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Wittmann’s law of iterated logarithm for tail sums of \(B\)-valued random variables. (English) Zbl 1056.60003
Let \(B\) be a separable Banach space with norm \(\|\cdot\|\) and dual space \(B^*\) and let \(\{X_n, n=1,2,\ldots\}\) be a sequence of independent \(B\)-valued weak second order centered random variables (i.e. \(EX_n=0\) and \(Ef^2(X_n)<\infty\) for each \(f\in B^*\)) and \(\sum_{i=1}^\infty X_i\) converges a.s. For \(n\geq1\) put \(t_n^2=\sup_{f\in B_1^*}\sum_{i=n}^\infty Ef^2(X_i)\) and \(u_n=(2L_2t_n^{-2})^{1/2}\), where \(L_2x=L(Lx), Lx=\max(\log x,1)\) and \(B_1^*\) is the closed ball of \(B^*\). It is said that \(\{X_n\}\) satisfies the law of iterated logarithm (LIL) for tail sums if there exist a sequence of real numbers \(b_n\downarrow0\) and a constant \(\Lambda\) such that \[ \left.\lim\sup_{t\to\infty}\left\|\sum_{i=n}^\infty X_i\right\| \right /b_n=\Lambda \quad {\text{a.s.}} \] It is known that \(\Lambda=1\) for \(b_n=(2\sum_{i=1}^nEX_i^2L_2(\sum_{i=1}^nEX_i^2)^{-1})^{1/2}\) in the real-valued case [see A. Rosalsky, Bull. Inst. Math., Acad. Sin. 11, 185–208 (1983; Zbl 0517.60032)]. The LIL for tail sums of the infinite-dimensional Banach space-valued random variables, in particular, we can find in the papers of the author [Acta Sci. Nat. Univ. Jilinensis 1988, No. 2, 15–24 (1988; Zbl 0988.60501), Northeast. Math. J. 7, 265–274 (1991; Zbl 0754.60011), and Acta Sci. Nat. Univ. Jilinensis 1994, No. 3, 1–10 (1994; Zbl 0988.60513)].
The purpose of the paper under review is to consider the extension of R. Wittmann’s LIL [Z. Wahrscheinlichkeitstheorie Verw. Geb. 68, 521–543 (1985; Zbl 0547.60036)] to the case of tail sums in Banach space and to characterize an upper bound in this form of the LIL. The results in this direction are given in the following theorems.
Theorem 1. Let \(\{X_n\}\) be a sequence of independent \(B\)-valued weak second order centered random variables and \(\sum_{i=1}^\infty X_i\) a.s. converges. Let \(\{a_n\}\) be a positive real sequence with \(a_n\downarrow0\). Suppose that the following statements hold: \(\sum_{n=1}^\infty a_n^{-p}E\| X_n\|^p<\infty\) for some \(p>2\) and \((2t_n^2L_2t_n^{-2})^{1/2}\leq a_n\). Assume, moreover, that there exists some \(\beta\geq1\) such that \(1\leq(a_na_{n+1}^{-1})\leq(t_nt_{n+1}^{-1})^\beta\) for any \(n\geq1\) and \(a_n^{-1}\sum_{i=n}^\infty X_i\to0\) in probability as \(n\to \infty\). Then \[ \left.\Lambda=\lim\sup_{n\to\infty}\left\|\sum_{i=n}^\infty X_i\right\| \right/a_n\leq1 \quad {\text{a.s.}} \]
Theorem 2. Let \(\{X_n\}\) be a sequence of independent \(B\)-valued weak second order centered random variables and \(\sum_{i=1}^\infty X_i\) a.s. converges. Suppose that the following statements hold: \(\sum_{n=1}^\infty (2t_n^2L_2t_n^{_2})^{-p/2}E\| X_n\|^p<\infty\) for some \(p>2\); \(\lim_{n\to\infty}t_n=0\) and \(\lim\sup_{n\to\infty}t_n/t_{n+1}<\infty\); \((2t_n^2L_2t_n^{_2})^{-1/2}\sum_{i=n}^\infty X_i\to0\) in probability as \(n\to \infty\). Then \[ \left.\Lambda=\lim\sup_{n\to\infty}\left\|\sum_{i=n}^\infty X_i\right\|\right/ (2t_n^2L_2t_n^{_2})^{1/2}=1 \quad {\text{a.s.}} \]
Theorem 3. Let \(\{X_n\}\) be a sequence of independent \(B\)-valued centered random variables and \(\sum_{i=1}^\infty X_i\) converges a.s. Suppose that \(\{a_n\}\) is a positive, decreasing sequence of real numbers satisfying the following conditions: \(\sum_{n=1}^\infty a_n^{-p}E\| X_n\|^p<\infty\) for some \(1\leq p\leq2\); \(\lim_{n\to\infty}a_n=0\) and \(\lim\sup_{n\to\infty}a_n/a_{n+1}<\infty\). Then \[ \sum_{i=n}^\infty X_i/a_n \quad {\text{a.s.}}\Leftrightarrow\sum_{i=n}^\infty X_i/a_n \quad {\text{in probability.}} \]

MSC:
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15 Strong limit theorems
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