Return to Answer

In other words, your question is whether $A=[A_{i,j}]_{i,j=1}^n$ and $d=[d_1,\dots,d_n]^T$ are independent. This is of course so for $n=1$.

However, already for $n=2$ a numerical calculation suggests that $$E(A_{1,1}^2-1)d_1^2<-0.09,$$ so that $E(A_{1,1}^2-1)d_1^2\ne0=E(A_{1,1}^2-1)\,Ed_1^2$ and hence $A$ and $d$ are not independent.

Here are 10 (rounded) simulated values of $E(A_{1,1}^2-1)d_1^2$ using $10$ Monte Carlo samples each of size $10^6$ from the joint distribution of $A$ and $b$:
$$-0.123813, -0.125708, -0.125293, -0.125339, -0.124647, -0.125159, \ -0.125416, -0.125916, -0.124846, -0.124275.$$

These values can be compared with the value of the $6$-fold integral $E(A_{1,1}^2-1)d_1^2$ estimated by using the Mathematica command NIntegrate, which gave about $-0.125265\pm0.029215$.

(It will be assumed here that $A$ and $b$ are independent. Of course, without an assumption on the joint distribution of $A$ and $b$, hardly anything can be said about the joint distribution of $A$ and $d$.)

Your question, in other words, is whether $A=[a_{i,j}]_{i,j=1}^n$ and $d=[d_1,\dots,d_n]^T$ are independent. This is of course so for $n=1$.

However, already for $n=2$ a numerical calculation suggests that $$E(A_{1,1}^2-1)d_1^2<-0.09,$$ so that $E(A_{1,1}^2-1)d_1^2\ne0=E(A_{1,1}^2-1)\,Ed_1^2$ and hence $A$ and $d$ are not independent.

Here are 10 (rounded) simulated values of $E(A_{1,1}^2-1)d_1^2$ using $10$ Monte Carlo samples each of size $10^6$ from the joint distribution of $A$ and $b$:
$$-0.123813, -0.125708, -0.125293, -0.125339, -0.124647, -0.125159, \ -0.125416, -0.125916, -0.124846, -0.124275.$$

These values can be compared with the value of the $6$-fold integral $E(A_{1,1}^2-1)d_1^2$ estimated by using the Mathematica command NIntegrate, which gave about $-0.125265\pm0.029215$.

In other words, your question is whether $A=[A_{i,j}]_{i,j=1}^n$ and $d=[d_1,\dots,d_n]^T$ are independent. This is of course so for $n=1$.

However, already for $n=2$ a numerical calculation suggests that $$E(A_{1,1}^2-1)d_1^2<-0.09,$$ so that $E(A_{1,1}^2-1)d_1^2\ne0=E(A_{1,1}^2-1)\,Ed_1^2$ and hence $A$ and $d$ are not independent.

Here are 10 (rounded) simulated values of $E(A_{1,1}^2-1)d_1^2$ using $10$ Monte Carlo samples of size $10^6$ from the joint distribution of $A$ and $b$:
$$-0.123813, -0.125708, -0.125293, -0.125339, -0.124647, -0.125159, \ -0.125416, -0.125916, -0.124846, -0.124275.$$

In other words, your question is whether $A=[A_{i,j}]_{i,j=1}^n$ and $d=[d_1,\dots,d_n]^T$ are independent. This is of course so for $n=1$.

However, already for $n=2$ a numerical calculation suggests that $$E(A_{1,1}^2-1)d_1^2<-0.09,$$ so that $E(A_{1,1}^2-1)d_1^2\ne0=E(A_{1,1}^2-1)\,Ed_1^2$ and hence $A$ and $d$ are not independent.

Here are 10 (rounded) simulated values of $E(A_{1,1}^2-1)d_1^2$ using $10$ Monte Carlo samples each of size $10^6$ from the joint distribution of $A$ and $b$:
$$-0.123813, -0.125708, -0.125293, -0.125339, -0.124647, -0.125159, \ -0.125416, -0.125916, -0.124846, -0.124275.$$

These values can be compared with the value of the $6$-fold integral $E(A_{1,1}^2-1)d_1^2$ estimated by using the Mathematica command NIntegrate, which gave about $-0.125265\pm0.029215$.

In other words, your question is whether $A=[A_{i,j}]_{i,j=1}^n$ and $d=[d_1,\dots,d_n]^T$ are independent. This is of course so for $n=1$.

However, already for $n=2$ a numerical calculation suggests that $$E(A_{1,1}^2-1)d_1^2<-0.09,$$ so that $E(A_{1,1}^2-1)d_1^2\ne0=E(A_{1,1}^2-1)\,Ed_1^2$ and hence $A$ and $d$ are not independent.

In other words, your question is whether $A=[A_{i,j}]_{i,j=1}^n$ and $d=[d_1,\dots,d_n]^T$ are independent. This is of course so for $n=1$.

However, already for $n=2$ a numerical calculation suggests that $$E(A_{1,1}^2-1)d_1^2<-0.09,$$ so that $E(A_{1,1}^2-1)d_1^2\ne0=E(A_{1,1}^2-1)\,Ed_1^2$ and hence $A$ and $d$ are not independent.

Here are 10 (rounded) simulated values of $E(A_{1,1}^2-1)d_1^2$ using $10$ Monte Carlo samples of size $10^6$ from the joint distribution of $A$ and $b$:
$$-0.123813, -0.125708, -0.125293, -0.125339, -0.124647, -0.125159, \ -0.125416, -0.125916, -0.124846, -0.124275.$$