Why Gaussian Copulas Look Dense at Corners but Have Zero Tail Dependence

This is a question that confuses many people when first studying copulas: How can a Gaussian copula appear to have “mass in the tails” when plotting its density, yet have zero tail dependence coefficients? Let’s unpack this apparent paradox.

The Key Distinction

The confusion arises from mixing two different concepts:

1. Copula density $c(u,v)$ - tells you how probability mass is distributed relative to uniform marginals
2. Tail dependence coefficient $\lambda$ - measures the asymptotic probability of joint extremes

Both can be true simultaneously: local density can be elevated near corners while asymptotic dependence vanishes.

Why Gaussian Copula Density Looks “Peaked” at Corners

Finite Sample Perspective

When the correlation $\rho > 0$, variables $U_1$ and $U_2$ tend to be large or small together. This means:

• Around $(0,0)$ and $(1,1)$, the density $c(u,v)$ is higher than 1 (the independence case)
• Contour plots show apparent “mass in the tails”
• This creates the visual impression of corner dependence

This is purely a finite-sample phenomenon - the density function describes local concentration of probability mass.

The Asymptotic Definition of Tail Dependence

Tail dependence asks a fundamentally different question about limiting behavior.

Upper Tail Dependence

The upper tail dependence coefficient is defined as:

$$\lambda_U = \lim_{u \to 1^-} \Pr(U_2 > u \mid U_1 > u)$$

This asks: As you push the threshold $u \to 1$, what’s the limiting probability that $V$ is also beyond $u$?

Lower Tail Dependence

Similarly, the lower tail dependence coefficient is:

$$\lambda_L = \lim_{u \to 0^+} \Pr(U_2 \leq u \mid U_1 \leq u)$$

Why Gaussian Copulas Have $\lambda_U = \lambda_L = 0$

For a Gaussian copula with correlation $-1 < \rho < 1$:

• The density may “bunch up” near corners (finite-sample behavior)
• But when you push $u \to 1$, the joint probability $\Pr(U_1 > u, U_2 > u)$ decays faster than the marginal probability $\Pr(U_1 > u)$
• Therefore, their ratio (the conditional probability) goes to 0

The Mathematical Details

Let $X, Y$ be standard normal with correlation $\rho$. As $t \to \infty$:

Marginal upper tail:

$$\Pr(X > t) \sim \frac{\phi(t)}{t} = \frac{1}{t\sqrt{2\pi}} e^{-t^2/2}$$

Joint upper tail:

$$\Pr(X > t, Y > t) \sim \frac{1}{\sqrt{2\pi} t(1-\rho)} \exp\left(-\frac{t^2}{1+\rho}\right)$$

The conditional probability becomes:

$$\Pr(Y > t \mid X > t) = \frac{\Pr(X > t, Y > t)}{\Pr(X > t)} \sim \text{const} \times \exp\left(-t^2\left[\frac{1}{1+\rho} - \frac{1}{2}\right]\right)$$

Computing the exponent difference:

$$\frac{1}{1+\rho} - \frac{1}{2} = \frac{1-\rho}{2(1+\rho)}$$

When $-1 < \rho < 1$, this quantity is positive, so the exponential factor decays to 0 as $t \to \infty$.

Therefore: $\lim_{t \to \infty} \Pr(Y > t \mid X > t) = 0$, i.e., $\lambda_U = 0$.

The Intuitive Explanation

Even though positive correlation makes large values more likely to occur together (elevated density near corners), the joint probability of both being extremely large shrinks much faster than the marginal probability of one being extremely large.

This faster shrinkage makes the conditional probability vanish in the limit, despite the apparent “tail mass” you see in density plots.

Special Cases

The only exceptions for Gaussian copulas are the degenerate cases:

• $\rho = 1$: perfect dependence, $\lambda_U = 1$
• $\rho = -1$: perfect negative dependence, $\lambda_L = 1$

Key Takeaways

✅ Both observations are correct:

• Local density near corners is elevated (visual “peaks”)
• Asymptotic dependence vanishes ($\lambda_U = \lambda_L = 0$)

✅ The difference: Shape of the PDF vs. limit definition of tail dependence

✅ Practical implication: If you need models with genuine tail dependence for risk management or extreme value modeling, you’ll need copulas beyond the Gaussian family (e.g., t-copulas, Archimedean copulas).

This distinction between local density behavior and asymptotic tail properties is crucial for choosing appropriate copula models in applications where extreme co-movements matter.