Given a positive integer $n$, let $p$ be the largest prime less than or equal to $n$.

Let $N(n)=2^{C_2}\cdots p^{C_p}$ be uniformly distributed from $1$ to $n$, and $M(n)=2^{Z_2}\cdots p^{Z_p}$ where $Z_p's$ are independent geometric with $P(Z_p\ge k)=\frac{1}{p^k}$. It can be shown that $C_p$ has probability mass $P(C_p=k)=\frac{\big\lfloor \frac{n}{p^k}\big\rfloor}{n}.$

Consider the metric $d(M,N)=\sum_{p\le n}\vert C_p-Z_p\vert$

($d$ counts the number of prime insertions and deletions needed to convert $N$ to $M$).

Using this metric, we have the Wasserstein metric $d_W(M,N)=\inf_{\text{couplings}}\mathbb{E}d(M,N).$

Arratia (page 10 of https://arxiv.org/pdf/1305.0941.pdf) claims

- $d_W(M,N)=o(\log \log n)$ implies the
**Hardy-Ramanujan Theorem**(which states that for almost every positive integer $n$, $\omega(n) \approx \log \log n$, where $\omega(n)$ is the number of distinct prime divisors of $n$). - $d_W(M,N)=o(\sqrt{\log \log n})$ implies the
**Erdos-Kac Central Limit Theorem**(which states that $\frac{\omega(n)-\log \log n}{\sqrt{\log \log n}}$ has the standard normal distribution.

Further confirmation of this is stated by Bollobas on page 29 of "Contemporary Combinatorics":

How can these results be derived from the asymptotics?