On independent and stationary subsystems of the Walsh system and periodic multiplicative systems (Q1033930)

Let \((X_k)_1^{\infty}\) be a system of random variables, and let \textsl{E}\(X_k=0.\) Let \(m\geq 2\) be a fixed integer. A system \((X_k)\) is said to be independent in \(m\)-subsystems if any of its subsystems containing \(m\) elements is independent. A system \((X_k)\) is said to be \(m\)-multiplicative if \textsl{E}\((X_{k_1}\ldots X_{k_s})=0\) for \(s\leq m\) and \(1\leq k_1<\ldots <k_s.\) A system \((X_k)\) is said to be \(m\)-stationary if, for \(p\geq 1\) and \(i_1<\ldots <i_m,\) the distributions of the vectors \((X_{i_1},\ldots X_{i_m})\) and \((X_{i_1+p},\ldots, X_{i_m+p})\) coincide. In the first part the author proves that above-defined properties are equivalent for subsystems of the orthonormal Walsh system. Second part is devoted to analogues and distinctions for subsystems of periodic multiplicative Vilenkin systems.