Properties of the Wold Decomposition of Stationary Stochastic Processes

E. A. Robinson

Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 2, pp. 201-211

Résumé

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.