# (Non-limit) distribution of maxima from different univariate, discrete and stationary time series

Motivation: I'm currently studying the convergence of maxima from simulated time series to max-stable distributions, and in order to do so, I want to better understand the penultimate distribution of such maxima in non-limit cases. I tried to read Leadbetter et al. (1983) "Extremes and related properties of random sequences and processes", but did not get too far.

Starting point: I understand that maxima $$M_n$$ from a sequence of n iid r.v. $$X_1, X_2, ..., X_n$$ are distributed according to $$P(M_n\leq x)=P(X_1\leq x, X_2\leq x,...,X_n\leq x))=F^n(x)$$. Thus I manage to derive the penultimate distribution of maxima from a white-noise process with Gaussian innovations.

Question 0 (New): Is $$F^n(x)$$ even the penultimate distribution of the Gaussian, or is this a first misunderstanding? According to Cohen (1982), the penultimate distribution of maxima from a sequence of Gaussians is the Type III Extreme Value distribution / GEV with shape parameter $$\xi<0$$.

Question 1: Is there are closed-form distribution of maxima from an ARMA(2,2) process with Gaussian innovations? I found something on arxiv on maxima of ARMA(1,1) processes. But if there is not such distribution, ...

Question 2: ... what is the unconditional (i.e. $$P(X_i, not $$P(X_i) distribution of ARMA(2,2)-generated r.v., such that one could employ $$F^n(x)$$ (either ignoring that fact that these r.v. are not independent, or adjusting $$n$$ such that it accounts for the dependence, e.g. such that $$n^*=n/k$$, where $$k$$ is the lag where the ACF/PACF are close enough to zero).

I'm thankful for both every hint you can give me, but also for pointing out if my questions are nonsense.

• One immediate point worth making would be to amend your question to read "distribution(s) of maxima." That said and as Yves points out in his answer, Embrechts, et al., provide the single best introduction and overview wrt Modeling Extremal Events. The workhorse distribution for EVT models is the GPD (generalized Pareto distribution) but many other distributional assumptions are possible. The choice is as much a function of field, specialization and training as anything since it can be demonstrated that many distributions will fit the same information. – Mike Hunter Aug 9 at 18:49

Question 2. You want the stationary distribution of the Gaussian AR process $$X_t$$ $$(1 - \phi_1 B - \dots - \phi_p B^p) \, X_t = (1 + \theta_1 B + \dots + \theta_q B^q) \,\varepsilon_t$$ for the special case $$p=q=2$$. This distribution a.k.a. the invariant distribution is a Gaussian distribution: its mean $$\mu_X$$ and sd $$\sigma_X$$ can be found. In the case where $$\varepsilon_t$$ has mean zero we have $$\mu_X = 0$$ and $$\sigma_X^2 = \sigma_\zeta^2 \sum_{k \geq 0} \psi_k^2$$ where the coefficients $$\psi_k$$ are the "psi weights" of the $$\text{MA}(\infty)$$ representation $$X_t = \sum_{k \geq 0} \psi_k \zeta_{t-k}$$ where $$\zeta_t$$ is a Gaussian white noise. The "psi weights" are computed by many R packages. An alternative derivation uses the ARMA model in state-space form: the state equation defines a vector AR(1) process with $$r:= \max\{p, \, q + 1\}$$. We can assume that the observed series is the first component of the state $$\boldsymbol{\alpha}_t$$ in the model \begin{align*} \boldsymbol{\alpha}_t &= \mathbf{T} \boldsymbol{\alpha}_{t-1} + \boldsymbol{\eta}_t\\ X_t &= \alpha_{1,t} \end{align*} where both the $$r \times r$$ transition matrix $$\mathbf{T}$$ and the covariance of the Gaussian white noise $$\boldsymbol{\eta}_t$$ depend on the ARMA coefficients $$\phi_i$$, $$\theta_j$$. The stationary covariance of the state $$\boldsymbol{\alpha}_t$$ can be computed by solving a linear system. See e.g. Chap. 4 of Harvey A.C. Time Series Models. For the special case $$p = q= 2$$ you can find a closed form for the variance if needed.
Question 0. No, $$F_X^n(x)$$ is not the cited penultimate distribution, which is a a Generalized Extreme Value (GEV) with negative shape $$\xi_n < 0$$ depending on $$n$$. The penultimate approximation depending on $$n$$ improves the convergence rate compared to the ultimate distribution (here Gumbel). See p. 151 in Embrechts P., Klüppelberg C. and Mikosch T. for a discussion. In the article by Cohen (1982) cited in OP, a penultimate approximation is found for a sequence of i.i.d. normal and is shown to be such that an approximation with rate $$O\{(\log n)^{-2}\}$$ results instead of the $$O\{(\log n)^{-1}\}$$ rate known to hold for the Gumbel approximation. In Theorem 3, the case of a Gaussian stationary times series $$X_t$$ is considered; It is shown that under mild conditions on the autocorrelation sequence, the distribution of the maximum differs from that of the maximum $$n$$ i.i.d. rv.s with the same margin by $$O\{(\log n)^{-2}\}$$. So by triangle inequality the penultimate approximation still leads to the better rate of convergence when applied to the maximum of stationary Gaussian sequences.
Question 1. I doubt that a closed-form expression would be of a great practical interest. I think that a good approximation can be obtained as $$F_{M_n}(x)\approx F_X^{n\theta} (x)$$ where $$\theta \in (0,\,1)$$ depends on $$n$$ and on the ARMA coefficients. For a given size $$n$$ and given parameters we can find a $$\theta$$ that leads to a good approximation for $$x$$ large enough, say for $$x > 0.95$$. Indeed the Gaussian $$\text{ARMA}(p,\,q)$$ process with given coefficients is easy to simulate from and thus it is easy to simulate a sample of maxima $$M_n$$ and then find a good value for $$\theta$$ by censoring the small values of $$M_n$$.