Forget the concept of linear independence. The interrelationships between assets as they are currently modeled are linearly correlated. However, this linear correlation is only an assumption. You should give up on this assumption and realize that the relationship between assets is much more complex than that.
Familiarize yourself with the various statistical distributions. There is more than one copula. Although the standard copula is Gaussian (also known as “normal”), this easily understood distribution is not always the best for using copulas to analyze portfolios. Other copulas, such as the Achimedian copula, t-copula, elliptical copula and Marshall-Olkin copula provide greater accuracy in many situations. Become familiar with the different types of copulas and know when they are most likely the best suited for a given situation.
Begin seeing assets as a combination of two variable factors. In the copula method, each asset changes due to two main factors. First, is random variance, coming from factors only related to the asset in question. Second is the interdependence of that asset on other assets. Thus, you should begin to see each asset as a combination of random variance and interdependence on the other assets (the market as a whole).
Begin analyzing assets simultaneously. While you may be accustomed to analyzing assets individually, doing so neglects the utility of the copula. Start plotting assets together on graphs, placing asset model predictions side-by-side and comparing assets in terms of their accordance to a general pattern.