Suppose I have a sequence of real-valued random variables $X_{1},X_{2},X_{3},\ldots$ defined on a common probability space $(\Omega,\mathfrak{F},P)$.
Suppose that $|X_{n} - c| = o_{P}(1)$ for some constant $c$ (using the Wikipedia definition here) and let $a_{n}$ be a deterministic sequence decreasing monotonically towards zero.
We say that $X_{n}$ converges to $c$ at the rate $a_{n}$ if $|X_{n}-c| = O_{P}(a_{n})$.
My question is: in what sense is this actually "informative" about the rate at which $X_{n}$ converges to $c$ in probability?
It seems to me that this is completely uninformative. To make my point, consider two cases:
Suppose $b_{n}\downarrow 0$ is a sequence satisfying $a_{n}/b_{n} \to 0$. Then $|X_{n}-c| = O_{P}(a_{n})$ implies $|X_{n}-c| = O_{P}(b_{n})$.
Suppose $b_{n}\downarrow 0$ is a sequence satisfying $b_{n}/a_{n} \to 0$. Then the fact that $|X_{n}-c| = O_{P}(a_{n})$ seems to say nothing about whether $|X_{n}-c| = O_{P}(b_{n})$. In particular, it is possible that $|X_{n}-c| = O_{P}(b_{n})$.
So again... in what sense is $|X_{n}-c| = O_{P}(a_{n})$ informative about the "rate of convergence" of $X_{n}$ to $c$ in probability?
While I cannot speak to all sources, it seems to me that $|X_{n}-c| = O_{P}(a_{n})$ only expresses something about the "rate of convergence" in probability of $X_{n}$ to $c$ if we also know $|X_{n}-c| \neq o_{P}(a_{n})$.
In this case we cannot have case 2, since: $$\frac{|X_{n} - c|}{b_{n}} = \frac{|X_{n} - c|}{a_{n}}\frac{a_{n}}{b_{n}},$$ and the right quantity diverges in probability given the assumption that $b_{n}/a_{n} \to 0$ and $|X_{n}-c| \neq o_{P}(a_{n})$.
Case 1 can never be ruled out if $|X_{n}-c| = O_{P}(a_{n})$ since in this case, if $a_{n}/b_{n}\to 0$, then we always have $|X_{n}-c| = O_{P}(b_{n})$ (in fact we have $|X_{n}-c| = o_{P}(b_{n})$).