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Short Communications

Properties of the Wold Decomposition of Stationary Stochastic Processes

E. A. Robinson

Institute of Statistics, Uppsala University

Abstract: The basic results of the paper (Theorems 11–13) treat the representation of the quantities $\hat x_{t+\alpha}$ –the best predictors of the quantities $x_{t+\alpha}$ of a process, which is stationary in the wide sense, from the quantities $x_s,s\leq t$ – in the form of a series
$$\hat x_{t+\alpha}\sim\sum\limits_{s=0}^\infty{k_s x_{t-s}},$$
where the coefficients ${k_s}$ satisfy the condition $\sum|k_s|^2<\infty$. Certain properties of the sequences $\{w_t\},\sum{|w_t|}^2<\infty$, are derived first.

Received: 06.05.1961

Language: English

 English version: Theory of Probability and its Applications, 1963, 8:2, 187–195