Why Both Probability Terms Use the Same Copula

Summary

The document considers the probability that one variable falls below a threshold while another lies in an interval. It expresses this event as the difference between two joint cumulative probabilities, each evaluated at a different upper bound for the second variable. The question is whether those expressions might involve different copulas or a special convolution copula.

The answer is that, when the marginal distributions are continuous, both terms come from the same joint distribution and therefore use the same copula under Sklar’s theorem. The interval probability is the difference of joint cumulative distribution function values, not a reason to introduce a second dependence structure. The response relies on the uniqueness of the copula for continuous marginals. It notes that continuity and the absence of atoms matter; for distributions with discontinuities, copula uniqueness may not hold everywhere. No data, empirical test, or further treatment of convolution copulas is provided.

Key ideas

• An interval probability can be written as the difference of two joint cumulative probabilities.
• For continuous marginals, Sklar’s theorem gives a unique copula for the joint distribution.
• Changing the threshold for one variable does not create a new copula.
• Copula uniqueness has qualifications when marginal distributions contain atoms.

Tags

# Convolution copula?


# Convolution copula?












Using copula formulation for the following probability:

$$\mathbb{P}(X\leq x,y_{1}\leq Y\leq y_{2})=\mathbb{P}(X\leq x,Y\leq y_{2})-\mathbb{P}(X\leq x,Y\leq y_{1})$$ $$=C(F_{X}(x),F_{Y}(y_{2}))-C(F_{X}(x),F_{Y}(y_{1}))$$

There is no need or requirement for the two copulas above to be the same. Is there a link between these two copulas?

I read somewhere that there should be a link of the type of convolution copula.

Does anyone know if there is a link or perhaps has some references of research concerning this topic?

## Answer by Richi Wa (score 4)

https://quant.stackexchange.com/a/10911

Is'nt it true that your first line can be written as $$ F_{X,Y}(x,y_2) - F_{X,Y}(x,y_1), $$ where $F_{X,Y}$ is the joint cdf of $(X,Y)$. If we assume that the distributions of $X$ and $Y$ are continuous without atoms (I have to check the exact formulation), then it is clear from Sklar's theorem that there is exactly one copula $C$ such that $$F_{X,Y}(x,y) = C(F_X(x),F_Y(y)).$$ So the answer is: no the "two" copulas are just one.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.