I would like to prove the following statement:
Proposition. If $X_{n,m}\to X$ in probability then there exist a subsequence $X_{k_n, l_m}$ of $X_{n,m}$ such that $X_{k_n,l_m}\to X$ almost surely.
Note: By double subsequence I mean two strictly increasing sequences of natural numbers $k_n$ and $l_m$ and by convergence of a double sequence I mean convergence as $n,m\to\infty$ jointly.
This is a well-known result for single sequences. Am not sure it holds for double sequences because the Borel-Cantelli lemma dosen't seem adaptable to double sequences.
However I think it is possible to prove the weaker result that there exist a subsequence $X_{k_n, l_n}$ of $X_{n,m}$ such that $X_{k_n,l_n}\to X$ almost surely. The difference here is that the double sequence $X_{k_n, l_n}$ is now regarded as a single sequence with single index $n$.
Is my reasoning correct? And what about the converse statement that $X_{n,m}\to X$ almost surely implies $X_{n,m}\to X$ in probability?
Any help is greatly appreciated.
Best Answer
There exist integers $N_k$ such that $P(|X_{nm}-X|>\frac 1 {2^{k}}) <\frac 1 {2^{k}}$ whenever $n,m \geq N_k$ (and in particular for $n=m=N_k$) and we may suppose that $N_k$ is increasing. It follows by Borel Cantelli Lemma that $X_{N_k,N_k} \to X$ almost surely.
The converse follows immediately from the following elementary fact:
Let $(a_{nm}) \subset \mathbb R$ and $a \in \mathbb R$ . Then $a_{nm} \to a$ as $n,m \to \infty$ iff $a_{n_k,m_k} \to a$ for any two sequence $(n_k),(m_k)$ increasing t $\infty$.