbl 0141.16501
[13] Wichura, M. J., Some Strassen type laws of the iterated logarithm for multiparameter stochastic processes with independent increments, Ann. Probab., 1, 272-296, (1973) · Zbl 0288.60030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.