Maxima and high level excursions of stationary Gaussian processes

Abstract:

Let $X(t),t \geqq 0$, be a stationary Gaussian process with mean 0, variance 1 and covariance function $r(t)$. The sample functions are assumed to be continuous on every interval. Let $r(t)$ be continuous and nonperiodic. Suppose that there exists $\alpha , 0 < \alpha \leqq 2$, and a continuous, increasing function $g(t),t \geqq 0$, satisfying \[ (0.1)\quad \lim \limits _{t \to 0} \frac {{g(ct)}}{{g(t)}} = 1,\quad for\;every\;c > 0,\] such that \[ (0.2)\quad 1 - r(t) \sim g(|t|)|t{|^\alpha },\quad t \to 0.\] For $u > 0$, let $v$ be defined (in terms of $u$) as the unique solution of \[ (0.3)\quad {u^2}g(1/v){v^{ - \alpha }} = 1.\] Let ${I_A}$ be the indicator of the event $A$; then \[ \int _0^T {{I_{[X(s) > u]}}ds} \] represents the time spent above $u$ by $X(s),0 \leqq s \leqq T$. It is shown that the conditional distribution of \[ (0.4)\quad v\int _0^T {{I_{[X(s) > u]}}ds,} \] given that it is positive, converges for fixed $T$ and $u \to \infty$ to a limiting distribution ${\Psi _\alpha }$, which depends only on $\alpha$ but not on $T$ or $g$. Let $F(\lambda )$ be the spectral distribution function corresponding to $r(t)$. Let ${F^{(p)}}(\lambda )$ be the iterated $p$-fold convolution of $F(\lambda )$. If, in addition to (0.2), it is assumed that \[ (0.5)\quad {F^{(p)}}\;is\;absolutely\;continuous\;for\;some\;p > 0,\] then $\max (X(s):0 \leqq s \leqq t)$, properly normalized, has, for $t \to \infty$, the limiting extreme value distribution $\exp ( - {e^{ - x}})$. If, in addition to (0.2), it is assumed that \[ (0.6)\quad F(\lambda )\;is\; absolutely \;continuous\; with\; the\; derivative\; f(\lambda ),\] and \[ (0.7)\quad \lim \limits _{h \to 0} \log h\int _{ - \infty }^\infty {|f(\lambda } + h) - f(\lambda )|d\lambda = 0,\] then (0.4) has, for $u \to \infty$ and $T \to \infty$, a limiting distribution whose Laplace-Stieltjes transform is \[ (0.8)\quad \exp [{\text {constant}}\int _0^\infty {} ({e^{ - \lambda \xi }} - 1)d{\Psi _\alpha }(x)],\quad \lambda > 0.\]