Some Interesting Limits

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  $i=1,...,N$ and points $z_i$ s are chosen in a way that $F(z_i)=i/N$ where N is the number of observations. Then we have the following limit as $N \rightarrow \infty$.

•  $N\sum_{i=0}^{N-1} \big\{\widehat{F} (z_{i+1})-F(z_{i+1})-\widehat{F} (z_{i})+F(z_i))\big\}^2\rightarrow 1$

It can be shown that this is equivalent to this limit

•  $[\widehat {\bold c}-{F}(\bold z)]'\bold\Omega_1^{-1} [\widehat {\bold c}-{F}(\bold z)]\rightarrow 1$

where $\widehat {\bold c}=\big\{\widehat{F} (z_i),i=1,...,N\big\}$ and $\Omega_1$ is the covariance matrix of $\widehat {\bold c}$.

Now let $c_i=i/N$. Then we can have the following limit involving quantiles

•  $N\sum_{i=0}^{N-1} \big\{\dfrac{\widehat{F}^{-1} (c_{i+1})-{F}^{-1}(c_i+1)}{G_{i+1}}-\dfrac{\widehat{F}^{-1} (c_{i})-F^-1(c_i))}{G_i}\big\}^2 \rightarrow 1$

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

•  $[\widehat {\bold z}-{F}^{-1}(\bold c)]'\bold\Omega_2^{-1}[\widehat {\bold z}-{F}^{-1}(\bold c)] \rightarrow 1$

where $\widehat {z_i}=\widehat{F^{-1}} (c_i)$  and  $\Omega_2$ 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

•  $[\widetilde{\bold y} -L(\bold c)]'\bold\Omega_L^{-1}[\widetilde{\bold y} -L(\bold c)] \rightarrow 1.73$

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