Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 29 / 1 / ii / Solution

An invertible time-series representation recovers the driving white noise from current and past observations. In the inverse series the support condition is thereforeAgain the series must converge. Stable invertibility uses ${\sum }_{r\ge 0}|b_r|<\infty$, which ensures mean-square convergence when $X_t$ has finite variance. Merely writing a bilateral inverse is not invertibility in this one-sided sense: it may require future observations. For a general correlated input $X$, square summability of $b_r$ alone is not the same sufficient condition as it is for a white noise input.