Limit Theorems for the Maximum Term in Stationary Sequences

Let $\{X_n, n = 0, \pm 1, \cdots\}$ be a real valued discrete parameter stationary stochastic process on a probability space $(\Omega, \mathscr{F}, P);$ for each $n = 1, 2, \cdots$, let $Z_n = \max (X_1, \cdots, X_n)$. We shall find general conditions under which the random variable $Z_n$ has a limiting distribution function (d.f.) as $n \rightarrow \infty$; that is, there exist sequences $\{a_n\}$ and $\{b_n\}, a_n > 0$, and a proper nondegenerate d.f. $\Phi(x)$ such that \begin{equation*}\tag{1.1}\lim_{n \rightarrow \infty} P\{Z_n \leqq a_nx + b_n\} = \Phi(x)\end{equation*} for each $x$ in the continuity set of $\Phi(x)$. The simplest type of stationary sequence $\{X_n\}$ is one in which the random variables are mutually independent with some common d.f. $F(x)$. In this case, $Z_n$ has the d.f. $F^n(x)$ and (1.1) becomes \begin{equation*}\tag{1.2}\lim_{n \rightarrow \infty} F^n(a_nx + b_n) = \Phi(x).\end{equation*} It is well known that in (1.2) $\Phi(x)$ is of one of exactly three types; necessary and sufficient conditions on $F$ for the validity of (1.2) are also known [9]. The three types are usually called extreme value d.f.'s [10]. Theorem 2.1 gives the limiting d.f. of $Z_n$ in a stationary sequence satisfying a certain condition on the upper tail of the conditional d.f. of $X_1$, given the "past" of sequence: the limiting d.f. is a simple mixture of extreme value d.f.'s of a single type. These are the same kind of d.f.'s found by us [3] to be the limiting d.f.'s of maxima in sequences of exchangeable random variables. The conditions of Theorem 2.1 are specialized to exchangeable and Markov sequences, and Theorem 2.2 extends the methods of Theorem 2.1 to general (not necessarily stationary) Markov sequences. It is shown that stationary Gaussian sequences, except for the trivial case of independent, identically distributed Gaussian random variables, do not obey the requirements of the hypothesis of Theorem 2.1: hence, Sections 3, 4, and 5 are devoted to a detailed study of the maximum in a stationary Gaussian sequence. Theorem 3.1 provides conditions on the rate of convergence of the covariance sequence to 0 which are sufficient for $Z_n$ to have the same extreme value limiting d.f. as in the case of independence, namely, $\exp (-e^{-x})$. The relation of these conditions to the spectral d.f. of the process is also discussed. A weaker condition on the covariance sequence ensures the "relative stability in probability" of $Z_n$ (Theorem 4.1). Theorem 5.1 describes the behavior of $Z_n$ when the spectrum has a discrete component with "not too many large jumps" and a "smooth" continuous component: when properly normalized, $Z_n$ converges in probability to a random variable representing the maximum of the process corresponding to the discrete spectral component. A special case was given by us in [2]. We now summarize some known results used in the sequel. The extreme value d.f.'s are continuous, so that (1.2) holds for all $x$; furthermore, this holds if and only if it holds for all $x$ satisfying $0 < \Phi(x) < 1.$ (1.2) implies that for all such $x$ $0 < F^n (a_nx + b_n) < 1,\quad\text{for all large} n,$ and \begin{equation*}\tag{1.3}\lim_{n \rightarrow \infty} F(a_nx + b_n) = 1.\end{equation*} Let $x_\infty$ be the supremum of all real numbers $x'$ for which $F(x') < 1$; then, for all $x$ satisfying $0 < \Phi(x) < 1$, we have \begin{equation*}\tag{1.4}\lim_{n \rightarrow \infty} a_nx + b_n = x_\infty.\end{equation*} From (1.3), and the asymptotic relation $-\log F \sim (1 - F), F \rightarrow 1$, we see that (1.2) holds if and only if \begin{equation*}\tag{1.5}\lim_{n \rightarrow \infty} n\lbrack 1 - F (a_nx + b_n)\rbrack = -\log \Phi(x).\end{equation*}