Almost sure convergence implies convergence in probability with arbitrary rate (up to constant)? I have what seems like a simple question, but I am not quite able to prove it. Suppose that $A_n$ is a sequence of random variables, and we know that $A_n$ strongly converges to zero; i.e. almost surely. I am aware that this implies that $A_n$ converges in probability to $0$, but I am wondering about the rate of convergence. 

Suppose that I wish to establish that $A_n = \mathcal O_P(f(n))$, where $f(n)$ is some positive, strictly monotonically decreasing function which decays to zero at some rate. My understanding is that strong convergence implies that for any $\epsilon$ there are finitely many $n$ such that $A_n &gt; \epsilon$. But does this condition imply that there is some constant $C_f$, potentially quite large!, such that for $n&gt;&gt;0$,
$$\Pr\left(A_n &gt; C_f f(n)\right) \to 0 ?$$
Expanding on @saz's comment about the rate of decay, for any function $f(n)$ that you pick, I can find a function $g(n)$ which still decays to $0$, but which does so at a much slower rate. In other words, we want $g(n) &gt;&gt; Cf(n)$ for any constant $C$ and large enough $n$, or, equivalently, $\lim_{n \to \infty} \frac{g(n)}{f(n)} = \infty$. Using this $g(n)$, I can construct a specific counterexample as follows.

Let $\Omega$ be a probability space with only one outcome, $\omega$ (for instance, flipping a trick coin that always comes up heads). Then, let $A_n(\omega) = g(n)$ for all $n$. Saying that $A_n$ converges to $0$ almost surely in this simple probability space is the same as saying that $g(n) \to 0$, which it does. However, $\mathrm{Pr}(A_n &gt; Cf(n))$ is equal to $1$ for an infinite number of $n$, no matter what $C$ is, so $\mathrm{Pr}(A_n &gt; Cf(n)) \not\to 0$.
My understanding is that strong convergence implies that for any $\epsilon$ there are finitely many $n$ such that $A_n&gt;\epsilon$.


This is quite simply called convergence, assuming $A_n&gt;0$.  Unconditional convergence, if you will. Strong convergence does not necessarily imply unconditional convergence.

"Strong convergence," i.e., convergence with probability one, only implies that "almost surely," i.e., with probability one, there are no more than finitely many $n$ such that $|A_n|&gt;\epsilon$.

Nor is "strong convergence" necessary to establish convergence in probability of the order of magnitude a particular function $f$.



If $A_n = \mathcal O_P(f(n))$, where $f(n)&gt;0,f(n)\longrightarrow 0$, then by definition [Big O in probability notation], there is a constant $C_f&gt;0$, and for every $\epsilon&gt;0$ there exists $N&gt;0$ such that for all $n&gt;N$,

$$ \mathrm{Pr}(|A_n|&gt;C_ff(n)) &lt; \epsilon$$

Moreover, for every such $\epsilon$, there is a least such $N$, which increases monotonically as $\epsilon\longrightarrow 0, \epsilon&gt;0$. Conversely for each such $N$, there is a least such $\epsilon$, which is exactly

$$ \sup_{n&gt;N}\mathrm{Pr}(|A_n|&gt;C_ff(n)) $$

This expression must converge to zero as $N\longrightarrow\infty$, so the answer to your last question is yes.