Limit Behaviour of Convolution Products of Probability Measures

Limit Behaviour of Convolution Products of Probability Measures Positivity 5: 51–63, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands. Limit Behaviour of Convolution Products of Probability Measures 1 2 WOLFHARD HANSEN and IVAN NETUKA Universität Bielefeld, Fakultät für Mathematik, Postfach 100 131, 33501 Bielefeld, Germany. E-mail: Mathematical Institute, Charles University, Sokolovská 83, 186 75 Praha, Czech Republic. E-mail: ( Corresponding author) (Received 18 May 1999; accepted 7 November 1999) AMS subject classifications: 60B10; 28A33 Key words: Convergence of probability measures, convolution 1. Introduction For r> 0let m denote normalized Lebesgue measure on the ball B.0;r/ with centre at the origin and radius r . The following question asked by Choquet (see [2], p. 478) on the occasion of the International Conference on Potential Theory (1994, Kouty, Czech Republic) was the starting point of our investigations of the limit behaviour of successive convolutions of probability measures: Let f be a con- tinuous real function on R and let r ;r ;r ;::: be strictly positive real numbers. 1 2 3 Under what conditions on f and the sequence .r / does .f m m  m / n r r r 1 2 n converge to a harmonic function? An answer to this question Positivity Springer Journals

Limit Behaviour of Convolution Products of Probability Measures

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Kluwer Academic Publishers
Copyright © 2001 by Kluwer Academic Publishers
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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