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Suppose samples $x_1,x_2,\ldots,x_n$ in space $G$ follow a (multivariate) gaussian (normal) distribution with specified mean and variance, i.e., $x_1,x_2,\ldots,x_n \overset{i.i.d}{\sim} f_X\sim N(\mu,\Sigma)$, where $f_X$ means corresponding probability density function, and $\mu,\Sigma$ are known. The cardinality of $G$ is $\#G=n$. Now we partition $G$ into subspaces $G_1$ and $G_2$, namely, $G=G_1\bigoplus G_2$. We suppose $\#G_1=s$ and $s\ll n$. Next we construct $$ \widetilde{x}\overset{\mbox{def}}{=}x^{jk}=x_j-x_k,\quad j,k\in G_1 $$ and let $K=\{\widetilde{x}\}$. Then we easily have that $\widetilde{x}$ is normal (it is the linear combination of $x$ restricted on $G_1$) and $\#K = s^2$. We assume that points of $\widetilde{x}$ are very sparse in the whole space $G$ (the condition $s\ll n$ can also guarantee this). In particular, WLOG, we set $n=20000$ and $s=10$. We investigate the independence between $\widetilde{X}$ and $X$. Intuitively and theoretically, $\widetilde{X}$ and $X$ are not independent, since they are correlated by noting that $G_1\subset G$ (independence and correlation are equivalent for normal variables). Let $C=\mbox{Cov}(\widetilde{X},X)$. By the construction as well as the sparsity assumption of $\widetilde{x}$, we have that $C$ is nearly a zero matrix, which means $\widetilde{X}$ and $X$ are somewhat (maybe) weak independence. In addition, the simulation studies on assumption $\widetilde{X}$ and $X$ are independent works well (since then we have $f(\widetilde{x},x)=f_{\widetilde{X}}(\widetilde{x})f_X(x)$). However, I still cannot convince myself without literature support (papers or books).

Another construction on $G=\{x_1,x_2,\ldots,x_n\}$ given as $$ \breve{x}\overset{\mbox{def}}{=}x_i,\quad i\in H, $$ where $H=\{x\in G: \mbox{mean}(x)=1\}$. Then we also want to check the independence between $\breve{X}$ and $X$. This time we have a help link to Hint, but I cannot fully understand the discussion there.

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  • $\begingroup$ Could you please clarify what your question is? $\endgroup$ – JeremyC Nov 7 '18 at 6:35
  • $\begingroup$ Sorry for the unclarity @JeremyC. My question are two aspects: (1) can we treat $\widetilde{X}$ and $X$ is independent? (2) can we treat $\breve{X}$ and $X$ is independent? $\endgroup$ – John Stone Nov 8 '18 at 13:29

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