How to Detect Linear Dependence on the Copula Level?

In many practical situations, the dependence between the quantities is linear or approximately linear. Knowing that the dependence is linear simplifies computations; so, is is desirable to detect linear dependencies. If we know the joint probability distribution, we can detect linear dependence by computing Pearson’s correlation coefficient. In practice, we often have a copula instead of a full distribution; in this case, we face a problem of detecting linear dependence based on the copula. Also, distributions are often heavy-tailed, with infinite variances, in which case Pearson’s formulas cannot be applied. In this paper, we show how to modify Pearson’s formula so that it can be applied to copulas and to heavy-tailed distributions. 1 Detecting Linear Dependence: Formulation of the Problem Locally, linear dependencies are ubiquitous. Dependencies between quantities are often described by smooth (even analytical) functions y = f(x1, . . . , xn). An analytical function can be expanded in Taylor series around each point x = (x (0) 1 , . . . , x (0) n ):