How can a stationary, invertible ARMA(1,1) process be represented as either an infinite order AR or infinite order MA process?
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Finite moving average processes have infinite autoregressive representations if they satisfy the invertibility conditions given in Box, Jenkins and Reinsel (1994) page 70. Mixed models such as ARMA(1,1) have both infinite moving average and infinite autoregressive representations if they are stationary and satisfy invertibility conditions. This is also shown in Box, Jenkins and Reinsel (1994) pp. 77-78 where you can see how the representations are constructed.
Check out Time Series Modeling, Inference and Forecasting, page 65. "If the ARMA process is causal and invertible, it can be written either as AR (MA) process of infinite order."