# When does the average of R.V.s converge to the average of their means?

When does the average of $$n$$ independent random variables concentrate around the average of their means as $$n$$ tends towards infinity?

I'm having trouble understanding Theorem 11 in these class notes from Wade Trappe's class. It gives the following sufficient condition:

However, I think this might be incorrect. Consider $$X_2,X_3,X_4,\ldots=0$$. This scenario does fulfill the theorem's condition and it also implies that the probability of $$X_1$$ being equal to its expected value is 1, which is applicable to any random variable $$X_1$$ that has a finite variance. This does not seem right.

So, what is the accurate condition for this convergence to occur?

This related post provides a formula for the variance of a linear transformation of a 'd'-dimensional Dirichlet random variable. This might suggest that the sum of the squares of the differences between the variances of all pairs of these random variables (i.e., $$\sum_{i) needs to grow at a rate slower than $$O(n)$$ for this convergence to happen.

In your example, $$M_n=\frac {X_1} n$$ so $$\lim M_n=0$$. Also, $$\lim \frac 1n \sum\limits_{i=1}^{n}EX_i=\lim \frac {EX_1} n=0$$. So there is no contradiction.