I have encountered an interesting statement related to the law of large numbers. Namely if any function of $n$ variables has finite gradient for $n \to \infty$ then it is asymptotically constant. (I cite from memory).
As one example the ratio of arithmetic to geometric means was stated to approach a constant when $n \to \infty$:
$$\lim_{n \to \infty} \frac{x_1+x_2+\dots_+x_n}{n \sqrt[n]{x_1x_2\cdots x_n}}=\lambda < \infty \tag{1}$$
I don't remember what kind of restrictions were placed on the set $\{x_k\}$. However, I considered some examples:
$x_k=1$, then it's easy to see $\lambda=1$. More general, if $x_k=a$, then $\lambda=1$.
$x_k=k$, then we have: $$\lambda=\lim_{n \to \infty} \frac{n(n+1)}{2n \sqrt[n]{n!}}=\frac{1}{2} \lim_{n \to \infty} \sqrt[n]{\frac{n^n}{n!}}=\frac{e}{2}$$
Here we encounter another interesting definition for the number $e$ as twice the ratio of AM to GM for the sequence of natural numbers.
Similarly, $x_k=k^2$ leads to: $$\lambda=\lim_{n \to \infty} \frac{n(n+1)(2n+1)}{6n \sqrt[n]{(n!)^2}}=\frac{e^2}{3}$$
$x_k=\frac{1}{k}$, then we have: $$\lambda=\lim_{n \to \infty} \frac{H_n \sqrt[n]{n!}}{n}=\infty$$
Thus obviously for this latter sequence the rule (1) doesn't work.
I have the following questions:
What properties the set $\{x_k\}$ needs to have for (1) to be true?
Do any other famous constants aside from $e$ arise (nontrivially) in this limit for some sequences?