Abstract
In Chapter 13 we turn to the old problem of characterizing the sequences (a m ) such that for each orthonormal sequence (φ m ) the series ∑ m≥1 a m φ m converges a.s., which has recently been solved by A. Paszkiewicz. Using a more abstract point of view, we present a very much simplified proof of his results (due essentially to W. Bednorz). This leads us to the question of discussing when a certain condition on the “increments” of a process implies its boundedness. When the increment condition is of “polynomial type”, this is more difficult than in the case of Gaussian processes, and requires the notion of “majorizing measure”. We present several elegant results of this theory, in their seemingly final form recently obtained by W. Bednorz.
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Talagrand, M. (2014). Convergence of Orthogonal Series; Majorizing Measures. In: Upper and Lower Bounds for Stochastic Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54075-2_13
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DOI: https://doi.org/10.1007/978-3-642-54075-2_13
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