Some properties of trigonometric series whose terms have random signs

Trigonometric series of the type $$ \sum\limits_1^{\infty } {{\varphi_n}(t)\left( {{a_n}\;\cos nx + {b_n}\;\sin nx} \right)} $$ (0.1) where \( \left\{ {{\varphi_n}(t)} \right\} \) denotes the system of Rademacher functions, have been extensively studied in order to discover properties which belong to “almost all” series, that is to say which are true for almost all values of t.1 We propose here to add some new contributions to the theory.