contains every two-dimensional normed space
David Yost (1988)
Annales Polonici Mathematici
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David Yost (1988)
Annales Polonici Mathematici
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Sreela Gangopadhyay (1990)
Colloquium Mathematicae
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Y. Gordon, A. E. Litvak, A. Pajor, N. Tomczak-Jaegermann (2007)
Studia Mathematica
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We show that, given an n-dimensional normed space X, a sequence of independent random vectors , uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map defined by embeds X in with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into with asymptotically best possible relation between N, n, and ε.
Andrew Rosalsky, Yongfeng Wu (2015)
Applications of Mathematics
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Let be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let . Conditions are given for completely and for completely. As an application of these results, we obtain a complete convergence theorem for the row sums of the dependent bootstrap samples arising from a sequence of i.i.d. random variables .
Larry Kitchens, Charles Swartz (1974)
Colloquium Mathematicae
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Félix Cabello Sánchez (1999)
Studia Mathematica
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Let X be a normed space and the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if acts transitively on the unit sphere then X must be an inner product space.
Gennadiy Averkov, Horst Martini (2009)
Colloquium Mathematicae
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Let be a d-dimensional normed space with norm ||·|| and let B be the unit ball in . Let us fix a Lebesgue measure in with . This measure will play the role of the volume in . We consider an arbitrary simplex T in with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of are determined. For d ≥ 3 it is noticed that the tight lower bound of is zero.
Yongfeng Wu, Jiangyan Peng (2018)
Kybernetika
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The authors first establish the Marcinkiewicz-Zygmund inequalities with exponent () for -pairwise negatively quadrant dependent (-PNQD) random variables. By means of the inequalities, the authors obtain some limit theorems for arrays of rowwise -PNQD random variables, which extend and improve the corresponding results in [Y. Meng and Z. Lin (2009)] and [H. S. Sung (2013)]. It is worthy to point out that the open problem of [H. S. Sung, S. Lisawadi, and A. Volodin (2008)] can be...
David Brink (2015)
Acta Arithmetica
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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula with convergence as , in much the same way as the Euler transformation gives with convergence as . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...
Alain Faisant, Georges Grekos, Ladislav Mišík (2016)
Mathematica Bohemica
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Let be a convergent series of positive real numbers. L. Olivier proved that if the sequence is non-increasing, then . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having ; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the -convergence, that is a convergence according to an ideal of subsets of . Again, Olivier’s theorem is a consequence...
S. V. Konyagin, V. N. Temlyakov (2003)
Studia Mathematica
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We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as , where . We study a generalized version of that we call the weak thresholding approximation. We modify the in the following way. For ε > 0, t ∈ (0,1) we set and consider...
Ferenc Móricz (2005)
Colloquium Mathematicae
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The concepts of statistical convergence of single and double sequences of complex numbers were introduced in [1] and [7], respectively. In this paper, we introduce the concept indicated in the title. A double sequence is said to be regularly statistically convergent if (i) the double sequence is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence is statistically convergent to some for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence is statistically convergent...
Gh. Constantin (1973)
Matematički Vesnik
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Rafał Kapica (2003)
Colloquium Mathematicae
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Given a probability space (Ω,, P) and a closed subset X of a Banach lattice, we consider functions f: X × Ω → X and their iterates defined by f¹(x,ω) = f(x,ω₁), , and obtain theorems on the convergence (a.s. and in L¹) of the sequence (fⁿ(x,·)).
W. Szczotka (2006)
Applicationes Mathematicae
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For each n ≥ 1, let and be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let , , t ≥ 0, and . The main result gives conditions under which the weak convergence , where X is a Lévy process, implies and , where and are mutually independent Lévy processes and .
Alexander R. Pruss (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
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Let Ω be a countable infinite product of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have , where and E* are the lower and upper expectations....
I. Assani (2005)
Colloquium Mathematicae
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We answer a question of H. Furstenberg on the pointwise convergence of the averages , where U and R are positive operators. We also study the pointwise convergence of the averages when T and S are measure preserving transformations.
Itai Benjamini, Alain-Sol Sznitman (2008)
Journal of the European Mathematical Society
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We consider random walk on a discrete torus of side-length , in sufficiently high dimension . We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time . We show that when is chosen small, as tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const . Moreover, this connected component occupies a...
Andrzej Kłopotowski
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CONTENTSIntroduction.......................................................................................................................................................................... 5 I. Infinitely divisible probability measures on ....................................................................................... 6 II. The classical limit theorems for sums of independent random vectors................................................ 14 III. Convergence in law to ℒ (,...