Convergence in probability is weaker than almost sure convergence.
I wonder if the property "$X_n$ converges to $0$ in probability" can be expressed as: "For almost all $\omega$, the sequence $X_n(\omega)$ has a certain property (to determine)"
In other words, what are the consequences of convergence in probability that can be observed on each individual realization of the sequence. And do they constitute a characterization?
More formally, is there a property $\phi$ whose argument is a sequence of real numbers (or if you prefer $\phi$ is a function from $\mathbb{R}^\mathbb{N}$ to $\{true,false\}$) such as for all sequences of random variables $(X_n)_{n\in\mathbb{N}}$:
"$(X_n)_{n\in\mathbb{N}}$ converges to $0$ in probability" $\Longleftrightarrow$ "for almost all $\omega$, $\phi((X_n(\omega))_{n\in\mathbb{N}})$ holds"
(and find $\phi$)
A candidate for $\phi$ I'm thinking of would be something like that:
$$\phi((x_n)_{n\in\mathbb{N}}): \forall\epsilon>0\quad\lim_{n\rightarrow+\infty}\frac{\#\{k<n\mid |x_k|>\epsilon\}}{n}=0$$
Best Answer
The answer is no: there is no such property. Any property of the form "a.s. something" that implies convergence in probability also implies a.s. convergence, hence cannot be equivalent to convergence in probability.
About about this property:
$$\phi((x_n)_{n\in\mathbb{N}}): \forall\epsilon>0\quad\lim_{n\rightarrow+\infty}\frac{\#\{k<n\mid |x_k|>\epsilon\}}{n}=0$$
"Almost surely $\phi$" is not equivalent to convergence in probability.
First, it is not implied by convergence in probability. A counter example is not so easy to find. The implication holds for independent variables. A counter example can be constructed by creating successive independent groups of 0/1 variables. All variables in the same group are equal: their common value is called the value of the group. And we have:
The last two points can be be obtained by saying that the probability for the $k^{th}$ group to have value 1 is $1/k$ and using the second Borrel-Cantelli lemma.
"Almost surely $\phi$" does not imply convergence in probability either. But it's not far from it. "For all subsequences of $X$, almost surely $\phi$ for this subsequence" implies convergence in probability and this is quite easy to prove.
Going further than this seems to raise key theoretical difficulties: how to define asymptotic empirical frequency in a more general way than this.