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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?