I have a random vector $\mathbf{X} \in \mathbb{R}^3$ of supposedly i.i.d. random variables and I can generate a histogram for one of them, giving me an estimate of its pdf. Suppose I have an orthonormal transformation $\mathbf{U}$. Are the components of $\mathbf{U} \mathbf{X}$ also i.i.d. with the same pdf? Is there anything I can say about $\mathbf{U}\mathbf{X}$?
2026-03-30 17:05:28.1774890328
Given set of i.i.d. RVs with distribution $f$, find distribution of orthonormal transformation
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$$f_{\mathbf {UX}}(\mathbf x)=\dfrac{1}{|\det \mathbf U|}f_{\mathbf X}(\mathbf U^{-1}\mathbf x)=f_{\mathbf X}(\mathbf U^T \mathbf x), $$ where $\mathbf x=(x_1,x_2,x_3)\in\mathbb R^3$.
The distribution of $\mathbf {UX}$ remains the same as the distribution of $\mathbf X$ iff the distribution of $\mathbf X$ is spherically symmetric. The distribution of vector consisting of independent r.v. is spherically symmetric iff the distribution of $X_i$ is Gaussian with zero mean.
There are a lot of links on the subject, here's the first one.