rlfsm (version 1.1.2)

U_ghuv: A dependence structure of 2 random variables.

Description

It is used when random variables do not have finite second moments, and thus, the covariance matrix is not defined. For \(X= \int_{\R} g_s dL_s\) and \(Y= \int_{\R} h_s dL_s\) with \(\| g \|_{\alpha}, \| h\|_{\alpha}< \infty\). Then the measure of dependence is given by \(U_{g,h}: \R^2 \to \R\) via $$U_{g,h} (u,v)=\exp(- \sigma^{\alpha}{\| ug +vh \|_{\alpha}}^{\alpha} ) - \exp(- \sigma^{\alpha} ({\| ug \|_{\alpha}}^{\alpha} + {\| vh \|_{\alpha}}^{\alpha}))$$

Usage

U_ghuv(alpha, sigma, g, h, u, v, ...)

Arguments

alpha

self-similarity parameter of alpha stable random motion.

sigma

Scale parameter of lfsm

g, h

functions \(g,h: \R \to \R\) with finite alpha-norm (see Norm_alpha).

v, u

real numbers

...

additional parameters to pass to U_gh and U_g

Examples

g<-function(x) exp(-x^2)
h<-function(x) exp(-abs(x))
U_ghuv(alpha=1.5, sigma=1, g=g, h=h, u=10, v=15,
rel.tol = .Machine$double.eps^0.25, abs.tol=1e-11)