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**Stability results and strong invariance principles for partial sums of Banach space valued random variables.**
*(English)*
Zbl 0669.60035

Let \(\{X_ n\}\) be a sequence of independent and identically distributed random variables taking their values in a separable Banach space, \((B,\| \cdot \|)\). Suppose \(\{X_ n\}\) is in the domain of attraction of a Gaussian law, i.e. \(L(a^{-1}(n)\sum^{n}_{1}X_ k)\to L(Y)\) weakly for some sequence a(n)\(\uparrow \infty\). Here Y is a non-degenerate B-valued mean zero Gaussian random variable. After briefly reviewing recent results on the law of the iterated logarithm and stability results, this paper proves three main theorems.

Theorem 1 is a stability theorem for \(\{X_ n\}\). Specifically, if \(\{c_ n\}\) is a sequence of positive real numbers such that \(c_ n/\sqrt{n}\) is nondecreasing, and if for some \(q\in R\), \[ c_ n/\sqrt{\log \log n}a(n(\log \log n)^ q)\to \infty, \] then with probability one, \[ \limsup_{n\to \infty}\| \sum^{n}_{1}X_ k/c_ n\| =0\quad or\quad =\infty \] according as \(\sum^{\infty}_{1}P\{\| X_ 1\| >c_ n\}\) is finite or infinite. This improves previous results in that q need not be equal to 1.

Theorem 2 provides a strong invariance principle for \(\{X_ n\}\). Let \(q\in R\). If the probability space is rich enough, one can define a sequence \(\{Y_ n\}\) of independent copies of Y such that \[ \max_{1\leq m\leq n}\| \sum^{m}_{1}X_ k-a(n(\log \log n)^ q)\sum^{m}_{1}Y_ k/\sqrt{n}\| =o(\sqrt{\log \log n}a(n(\log \log)^ q))\quad a.s. \] if and only if \(\sum^{\infty}_{1}P\{\| X\| >\sqrt{\log \log n}a(n(\log \log n)^ q)\}<+\infty.\)

This result, and the compact (functional) law of the iterated logarithm for \(\{Y_ n\}\), immediately yields that the corresponding LIL for \(\{X_ n\}\) holds if and only if the above condition holds.

The proof of Theorem 2 is accomplished in three parts. If B is finite- dimensional and \(q\leq 1\), the argument follows the proof of Theorem 2 of the author in Probab. Theory Related Fields 77, No.1, 65-85 (1988; Zbl 0618.60010). For \(q\geq 1\), Theorem 2 of the author (loc. cit.) is used directly. Arguments on slowly varying functions complete the proof of the second part. The general case is reduced to the previous two cases by means of the maps of Lemma 2.1 of J. Kuelbs [Ann. Probab. 4, 744- 771 (1976; Zbl 0365.60034)], Theorem 3, Lemma A. 1 of I. Berkes and W. Philipp [ibid. 7, 29-54 (1979; Zbl 0392.60024)], and a known technique of P. Major [Z. Wahrscheinlichkeitstheor. Verw. Gebiete 35, 221-229 (1976; Zbl 0338.60032)].

Theorem 3 is a technical stability result related to Theorem 5 of J. Kuelbs and J. Zinn [Ann. Probab. 11, 506-557 (1983; Zbl 0518.60009)]. Its proof rests on two exponential inequalities essentially due to V. V. Yurinskij [J. Multivariate Anal. 6, 473-499 (1976; Zbl 0346.60001)] and is simpler than the proof of the Theorem of Kuelbs and Zinn.

Theorem 1 is a stability theorem for \(\{X_ n\}\). Specifically, if \(\{c_ n\}\) is a sequence of positive real numbers such that \(c_ n/\sqrt{n}\) is nondecreasing, and if for some \(q\in R\), \[ c_ n/\sqrt{\log \log n}a(n(\log \log n)^ q)\to \infty, \] then with probability one, \[ \limsup_{n\to \infty}\| \sum^{n}_{1}X_ k/c_ n\| =0\quad or\quad =\infty \] according as \(\sum^{\infty}_{1}P\{\| X_ 1\| >c_ n\}\) is finite or infinite. This improves previous results in that q need not be equal to 1.

Theorem 2 provides a strong invariance principle for \(\{X_ n\}\). Let \(q\in R\). If the probability space is rich enough, one can define a sequence \(\{Y_ n\}\) of independent copies of Y such that \[ \max_{1\leq m\leq n}\| \sum^{m}_{1}X_ k-a(n(\log \log n)^ q)\sum^{m}_{1}Y_ k/\sqrt{n}\| =o(\sqrt{\log \log n}a(n(\log \log)^ q))\quad a.s. \] if and only if \(\sum^{\infty}_{1}P\{\| X\| >\sqrt{\log \log n}a(n(\log \log n)^ q)\}<+\infty.\)

This result, and the compact (functional) law of the iterated logarithm for \(\{Y_ n\}\), immediately yields that the corresponding LIL for \(\{X_ n\}\) holds if and only if the above condition holds.

The proof of Theorem 2 is accomplished in three parts. If B is finite- dimensional and \(q\leq 1\), the argument follows the proof of Theorem 2 of the author in Probab. Theory Related Fields 77, No.1, 65-85 (1988; Zbl 0618.60010). For \(q\geq 1\), Theorem 2 of the author (loc. cit.) is used directly. Arguments on slowly varying functions complete the proof of the second part. The general case is reduced to the previous two cases by means of the maps of Lemma 2.1 of J. Kuelbs [Ann. Probab. 4, 744- 771 (1976; Zbl 0365.60034)], Theorem 3, Lemma A. 1 of I. Berkes and W. Philipp [ibid. 7, 29-54 (1979; Zbl 0392.60024)], and a known technique of P. Major [Z. Wahrscheinlichkeitstheor. Verw. Gebiete 35, 221-229 (1976; Zbl 0338.60032)].

Theorem 3 is a technical stability result related to Theorem 5 of J. Kuelbs and J. Zinn [Ann. Probab. 11, 506-557 (1983; Zbl 0518.60009)]. Its proof rests on two exponential inequalities essentially due to V. V. Yurinskij [J. Multivariate Anal. 6, 473-499 (1976; Zbl 0346.60001)] and is simpler than the proof of the Theorem of Kuelbs and Zinn.