Do orthogonal projectors change the statistics of i.i.d. generated data? I am wondering the orthogonal projectors such as Hadamard transform change the statistics of i.i.d. generated data.
Consider, i.i.d. generated data $\mathbf{X}$ of length $N$,
$$\mathbf{X}_P=\mathbf{P}\mathbf{X}$$
where $\mathbf{P}$ is an orthogonal projector of size $N \times N$ and $\mathbf{X}_P$ is the projected data of length $N$.
It is explicit that $\mathbf{P}$ doesn't change the covariance matrix of i.i.d. data, but I am doubtful whether the projected data is still i.i.d. and the shape of distribution is the same, e.g., if $\mathbf{X}$ i.i.d. is generated from Laplacian, then $\mathbf{X}_P$ is still i.i.d. Laplacian?
Thanks a lot in advance
 A: The simplest non-trivial example I can construct uses an iid 2-vector of Bernoulli variables and the projection matrix {{1,1},{0,0}}.  That is,  $\mathbf{P}(\mathbf{x_1},\mathbf{x_2})' = \mathbf{x_1+x_2}$.  The projected random variable can take on the values 0, 1, and 2 with positive probability (it has a binomial distribution).  Even when you rescale this to be an orthogonal projection, there will still be three distinct values with positive probability.  Therefore, because the original Bernoulli variables can only have two distinct values, you cannot generally expect the projected components to have the same distribution as the original variables.
It may be worth making a few additional remarks:


*

*In most cases, any proper projection (i.e., not the identity) does change the covariance matrix.  This is obvious, because the resulting covariance matrix must have a nonzero kernel, implying it is not positive definite.

*When the original distribution is normal, the components of $\mathbf{X}_P$ will be normal or degenerate (that is, constant).  If the projection is proper, the components cannot possibly be iid, because the covariance is degenerate.
