Suppose you have a sequence of random variables $ \left\lbrace X_{i}\right\rbrace_{i=1,...,n}$ which converges in probability to a random variable $X$, shown by $ X_n \ \xrightarrow{p}\ X$ as n goes to $\infty$.
$$\lim_{n\rightarrow\infty} P(|X_n-X|>\varepsilon) = 0$$
Now, suppose we can furteher obtain a sequence of random variables for each n $ \left\lbrace X_{j,n}\right\rbrace_{j=1,...,m}$ such that $ X_{m,n} \ \xrightarrow{p}\ X_n$ as m goes to $\infty$.
$$\lim_{m\rightarrow\infty} P(|X_{m,n}-X_{n}|>\varepsilon) = 0$$
My question is, may I define $ X_{m,n} \ \xrightarrow{p}\ X$ as both m and n go to $\infty$ at the same time (double limit)?
$$ \lim \limits_{\substack{% m \to \infty\\ n \to \infty}} P(|X_{m,n}-X|>\varepsilon) = 0$$
