# Almost sure convergence definitions

I've seen these two definitions of almost sure convergence:

• $$\mathbb{P}\left(\lim _{n \rightarrow \infty} X_n=X\right)=1$$
• The sequence $$X_n$$ converges almost surely to $$X$$ if there exists a sequence of random variables $$\Delta_n$$ such that $$d\left(X_n, X\right) \leq \Delta_n$$ and $$\Delta_n \stackrel{\text { as }}{\rightarrow} 0$$.

The first of these came from Wikipedia and the second came from Asymptotic Statistics by A.W. Van der Vaart. Are these equivalent? I'm a little confused.

• If you read the text carefully, the second definition is the definition of $X_n \overset{as*}{\to} X$ with respect to the outer probability measure $P^*$, not the standard definition of converge almost surely (with respect to the original probability measure $P$). On the other hand, if $X_n$ and $X$ are all real-valued random variables, then these two definitions are indeed equivalent. Mar 30 at 2:32
• @Zhanxiong, if you wish, you can make it as an answer, for I think it directly hits at the poster's confusion. Mar 30 at 2:40
• @User1865345 Followed your request, my friend. Mar 30 at 3:34

As I commented under the question, the second definition looks more contrived because it wants to accommodate the situation that $$X_n$$ and $$X$$ are not measurable or $$d(X_n, X)$$ is not measurable (for more details, see Section 18.2 in Asymptotic Statistics). In this situation, the set $$[\lim_{n \to \infty}X_n = X]$$ may not be an event so that the notation "$$P[\lim_{n \to \infty}X_n = X]$$" may become meaningless.
That said, the two definitions are indeed equivalent when $$X_n$$, $$X$$, and $$d(X_n, X)$$ are all measurable. For example, when $$X_n$$ and $$X$$ are real-valued random variables and $$d(x, y) = |x - y|$$. This case can be proved as follows.
In this thread, it was shown that $$Y_n$$ converges to $$Y$$ almost surely if and only if for any $$\epsilon > 0$$, it holds that \begin{align} P[\cap_{k \geq 1}\cup_{n \geq k}[|Y_n - Y| \geq \epsilon]] = 0. \tag{1} \end{align}
Suppose $$|X_n - X| \leq \Delta_n$$ and $$\Delta_n$$ converges to $$0$$ almost surely, it then follows by $$(1)$$ that (set $$Y_n = \Delta_n$$, $$Y = 0$$) for any $$\epsilon > 0$$:
\begin{align} P[\cap_{k \geq 1}\cup_{n \geq k}[|X_n - X| \geq \epsilon]] \leq & P[\cap_{k \geq 1}\cup_{n \geq k}[\Delta_n \geq \epsilon]] = 0, \end{align} which implies (by the other direction of $$(1)$$) that $$X_n$$ converges to $$X$$ almost surely. This shows that the second definition implies the first definition.
Conversely, suppose $$X_n$$ converges to $$X$$ almost surely, then simply taking $$\Delta_n = |X_n - X|$$ (or if you don't want to be that extreme, take $$\Delta_n = 2|X_n - X|$$, say) meets all the requirements in the second definition. This shows the first definition implies the second definition. In conclusion, these two definitions are equivalent.