Notations for Random Variables Can someone explain what it means when we use random variables of the form $X_{n,m}$ given $1\le m \le n$. I understand what it means for random variables of the form $X_1,X_2, ..., X_n $ which is that some event X is being repeated n times (I am assuming that these are all i.i.d) but I can't quite wrap my head around the other notation.
 A: This notation represents a doubly-indexed array of random variables. The notation $1\le m\le n$ suggests that index $n$ is the "main" index while the index $m$ is the "subsidiary" index within $n$. Writing this out, you can arrange the variables $X_{n,m}$ into a triangular array as $n$ varies from $1,2,\ldots$:
$$
(n=1):\qquad X_{1,1}\\
(n=2):\qquad X_{2,1}, X_{2,2}\\
(n=3):\qquad X_{3,1}, X_{3,2}, X_{3,3}\\
(n=4):\qquad X_{4,1}, X_{4,2}, X_{4,3}, X_{4,4}\\
\vdots\\\
(n=k):\qquad X_{k,1}, X_{k,2}, \ldots, X_{k,k-1}, X_{k,k}
$$
and so on. If you are studying advanced probability theory, the most general version of the Central Limit Theorem is stated in terms of a triangular array, with the notion being that your observed sample of random variables, with a fixed value for $n$, is one particular row of a triangular array. You might then invoke the CLT to justify why the sum of your random variables has approximately a normal distribution.
The reason why there are two indices is that this allows the distribution of the $X$'s to differ from one row to the next, for example in row $n$ the $X_{n,1},\ldots,X_{n,n}$ are an IID sample from the Bernoulli($p_n$) distribution -- you're tossing a coin $n$ times where the coin has probability $p_n$ of landing heads. $X_{n,m}$ is then the 0/1 result of the $m$th toss.
