Gebelein's inequality and its consequences

M. Beśka; Z. Ciesielski

M. Beśka ; Z. Ciesielski

Banach Center Publications, Tome 72 (2006), p. 11-23 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $(X_{i}, i = 1, 2, . . .)$ be the normalized gaussian system such that $X_{i} \in N (0, 1)$ , i = 1,2,... and let the correlation matrix $ρ_{i j} = E (X_{i} X_{j})$ satisfy the following hypothesis: $C = s u p_{i \geq 1} \sum_{j = 1}^{\infty} | ρ_{i, j} | < \infty$ . We present Gebelein’s inequality and some of its consequences: Borel-Cantelli type lemma, iterated log law, Levy’s norm for the gaussian sequence etc. The main result is that (f(X₁) + ⋯ + f(Xₙ))/n → 0 a.s. for f ∈ L¹(ν) with (f,1)ν = 0.

Publié le : 2006-01-01

Zbl 1109.60025

EUDML-ID : urn:eudml:doc:282570

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     author = {M. Be\'ska and Z. Ciesielski},
     title = {Gebelein's inequality and its consequences},
     journal = {Banach Center Publications},
     volume = {72},
     year = {2006},
     pages = {11-23},
     zbl = {1109.60025},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc72-0-1}
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M. Beśka; Z. Ciesielski. Gebelein's inequality and its consequences. Banach Center Publications, Tome 72 (2006) pp. 11-23. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc72-0-1/