Suppose that we have $n$ dependent random variables distributed as:

$$ X_1, \ldots, X_n \overset{id}{\sim} F $$

Is there a function $h$ such that:

$$ h(X_1), \ldots, h(X_n) \overset{iid}{\sim} G $$

for a new distribution $G$ where each of the $h(X_i)$ are now independent as well? I thought about the following:

$$ h(x) = 0\cdot x + \epsilon $$

where $\epsilon \sim N(0,1)$. However, this relies on a random variable, which I am not sure if it would be valid.

  • 1
    $\begingroup$ Is $G$ given to you? $\endgroup$ – Minus One-Twelfth Mar 31 at 1:34

$h= 1$ is such a function. There are cases where $h$ is necessarily constant: e.g., $X_1=X_2$ with standard normal distribution.


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