First and second moments for self-similar couplings and Wasserstein distances

Jonathan M. Fraser

    Research output: Contribution to journalArticlepeer-review


    We study aspects of the Wasserstein distance in the context of self-similar measures. Computing this distance between two measures involves minimising certain moment integrals over the space of couplings, which are measures on the product space with the original measures as prescribed marginals. We focus our attention on self-similar measures associated to equicontractive iterated function systems consisting of two maps on the unit interval and satisfying the open set condition. We are particularly interested in understanding the restricted family of self-similar couplings and our main achievement is the explicit computation of the 1st and 2nd moment integrals for such couplings. We show that this family is enough to yield an explicit formula for the 1st Wasserstein distance and provide non-trivial upper and lower bounds for the 2nd Wasserstein distance for these self-similar measures.

    Original languageEnglish
    Pages (from-to)2028-2041
    Number of pages14
    JournalMathematische Nachrichten
    Issue number17-18
    Publication statusPublished - 1 Dec 2015


    • Bernoulli convolution
    • self-similar coupling
    • self-similar measure
    • Wasserstein metric


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