Symmetrization of probability measures, pushforward of order 2 and the Boolean convolution

Volume 96 / 2011

Wojciech Młotkowski, Noriyoshi Sakuma Banach Center Publications 96 (2011), 271-276 MSC: 46L53, 60E10. DOI: 10.4064/bc96-0-18


We study relations between the Boolean convolution and the symmetrization and the pushforward of order 2. In particular we prove that if $\mu_1,\mu_2$ are probability measures on $[0,\infty)$ then $\left(\mu_1\uplus\mu_2\right)^{\mathbf{s}} =\mu_1^{\mathbf{s}}\uplus\mu_2^{\mathbf{s}}$ and if $\nu_1,\nu_2$ are symmetric then $\left(\nu_1\uplus\nu_2\right)^{(2)} =\nu_1^{(2)}\uplus\nu_2^{(2)}$. Finally we investigate necessary and sufficient conditions under which the latter equality holds.


  • Wojciech MłotkowskiMathematical Institute, University of Wrocław
    Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
  • Noriyoshi SakumaDepartment of Mathematics, Aichi University of Education
    1 Hirosawa, Igaya-cho, Kariya-shi, 448-8542, Japan

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