The essence of this paper is the exact evaluation of var S,, where S, is the number of peaks in a segment of n readings from a stationary Gaussian autoregressive moving-average (ARMA) process, and of the asymptotic normality of (S, -ES,)/(var S,,)'P. The emphasis is on the AR(1) and MA(1) cases, motivated by Stigler (Estimating serial correlation by visual inspection of diagnostic plots. Am.Statistician 40 (1986), 111-16). The evaluation of varS, is based on a discussion of closed-form calculation of orthant probabilities for a zero mean quadrivariate normal with correlation structure new to the literature. Related issues are the power of the peaks test against stationary alternatives and the good fit of the normal even for small n, validated by simulation.

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Limit theorem, sample autocorrelations, sample partial autocorrelations, Quenouille's test,

We shall show that for any M A(2) process (apart from those with coefficients θ 1 , θ 2 lying on certain line-segments) there is one and only one invertible M A(2) process with the same autocovariances γ 0 , γ 1 , γ 2 . It is this invertible version which computer-packages fit, regardless, even if data came from a non-invertible M A(2) process. This has consequences for prediction from a fitted process, inasmuch as such prediction would seem to be inappropriate. We express the coefficients θ 1 , θ 2 of the invertible version in terms of γ 0 , γ 1 , γ 2 explicitly using analytical reasoning, following a graphical approach of Sbrana (2012) which indicates this result within the invertibility region. We also express (θ 1 , θ 2 ) in the non-invertibility region.