independence of sparse Gaussian vector

independence of sparse Gaussian vector

A known fact is that if $v$ is a vector of i.i.d Gaussian random
variables, and $Q$ is an orthogonal matrix, then the vector $Qv$ is also a
vector of i.i.d Gaussian random variables.
Now, suppose that $v$ is an i.i.d vector, such that with certain
probability, say $p$ it is a Gaussian RVs and with probability $1-p$ it is
$0$. In that case, each element can be thought of as a Gaussian variable
multiplied by a random variable which is $0$ or $1$ (Bernoulli
distribution).
My question is, are the elements of $Qv$ i.i.d (in the case where $v$
constains some zeros) ?
Thanks, Gil