For an arbitrary probability space $(X,\mu)$, a sequence $(x_n)$ in $X$ is said to be equidistributed with respect to $\mu$ if the measures $\frac 1 n \sum_{1\le k\le n} \delta_{x_k}$ converges weakly to $\mu$ as $n \to \infty$.

The special case of a compact interval equipped with the Lebesgue measure is very well studied. One way of building equidistributed sequences is to construct an ergodic transformation preserving the measure. This post sketches such a construction for the Lebesgue measure on the real line.

Is there any known explicit construction of an equidistributed sequence for the measure associated to an exponential or Gaussian random variable, preferably that doesn’t involve applying the cumulative distribution function/its inverse?