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### $\int_{x_i>0} f(x_1+x_2, x_3+x_4)^* f(x_1+x_3, x_2+x_4) \geq0$ for any function $f$?

We consider the change of variables to $x_1, a, b, y$, where $$a = x_1 + x_2, b = x_1 + x_3, y = x_1 + x_2 + x_3 + x_4.$$ The Jacobian of this change of variable is $1$. The inverse of this change of ...
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• 7,712
1 vote

### Showing positive definiteness of joint process with cross-correlation

I use positive definite below to mean non-negative eigenvalues, because that's how I think the OP uses it. Your matrix is $\rho K^TK +(1-\rho) D$ with $K:=(L_1 \; L_2)$ and $D$ being block diagonal ...
• 7,973
1 vote

### $\int_{x_i>0} f(x_1+x_2, x_3+x_4)^* f(x_1+x_3, x_2+x_4) \geq0$ for any function $f$?

Too long for a comment on the post of @abacaba, hence this post. Let's integrate directly from the equations which describe the change of variables. Consider first $x_1$ as independent. Then, since \$...
• 15.7k

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