Rosenblatt, M. A comment on a conjecture of N. Wiener. (English) Zbl 1165.60016 Stat. Probab. Lett. 79, No. 3, 347-348 (2009). Let \(X_t\) be a stationary process and le \(\mathcal{B}_t=\mathcal{B}\{X_s,s<t\}\) be the \(\sigma\)-filed generated by the r.v.’s \(X_s, s<t\). N. Wiener [Nonlinear problems in random theory. Chapman and Hall (1958; Zbl 0121.12302)] posed the question of under what circumstances a stationary process \(X_t\) could have a one-sided representation \(X_t=f(\xi_t,\xi_{t-1},\cdots)\), in terms of iid r.v.’s. It was conjectured there that the necessary and sufficient condition for such a representation to hold was that the backward tail field \(\mathcal{B}_{-\infty}=\cap_t\mathcal{B}_t={\emptyset,\Omega}\) be trivial. This note shows that there are stationary sequences \(X_t\) with trivial tail field that cannot have such a one-sided representation in terms of iid r.v.’s. Reviewer: Pedro A. Morettin (São Paulo) Cited in 6 Documents MSC: 60G10 Stationary stochastic processes 37M10 Time series analysis of dynamical systems Keywords:stationary process; backward tail field; conjecture of Wiener Citations:Zbl 0121.12302 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Cornfield, I.; Sinai, Ya., Basic notions of ergodic theory and examples of dynamical systems, (Sinai, Ya., Dynamical Systems II (1989), Springer-Verlag), 2-27 · Zbl 1417.37021 [2] Hanson, D., On the representation problem for stationary stochastic processes with trivial tail field, J. Math. Mech., 12, 293-301 (1963) · Zbl 0139.34405 [3] Kalikow, S., \(T, T^{- 1}\) transformation is not loosely Bernoulli, Ann. Math., 115, 393-409 (1982) · Zbl 0523.28018 [4] Ornstein, D., Ergodic Theory, Randomness and Dynamical Systems (1974), Yale University Press · Zbl 0296.28016 [5] Rosenblatt, M., Stationary Markov chains and independent random variables, J. Math. Mech., 9, 945-950 (1960) · Zbl 0096.34004 [6] Wiener, N., Nonlinear Problems in Random Theory (1958), MIT Press, John Wiley · Zbl 0121.12302 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.