# Weak independence of gaussian distribution given a constraint on samples

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.

• Could you please clarify what your question is? – JeremyC Nov 7 '18 at 6:35
• 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? – John Stone Nov 8 '18 at 13:29