Tail triviality for sums of stationary random variables (Q910812)

Let \((X_ n)_{n\in {\mathbb{Z}}}\) be a strictly stationary sequence of real random variables and \(S_ n\) their partial sums, i.e. \(S_ 0=0\) and \(S_{n+1}-S_ n=X_ n\) (n\(\in {\mathbb{Z}})\). The authors study the relation between the following tail fields and shift-invariant \(\sigma\)- fields:\(\quad G_{\infty}=\cap_{M,N\geq 0}\sigma (S_ m,S_ n:\quad m\leq -M,\quad n\geq N),\quad G_{\infty}^{inv}=\cap_{M,N\geq 0}\sigma (S_ n-S_ m:\quad m\leq -M,\quad n\geq N),\quad F_{\infty}=\cap_{M,N\geq 0}\sigma (X_ n,X_ m:\quad m\leq -M,\quad n\geq N).\)It is first shown that \(G_{\infty}^{inv}=F_{\infty}\) if \(X_ 0\) has finite entropy and is integer-valued. Next, a zero-two law is established extending results of \textit{A. J. Stam} [Compos. Math. 17, 268-280 (1966; Zbl 0214.437); and ibid. 18, 201-228 (1967; Zbl 0201.202)], \textit{D. S. Ornstein} [Trans. Amer. Math. Soc. 138, 1-43 (1969; Zbl 0181.445)], and the first author [Random walks with stationary increments and renewal theory (1979; Zbl 0443.60083)]. Finally it is proved that the triviality of the positive tail field \(F^+_{\infty}\) (resp. \(F_{\infty})\) implies that of \(G^+_{\infty}\) (resp. \(G_{\infty})\), provided \(X_ 0\) is integer valued and \(S_ n\) is strongly aperiodic.