Causal time series

A causal time series can be expressed using present and past innovations, without future innovations. The autoregressive process of order one with $|\alpha |<1$ has the mean-square convergent causal representation $Y_t={\sum }_{j\ge 0}{\alpha }^j{\epsilon }_{t-j}$.

A causal time-series representation expresses a time series using present and past driving white noise. Its coefficients vanish at negative lags. Square summability ensures mean-square convergence for a white-noise input; absolute summability is the stronger stable-filter convention. Causality refers to the particular driving sequence, not merely to stationary existence.