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Haar systems and some results on the basis in a martingale space with mixed norm. (English. Russian original) Zbl 0923.60046

Theory Probab. Appl. 42, No. 3, 528-531 (1997); translation from Teor. Veroyatn. Primen. 42, No. 3, 627-633 (1997).
The main results of the paper deal with martingale spaces with mixed norm defined on the space \((\Omega, {\mathcal F}, ({\mathcal F}_n)_{n\geq 0}, {\mathbf P})\), where \(\Omega=[0,1)\), \({\mathcal F}\) is the Borel \(\sigma\)-algebra, \({\mathcal F}_n\) is generated by atoms \([2^{-n}k,\;2^{-n}(k+1)]\), \(0\leq k <2^n\), \({\mathbf P}\) is the Lebesgue measure. A condition on summation characteristics is given, which implies no unconditional bases in these spaces (a generalization of the classical result of Pelczynski for the space \(L_1(0,1)\)). A criterion of the existence of an unconditional basis in terms of the Paley function is obtained. The convergence theorem for almost all choices of signs is given.

MSC:

60G42 Martingales with discrete parameter
60G46 Martingales and classical analysis
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)