Extreme Value Theory for Moving Average Processes

This paper studies extreme values in infinite moving average processes Xₜ = ∑_λ cλ - t Z_λ defined from an i.i.d. noise sequence _λ\. In particular this includes the ARMA-processes often used in time series analysis. A fairly complete extremal theory is developed for the cases when the d.f. of the Z_λ's has a smooth tail which decreases approximately as exp\- zᵖ\ as z → ∞, for 0 < p < ∞, or as a power of z. The influence of the averaging on extreme values depends on p and the c_λ's in a rather intricate way. For $p = 2$, which includes normal sequences, the correlation function rₜ = ∑_λ cλ - tc_λ/∑_λ c²_λ determines extremal behavior while, perhaps more surprisingly, for p ≠ 2 correlations have little bearing on extremes. Further, the sample paths of ₜ\ near extreme values asymptotically assume a specific nonrandom form, which again depends on p and _λ\ in an interesting way. One use of this latter result is as an informal quantitative check of a fitted moving average (or ARMA) model, by comparing the sample path behavior predicted by the model with the observed sample paths.