Extremes of Gaussian process and the constant \(H_\alpha\) (Q1979091)

A continuous stationary Gaussian process \(X(t)\) is considered with \(E X(t)=0\), \(E X^2(t)=1\) and \(r(t)=E X(0)X(t)\) such that \((1-r(t))\sim C|t|^\alpha\) for \(|t|\to 0\) (\(0<\alpha\leq 2\)), and \(r(t)\log t\to 0\) as \(t\to\infty\). Let \(\nu_T=d_T+x/c_T\), \(q_T=\gamma\nu_T^{-2/\alpha}\), with \(c_T=\sqrt{2\log T}\), \[ d_T=c_T+c_T^{-1} \left\{ {2-\alpha\over 2\alpha}\log\log T +\log(2\pi)^{-1/2} 2^{(2-\alpha)/2\alpha}C^{1/\alpha}H_\alpha \right\}, \] where \(H_\alpha\) is some absolute constant. In this case for any \(\gamma>0\) \[ \Pr\left\{ \sup_{iq_T\leq T} X(iq_T)\leq\nu_T \right\} \to \exp(-\exp(-x)) \] as \(T\to\infty\). The constant \(H_\alpha\) can be defined as \[ H_\alpha=\lim_{T\to\infty}{1\over T}\int_{-\infty}^0 e^{-x} \Pr\left\{ \sup_{0\leq t\leq T}Y(t)>-x \right\}dx, \] where \(Y(t)\) is Gaussian with \(E Y(t)=-|t|^\alpha\), \(E Y(t)Y(s)=|s|^\alpha+|t|^\alpha-|t-s|^\alpha\). The author proposes a new characterization of \(H_\alpha\) in terms of the process \(X(t)\).