Similar probabilistic results for 2 different (discrete) random variables

Question

I have a general question, but I will motivate it with an example.

Let $X_s$ be a random variable in $\lbrace1,2,\ldots\rbrace$ that follows the Zeta distribution $\zeta(s):$ $$\mathbb{P}(X_s=k)=\frac{1}{\zeta(s)n^s}.$$

Let $Y_n\in \lbrace1,2,\ldots,n\rbrace$ have the uniform distribution: $\mathbb{P}(Y_n=k)=1/n$.

Although the variables $X_s$ and $Y_n$ have different distributions, there are some similarities - such as the following:

(i) If $\phi$ denotes Euler's totient, then Kac showed [pp/ 57-58 http://www.gibbs.if.usp.br/~marchett/estocastica/MarkKac-Statistical-Independence.pdf] $$\mathbb{E}\bigg(\frac{\phi(Y_n)}{Y_n}\bigg)\to \frac{6}{\pi^2},\text{ as }n\to \infty.$$

On the other hand, there's a Tauberian theorem which gives $\mathbb{E}\bigg(\frac{\phi(X_s)}{X_s}\bigg)\to 6/\pi ^2$ as $s\to 1^+$.

(ii) The Erdos-Kac central limit theorem states that $$\mathbb{P}\Bigg(a\le \frac{\omega(Y_n)-\log(\log n)}{\sqrt{\log (\log n))}}\le b\Bigg)\to \frac{1}{\sqrt{2\pi}}\int_a ^be^{-y^2/2}dy,$$ while the Lindenberg central limit theorem implies $$\frac{\omega(X_s)-\rho(s)}{\sqrt{\rho(s)}}$$ has the standard normal distribution.

Questions

1) What other significant similarities do these distributions share? (please provide references).

2) Are there similarities with any two (discrete) distributions, or is this just a coincidence for $X_s$ and $Y_n$?

Answer 1

The random variable $X_s$ is only well-defined (by your formula) for real $s>1$. As $s$ tends to $1^+$, the random variable $X_s$ has worse and worse regularity, having fewer and fewer moments ($\mathbb EX_s^p<\infty$ only for $s>p+1$), and the limiting case $p=0$ of a "zeroth moment" (i.e. existence of the distribution) ceases to exist at $s=1^+$. Meanwhile, the mass of $X_s$ shifts to larger and larger numbers (with the size scale depending on $(s-1)^{-1}$).

On the other hand, the distribution $Y_n$ as $n\to\infty$ becomes weakly "concentrated" on a scale of size $\Theta(n)$, by which I mean only that $\mathbb P(Y_n=o(n))=o(1)$. In both cases of $X_s$ and $Y_n$, there is a comparable distribution across the relevant scale and thus statistics that are coarse enough will "not notice the difference" between $X_s$ and $Y_n$ in the respective limits. This applies in the case of (1) and there are other such number theoretical results along the same lines (thought not that I could quote off top-of-head - see Terry Tao's blog for examples, however). So I would say that these type of results are specific to the $X_s$ and $Y_n$ being related to each other, and not to an arbitrary discrete distribution.

On the other hand, (2) is of a more general character - such results are called central limit theorems, and they abound across probability theory. Any time that a system is comprised of many small and largely non-interacting parts, it is expected that all suitably normalized statistics will obey CLTs. (This is known in probability theory and statistical physics as a universality class, and such systems as described in the previous sentence are said to be in the "basin of attraction" of the Gaussian model.) So (2) in itself would not be evidence of a special relationship between $X_s$ and $Y_n$.