While working on the Lorenz curve paper, we came up with several interesting limits involving CDF, Quantile and Lorenz functions which I haven’t seen in references.

Let and points s are chosen in a way that where N is the number of observations. Then we have the following limit as .

It can be shown that this is equivalent to this limit

where and is the covariance matrix of .

Now let . Then we can have the following limit involving quantiles

where is derivative of the quantile function with respect to . This formula can also be written in this matrix form

where and is the covariance matrix of empirical quantiles.

However, such quadratic forms do not always converge to one. For example, for a Lorenz curve it seems that we have

This is because becomes singular when N goes towards infinity. I don’t know how to prove this limit however!

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