Sequence of random variables vs random sample I am studying convergence theorems and consistency for estimators. I am confused on the difference between a sample of random variables and a sequence of random variables. Can someone explain the difference between these?
 A: I am sharing some notes (certainly too simplistic)... But I agree that some basic notions can, sometimes, be confusing.
Random sample
The random variables $X_1, ..., X_n$ are called a random sample of size n from the population $f(x)$ if $X_1,... , X_n$ are mutually independent random variables and the marginal $pdf$ or $pmf$ of each $X_i$ is the same function $f(x)$.
Alternatively, $X_1,...,X_n$ are called independent and identically distributed (iid) random variables with $pdf$ or $pmf$ $f(x)$.
Sequence of a random variables
As @whuber commented, a sequence is more general. It is mathematical concept: a sequence (Wikipedia) is an enumerated collection of objects in which repetitions are allowed and order matters. Unlike a set, the same elements can appear multiple times at different positions in a sequence. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position.
So, a random sample is a sequence of random variables but the converse is not true.
