Celebratio Mathematica — Rosenblatt

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M. Rosenblatt: “Non-Gaussian autoregressive and moving average schemes,” pp. 731–737 in Asymptotic methods in probability and statistics (Ottawa, 8–13 July 1997). Edited by B. Szyszkowicz. North-Holland (Amsterdam), 1998. A volume in honor of Miklós Csörgő on his 65th birthday. MR 1661514 Zbl 0946.62086 incollection

Consider the systems of equations \[ \sum_{k=0}^p\tilde{\phi}_kx_{t-k} = \sum_{j=0}^q \tilde{\theta}_j\xi_{t-j} \] where \( \xi_t \)’s are independent, identically distributed random variables with \( E\xi_t \equiv O \), \( E\xi_t^2 \equiv 1 \). Let \( \tilde{\phi}_0 = \tilde{\theta}_0 = 1 \) and \( \tilde{\phi}_p, \tilde{\theta}_q \neq 0 \). The polynomials \[ \phi(z) = \sum_{k=0}^p\tilde{\phi}_k z^k \quad\text{and}\quad \theta(z) = \sum_{j=0}^q\tilde{\theta}_j z^j \] are assumed to have no roots in common. There is then a stationary solution \( \{x_t\} \) of the system if and only if \( \phi(z) \) has no roots of absolute value one and this solution is uniqely determined. The solution is called an autoregressive moving average (ARMA) process. If \( \phi(z) \equiv 1 \) the solution is a moving average (MA) while if \( \theta(z) \equiv 1 \) it is an autoregressive scheme (AR). If all the roots of \( \phi(z) \) and \( \theta(z) \) are outside the unit disc \( |z| \leq 1 \) in the complex plane we shall call the solution minimum phase. It is well known that in the case of a stationary ARMA minimum phase process, the best predictor of \( x_t \) in mean square given the past is linear [Shepp et al. 1980]. A necessary condition for the best predictor to be linear is given in terms of a functional equation. This condition is then used in the case of autoregressive schemes and moving averages. A simple example is given of a nonminimum phase scheme in which the best predictor is nonlinear.

Barbara Szyszkowicz	Related
Miklós Csörgő	Related

M. Rosenblatt: “Short range and long range dependence,” pp. 283–294 in Asymptotic laws and methods in stochastics (Ottawa, 3–6 July 2012). Edited by D. Dawson, R. Kulik, M. Ould Haye, B. Szyszkowicz, and Y. Zhao. Fields Institute Communications 76. Fields Institute (Toronto), 2015. A volume in honour of Miklós Csörgő on the occasion of his 80th birthday. MR 3409836 Zbl 1368.60038 incollection

Rafał Kulik	Related
Barbara Szyszkowicz	Related
Miklós Csörgő	Related
Donald Dawson	Related
Yiqiang Q. Zhao	Related
Mohamedou Ould Haye	Related