Ergemlidze, Z.; Shangua, A.; Tarieladze, V. Sample behavior and laws of large numbers for Gaussian random elements. (English) Zbl 1054.60002 Georgian Math. J. 10, No. 4, 637-676 (2003). Let \((\xi_n)\) be a sequence of symmetric Gaussian random elements in a real Banach space \(X.\) Let \(R_{\xi_n}\) be the covariance operator of \(\xi_n,\,n=1,2,\dots\). It is not assumed that \((\xi_n)\) is Gaussian. In the first sections of the paper, conditions of a.s. boundedness and convergence to zero of the sequence \((\xi_n)\) in terms of individual parameters of the members are studied. The basic result is the following statement (cf. Theorem 3.1, p. 660): (SB) If \(\beta:=\limsup_n{\mathbb E}\,\| \xi_n\| ^2<\infty\) and \(\sum_{n=1}^{\infty}\exp(-\varepsilon/\| R_{\xi_n}\| )<\infty\) for some \(\varepsilon>0\), then \(\limsup_n\| \xi_n\| \leq \sqrt{\beta}+\sqrt{2\varepsilon}\) a.s.; in particular, \(\sup_n\| \xi_n\| <\infty\) a.s. (SC) \(\lim_n{\mathbb E}\,\| \xi_n\| ^2=0\) \(\forall\varepsilon>0\), \(\sum_{n=1}^{\infty}\exp(-\varepsilon/\| R_{\xi_n}\| )<\infty \Rightarrow \lim_n\| \xi_n\| =0\) a.s.Moreover, if \(\| \xi_n\| ,n=1,2,\dots\), are pairwise independent, then we also have: (ISB) If \(\sum_{n=1}^{\infty}\exp(-\varepsilon/\| R_{\xi_n}\|)=\infty\) for some \(\varepsilon>0\), then \(\limsup\| \xi_n\| \geq \sqrt{2\varepsilon}\) a.s. (ISB1) \(\sup_n\| \xi_n\| <\infty\) a.s. if and only if \(\sup_n{\mathbb E}\,\| \xi_n\| ^2<\infty\) and \(\sum_{n=1}^{\infty}\exp(-\varepsilon/\| R_{\xi_n}\|)<\infty\) for some \(\varepsilon>0.\) (ISC) \(\lim_n\| \xi_n\| =0\) a.s. if and only if \(\lim_n{\mathbb E}\,\| \xi_n\| ^2=0\) and \(\sum_{n=1}^{\infty}\exp(-\varepsilon/\| R_{\xi_n}\|)<\infty\) \(\forall \varepsilon>0.\) From this theorem the next corollary (cf. Corollary 3.3, p. 662) is derived:We have: (ssb) \(\exists \varepsilon>0\), \(\sum_{n=1}^{\infty}\exp(-\varepsilon/{\mathbb E}\,\| \xi_n\| ^2)<\infty \Rightarrow \sup_n\| \xi_n\| <\infty\) a.s. The converse is true provided \(X\) is finite-dimensional and \(\| \xi_n\| ,n=1,2,\dots\), are pairwise independent. (ssc) \(\forall \varepsilon>0\), \(\sum_{n=1}^{\infty}\exp(-\varepsilon/{\mathbb E}\,\| \xi_n\| ^2)<\infty \Rightarrow \lim_n\| \xi_n\| =0\) a.s. The converse is true provided \(X\) is finite-dimensional and \(\| \xi_n\| ,n=1,2,\dots\), are pairwise independent. It is also shown that in the converse parts of the corollary the assumption of finite-dimensionality of \(X\) is essential (cf. Proposition 3.8, p. 664). Note that the converse parts seem to be new even in the one-dimensional case since it is required only pairwise independence of the considered random variables which, in general, may not imply the joint independence because the sequence is not assumed to be jointly Gaussian (although in the paper no example of a sequence of pairwise independent Gaussian random variables is given which is not jointly Gaussian). In the general setting the sample properties of Banach space valued Gaussian random functions were considered in many works. There are sufficient conditions for the sample boundedness and sample continuity in terms of metric entropy [see Theorem 11.21 of M. Ledoux and M. Talagrand, “Probability in Banach spaces. Isoperimetry and processes” (1991; Zbl 0748.60004)]. The questions of a.s. boundedness and convergence to zero are treated by V. Buldygin and S. Solntsev [“Asymptotic behaviour of linearly transformed sums of random variables” (1997; Zbl 0906.60002)]. Theorem 3.1 and Corollary 3.3 can be considered as an infinite-dimensional version of the one obtained earlier for the case of scalar Gaussian sequences by N. N. Vakhania [“Probability distributions on linear spaces” (1981; Zbl 0481.60002)]. A result equivalent to Corollary 3.3 when \(X=l_p\), \(1\leq p<\infty,\) is contained in [the reviewer, Tr. Vychisl. Tsentra Im. N. I. Muskhelishvili 18, No. 2, 71–79 (1978)]. The statement (SB) of Theorem 3.1 is a refinement of Theorem 2.1 of X. Fernique [C. R. Acad. Sci., Paris, Sér. I 300, 315–318 (1985; Zbl 0574.60051)]. In the second part of the paper a general sequence \((\xi_n)\) of symmetric independent random elements in \(X\) is considered. Denoting \(S_n=\sum_{k=1}^n\xi_k\), the conditions of a.s. boundedness and convergence to zero of the sequence \((\alpha_n S_n)\) are studied, where \((\alpha_n)\) is a sequence of strictly positive real numbers tending to zero. It is assumed that the sequence \((\alpha_n)\) admits a subsequence \((\alpha_{k_n})\) with the properties: \[ \limsup_n\alpha_{k_n}\sum_{j=1}^{n}{{1}\over{\alpha_{k_j}}}<\infty \qquad {\text{and}} \qquad \limsup_n{{\max_{k_n<j\leq k_{n+1}}\alpha_j}\over{\alpha_{k_{n+1}}}}<\infty. \] The following reduction principle is obtained (cf. Proposition 4.3, p. 670): The sequence \((\alpha_n S_n)\) is a.s. bounded (respectively a.s. converging to zero) if and only if the sequence \((\alpha_{k_n}(S_{k_{n}}-S_{k_{n-1}}))\) consisting of independent random elements is a.s. bounded (respectively a.s. converging to zero). If \(\alpha_n=1/n\) and \(k_n=2^n\), then this result in case of real random variables was obtained by Yu. V. Prokhorov [Izv. Akad. Nauk SSSR, Ser. Mat. 14, 523–536 (1950; Zbl 0040.07302)]. The reduction principle obtained by other authors (M. Loeve; T. A. Azlarov and N. A. Volodin; V. V. Buldygin; V. Buldygin and V. Solntsev; A. I. Martikainen and V. V. Petrov; R. Wittmann; S. Chobanyan, S. Levental and V. Mandrekar) are commented in details (cf. Remark 4.4, p. 671). Using the reduction principle from Theorem 3.1 the strong laws of large numbers in the form of Yu. V. Prokhorov’s are obtained (cf. Theorem 4.5, p. 672).The paper is self-contained, most of auxiliary results are given with proofs. Reviewer: Vakhtang V. Kvaratskhelia (Tbilisi) MSC: 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60B11 Probability theory on linear topological spaces Keywords:almost sure convergence; Banach space; Gaussian measure; Gaussian random element; covariance operator; strong law of large numbers Citations:Zbl 0748.60004; Zbl 0906.60002; Zbl 0481.60002; Zbl 0574.60051; Zbl 0040.07302 PDFBibTeX XMLCite \textit{Z. Ergemlidze} et al., Georgian Math. J. 10, No. 4, 637--676 (2003; Zbl 1054.60002) Full Text: EuDML