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Why is Gaussian Copula's Tail Dependence Zero?

I am confused about the second equation. Why does the derivative of C(q,q) can be written in two parts? And why each part has a conditional probability form? Is it due to the full derivative?

enter image description here

Ericshaw
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    The expression "$\partial C(q,q)/\partial q$" is ill-formed (that is, nonsensical) because it does not indicate which variable is being differentiated. The formula suggests the *total derivative* $dC(q,q)/dq$ is intended. – whuber Oct 10 '19 at 13:16
  • The first limit is simply a short formulation of the tail dependence concerto. It could be easier to start with the definition and directly get to the final result – Aksakal Dec 03 '21 at 14:54

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I think maybe it can solve as follow:

By definition, enter image description here

And $$ \frac{\partial C(U,V)}{\partial x} = \frac{\partial C(U,V)}{\partial U}\frac{\partial U}{\partial x} + \frac{\partial C(U,V)}{\partial V}\frac{\partial V}{\partial x} $$ Since, $$ U\sim U(0,1) $$ $$ U(x)=P(U\leq x)=x $$ Then, $$ \frac{\partial U}{\partial x}=\frac{\partial x}{\partial x}=1 $$ Similar to V, I have: $$ \frac{\partial C(U,V)}{\partial x} = \frac{\partial C(U,V)}{\partial U} + \frac{\partial C(U,V)}{\partial V} $$ By the symmetry property of copula, $$ \frac{\partial C(U,V)}{\partial x} = 2\frac{\partial C(U,V)}{\partial V} $$

Then, I have to show: $$ \frac{\partial C(U,V)}{\partial V}=C_{1|2}(u|v) $$ where $$ C_{1|2}(u|v) = P(U\leq u|V=v) $$

Consider the conditional density of V given U=u is: $$ h_{1|2}(u|v)=\frac{h(u,v)}{f_{V}(v)}=h(u,v) $$ Then, on the one side, $$ C_{1|2}(u|v)=P(U\leq u|V=v)=\begin{matrix} \int_{0}^{u} h_{1|2}(x|v)\, dx \end{matrix}=\begin{matrix} \int_{0}^{u} h(x,v)\, dx \end{matrix} $$ On the other side, $$ C(u,v)=\int_{0}^{v} \int_{0}^{u}h(x,y) \,dy\,dx $$ so that: $$ \frac{\partial C(U,V)}{\partial V} = \begin{matrix} \int_{0}^{u} h(x,v)\, dx \end{matrix} $$ Thus, I get: $$ \frac{\partial C(U,V)}{\partial V}=C_{1|2}(u|v) = P(U\leq u|V=v) $$

To sum up, I can show that: $$ \frac{\partial C(q,q)}{\partial q} = P(U_2 \leq q|U_1 =q)+P(U_1 \leq q|U_2 =q) $$ And take the limit, equation in my question will be presented.

I am not sure whether it is correct or not. Let's discuss. Thx.

Ericshaw
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  • A lot of this notation, such as "$\partial C(U,V)/\partial x$" makes no sense (or, rather, means something different than you intended). – whuber Oct 10 '19 at 13:17