Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime number theorem conjectured by Gauss and Legendre. What I would like to know is whether other famous similar results are know both inside and outside the realm of number theory, and if general probability theorems can be used to obtain such results (like mimicking a deterministic quantity by a random variable that follows a given distribution law, the sum of expected values of which converges almost surely to the desired value of the considered limit, or the like).

Many thanks in advance.

Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime number theorem conjectured by Gauss and Legendre. What I would like to know is whether other famous similar results are known both inside and outside the realm of number theory, and if general probability theorems can be used to obtain such results (like mimicking a deterministic quantity by a random variable that follows a given distribution law, the sum of expected values of which converges almost surely to the desired value of the considered limit, or the like).

Many thanks in advance.