On tail probability of local times of Gaussian processes
Let \(X = \{X_t, t\geq 0\}\) be a Gaussian process with stationary increments, zero mean and such that \(X_0 = 0.\) Let \( \sigma ^2 (h) = E(X_{t+h} - X_t)^2 \) be the covariance function of the increments and \(l(t,x)\) be the local time of \(X\) defined by \[ l(t,x) = \lim _{\varepsilon \downarrow 0} \frac {1}{2\varepsilon } \int _0^t I_{[x-\varepsilon , x+\varepsilon ]} X_s ds . \] Asymptotic properties of \(l(1,0)\) are studied in connection with the behaviour of \(\sigma ^2.\) Especially, it is proved that under some conditions on \(\sigma ^2,\) the exponential rate of the tail probability \(P[l(0,1)> x] \) is asymptotically \(1/ {\sigma ^{-1}(1/x)}\) as \(x\to \infty ,\) where \(\sigma ^{-1}\) is the inverse function of \(\sigma .\) Results for Brownian and fractional Brownian processes follow then as corollaries.
scientific article; zbMATH DE number 1792509