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This note aims at providing some information about the concept of a strongly proximal compact transformation semigroup. In the affine case, a unified approach to some known results is given. It is also pointed out that a compact flow (X,𝓢) is strongly proximal if (and only if) it is proximal and every point of X has an 𝓢-strongly proximal neighborhood in X. An essential ingredient, in the affine as well as in the nonaffine case, turns out to be the existence of a unique minimal subset.

Some remarks on P. Levy's theorem for a net of discrete measures and the purity law on locally compact semigroups.

A. Janssen (1981)

Semigroup forum

Square variation of Brownian paths in Banach spaces.

Chang, Mou-Hsiung (1982)

International Journal of Mathematics and Mathematical Sciences

Stability of stochastic optimization problems - nonmeasurable case

Petr Lachout (2008)

Kybernetika

This paper deals with stability of stochastic optimization problems in a general setting. Objective function is defined on a metric space and depends on a probability measure which is unknown, but, estimated from empirical observations. We try to derive stability results without precise knowledge of problem structure and without measurability assumption. Moreover, $ε$ -optimal solutions are considered. The setup is illustrated on consistency of a $ε$ - $M$ -estimator in linear regression model.

Stability, sensitivity and sensitivity of characterizations

Bogusława Bednarek-Kozek, Andrzej Kozek (1980)

Banach Center Publications

Stone-Weierstrass type theorems for large deviations.

Comman, Henri (2008)

Electronic Communications in Probability [electronic only]

Strassen's law of the iterated logarithm

James D. Kuelbs (1974)

Annales de l'institut Fourier

Strassen’s functional form of the law of the iterated logarithm is formulated for partial sums of random variables with values in a strict inductive limit of Frechet spaces of Hilbert space type. The proof depends on obtaining Berry-Essen estimates for Hilbert space valued random variables.

Strong laws of large numbers for sequences of blockwise and pairwise $m$ -dependent random variables in metris spaces

Nguyen Van Quang, Pham Tri Nguyen (2016)

Applications of Mathematics

The aim of the paper is to establish strong laws of large numbers for sequences of blockwise and pairwise $m$ -dependent random variables in a convex combination space with or without compactly uniformly integrable condition. Some of our results are even new in the case of real random variables.

Strong laws of large numbers in certain linear spaces

Wojbor A. Woyczynski (1974)

Annales de l'institut Fourier

In this paper we are concerned with the norm almost sure convergence of series of random vectors taking values in some linear metric spaces and strong laws of large numbers for sequences of such random vectors. Section 2 treats the Banach space case where the results depend upon the geometry of the unit cell. Section 3 deals with spaces equipped with a non-necessarily homogeneous $F$ -norm and in Section 4 we restrict our attention to sequences of identically distributed random vectors.

Sums of independent Banach space valued random variables

Jørgen Hoffmann-Jørgensen (1974)

Studia Mathematica

Sums of independent Banach space valued random variables (after J. Hoffmann-Jørgensen)

S. Kwapien (1972/1973)

Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")

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