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On m-dimensional stochastic processes in Banach spaces.

Rodolfo De Dominicis, Elvira Mascolo (1981)

Stochastica

In the present paper the authors prove a weak law of large numbers for multidimensional processes of random elements by means of the random weighting. The results obtained generalize those of Padgett and Taylor.

On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets

Stanisław Kwapień, Jerzy Sawa (1993)

Studia Mathematica

The paper deals with the following conjecture: if μ is a centered Gaussian measure on a Banach space F,λ > 1, K ⊂ F is a convex, symmetric, closed set, P ⊂ F is a symmetric strip, i.e. P = {x ∈ F : |x'(x)| ≤ 1} for some x' ∈ F', such that μ(K) = μ(P) then μ(λK) ≥ μ(λP). We prove that the conjecture is true under the additional assumption that K is "sufficiently symmetric" with respect to μ, in particular it is true when K is a ball in Hilbert space. As an application we give estimates of Gaussian...

On some definition of expectation of random element in metric space

Artur Bator, Wiesław Zięba (2009)

Annales UMCS, Mathematica

We are dealing with definition of expectation of random elements taking values in metric space given by I. Molchanov and P. Teran in 2006. The approach presented by the authors is quite general and has some interesting properties. We present two kinds of new results:• conditions under which the metric space is isometric with some real Banach space;• conditions which ensure "random identification" property for random elements and almost sure convergence of asymptotic martingales.

On the best constant in the Khinchin-Kahane inequality

Rafał Latała, Krzysztof Oleszkiewicz (1994)

Studia Mathematica

We prove that if r i is the Rademacher system of functions then ( ʃ i = 1 n x i r i ( t ) 2 d t ) 1 / 2 2 ʃ i = 1 n x i r i ( t ) d t for any sequence of vectors x i in any normed linear space F.

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