Martingale methods in real analysis (Q2726253)

It became clear after the articles by \textit{A. V. Skorokhod} [Teor. Veroyatn. Primen. 1, 289-319 (1956; Zbl 0074.33802)] and by \textit{A. N. Kolmogorov} [ibid. 1, 239-247 (1956; Zbl 0074.34102)] on the now famous ``Skorokhod topology'' that functions with discontinuities of the first kind and especially piecewise constant functions are very convenient for the study of probability. This idea is central for the authors' approach to the problem of convergence for a class of ``lacunary-like'' functional series: the terms of these series are approximated by piecewise constant functions in such a way that the high accuracy of the approximation guarantees that both the initial series and the new one have the same convergence properties (are equiconvergent). At the same time the series with piecewise constant terms is a martingale or at least some object close to a martingale. In this situation the general martingale convergence theorem can be used. The class of examples includes the Haar (wavelets) series, lacunary Fourier-like series and more complicated constructions. These results are used for the evaluation of Lyapunov exponents for a product of purely deterministic matrices depending on a parameter. Other possible applications are related to spectral analysis.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].