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Let $ X=(X_1,\dots,X_n) $ and $ Y=(Y_1,\dots,Y_n) $ where the RVs $ X_1,\dots X_n, Y_1,\dots Y_n $ are independent and have the same limiting distribution (assume for simplicity that all moments exist). Furthermore let $ X_1 \leq X_2 \leq \dots \leq X_n $ and $ Y_1 \leq Y_2 \leq \dots \leq Y_n $ (so each vector is sorted). Can we say that $ \frac{1}{n} \left\Vert X-Y \right\Vert _1 \rightarrow 0 $, either in probability or almost surely?

As a simple example, let $ X_i \thicksim N(0,1) $ and $ Y_i \thicksim N(0,1)+\frac{1}{n}U(0,1) $, so that their asymptotic distributions are equivalent. Then it is evident from simulations that for large $n$ the scaled distance $ \frac{1}{n} \left\Vert X-Y \right\Vert _1 \rightarrow 0 $ only if the vectors $X$ and $Y$ are sorted.

Let $ X=(X_1,\dots,X_n) $ and $ Y=(Y_1,\dots,Y_n) $ where the RVs $ X_1,\dots X_n, Y_1,\dots Y_n $ are independent and have the same limiting distribution (assume for simplicity that all moments exist). Furthermore let $ X_1 \leq X_2 \leq \dots \leq X_n $ and $ Y_1 \leq Y_2 \leq \dots \leq Y_n $ (so each vector is sorted). Can we say that $ \frac{1}{n} \left\Vert X-Y \right\Vert _1 \rightarrow 0 $, either in probability or almost surely?

Let $ X=(X_1,\dots,X_n) $ and $ Y=(Y_1,\dots,Y_n) $ where the RVs $ X_1,\dots X_n, Y_1,\dots Y_n $ are independent and have the same limiting distribution (assume for simplicity that all moments exist). Furthermore let $ X_1 \leq X_2 \leq \dots \leq X_n $ and $ Y_1 \leq Y_2 \leq \dots \leq Y_n $ (so each vector is sorted). Can we say that $ \frac{1}{n} \left\Vert X-Y \right\Vert _1 \rightarrow 0 $, either in probability or almost surely?

As a simple example, let $ X_i \thicksim N(0,1) $ and $ Y_i \thicksim N(0,1)+\frac{1}{n}U(0,1) $, so that their asymptotic distributions are equivalent. Then it is evident from simulations that for large $n$ the scaled distance $ \frac{1}{n} \left\Vert X-Y \right\Vert _1 \rightarrow 0 $ only if the vectors $X$ and $Y$ are sorted.

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Convergence of scaled $L_1$ distance between two sorted random vectors with same limiting distribution

Let $ X=(X_1,\dots,X_n) $ and $ Y=(Y_1,\dots,Y_n) $ where the RVs $ X_1,\dots X_n, Y_1,\dots Y_n $ are independent and have the same limiting distribution (assume for simplicity that all moments exist). Furthermore let $ X_1 \leq X_2 \leq \dots \leq X_n $ and $ Y_1 \leq Y_2 \leq \dots \leq Y_n $ (so each vector is sorted). Can we say that $ \frac{1}{n} \left\Vert X-Y \right\Vert _1 \rightarrow 0 $, either in probability or almost surely?