Family indexed by $\mathbb{N}$ vs sequence

Question

Possibly a trivial question, but no harm in asking:

I am reading around and some notes mention a "family of random variables $\{X_n\}$ where $n\in \mathbb{N}$". Is this the same as the sequence of random variables $(X_n)_{n\ge 1}$?

The curly brackets and use of the word family seem to indicate that there is no specific order (since in other sections he talks about sequences of random variables), but this seems to not make much sense because in the end the author ends up proving limiting results like the central limit theorem etc.

Thanks!

Answer 1

For me the braces $\{X_n\}$ would suggest that the order is irrelevant (and so we're not taking a limit, say), while brackets $(X_n)_{n \ge 1}$ is much more suggestive that the order matters. "Family" normally suggests a set (so no order). Just my 2 cts.