Convergence in distribution

Convergence in distribution is like when you and your friend both have a bunch of marbles, but some of your marbles are different from your friend's marbles. You each count how many of your marbles are red, and how many are blue. You then compare your answers and notice that even though you have different marbles, the proportion of red and blue marbles is actually very similar.

In math, we call the proportion of red and blue marbles the "distribution" of marbles. When we say that two sequences of numbers are converging in distribution, it means that as the sequences get bigger and bigger, the distributions of the numbers in those sequences start to look more and more alike, even if the individual numbers in the sequences are different.

For example, let's say we have two sequences of random numbers: (1, 2, 3, 4, 5) and (2, 4, 6, 8, 10). Even though the individual numbers in these sequences are different, if we calculate their means (which is just the sum of the numbers divided by the total number of numbers), we see that the means are both equal to 3.

As the sequences get bigger and bigger, the mean of the first sequence and the mean of the second sequence will start to converge to 3. This is an example of convergence in distribution - even though the individual numbers in the sequences are different, the distributions of those numbers are becoming more and more similar as the sequences get longer.