Level crossings and local time for regularized Gaussian processes

Corinne Berzin (1) , José R. León (2) , Joaquín Ortega (3)

1 Laboratoire de probabilités et statistiques d'Orsay
2 UCV - Universidad Central de Venezuela
3 IVIC - Departamento de Matematica

Let $\{X_{t}, t in [0, 1]\}$ be a centred stationary Gaussian process defined on (Ω, A , P) with covariance function satisfying $r(t) \sim 1 -C \vert t \vert ^{2 \alpha}$, $0 < \alpha <1$, as $t \to 0$. Define the regularized process $X^{\varepsilon} = \varphi_{\varepsilon}* X$ and $Y^{\varepsilon} = X^{\varepsilon}/\sigma_{\varepsilon}$, where $\sigma_{\varepsilon}^{2} = var X_{t}^{\varepsilon}$, $\varphi_{\varepsilon}$ is a kernel which approaches the Dirac delta function as $\varepsilon \to 0$ and * denotes the convolution. We study the convergence of $$ Z_{\varepsilon}(f)= \varepsilon^{-a(\alpha)} \int_{-\infty}^{\infty} [N^{Y^{\varepsilon}}(x) / c(\varepsilon) - L_{X}(x)] f(x) dx $$ as $\varepsilon \to 0$, where $N^{V}(x)$ and $L_{V}(x)$ denote, respectively, the number of crossings and the local time at level x for the process V in [0, 1] and $$ c(\varepsilon) = (2 var (\ddot{X}_{t}^{\varepsilon})/(\pi var (X_{t}^{\varepsilon}))^{\frac{1}{2}}. $$ The limit depends on the value of $\alpha$.

Domaines: Probabilités [math.PR]

Corinne Berzin, José R. León, Joaquín Ortega. Level crossings and local time for regularized Gaussian processes. Probability and Mathematical Statistics, 1998, 18 (1), pp.39-81. ⟨hal-00319151⟩