Generalization of some classical inequalities in the theory of orthogonal series

Abstract

Let {X_i} ^∞_−∞ be a sequence of random variables, E(X_i) ≡ 0. If ν ≥1, estimates for the ν-th moments$\max _{1 \leqslant k \leqslant n} \left| {\sum\nolimits_{a + 1}^{a + k} {X_i } } \right|$ can be derived from known estimates$\left| {\sum\nolimits_{a + 1}^{a + n} {X_i } } \right|$ of the ν-th moment. Here we generalized the Men'shov-Rademacher inequality for ν=2 for orthonormal X_i, to the case ν≥1 and dependent random variables. The Men'shov-Payley inequality ν>2 for orthonormal X_i) is generalized for ν>2 to general random variables. A theorem is also proved that contains both the Erdös -Stechkin theorem and Serfling's theorem withv > 2 for dependent random variables.