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Currently displaying 1 – 2 of 2

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Asymptotic formulas for the error in linear interpolation

M. Beśka; K. Dziedziul — 2005

Banach Center Publications

We give the asymptotic formula for the error in linear interpolation with arbitrary knots.

Gebelein's inequality and its consequences

M. Beśka; Z. Ciesielski — 2006

Banach Center Publications

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.

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