Almost Everywhere Convergence and SLLN Under Rearrangements
The almost everywhere (a.e.) convergence of trigonometric Fourier series for L 2(0, 1) functions was conjectured by Luzin (1922) and was partially solved by Kolmogorov and Silvestrov in (1925). The full solution was given by Carleson (1966). In the work of Garsia (1964), (see Garsia (1970)) almost everywhere convergence of a rearrangement of series of orthogonal functions was initiated using the so-called Garsia Inequality (GI). His convergence result was generalized by Nikishin (1967) who removed the assumption of orthogonality. We prove an inequality which generalizes GI using the technique introduced in Chobanyan (1990) (for other references see Chobanyan (1994)). As a consequence of this inequality we derive GI and a generalization of Nikishin’s result to a series of Banach space valued random variables.
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