Given $n$ i.i.d. random variables $\{X_1, X_2, \dots , X_n\}$, each with mean $M$ and variance $V$, both strong and week LLNs seem to say that the average of the $n$ random variables, $S_n = \frac{X_1 + X_2 + \dots + X_n }{n}$, approaches $M$, as $n \to \infty$. The CLT seems to say that, as $n \to \infty$, the distribution of this average $S_n$ approaches a normal distribution with mean $M$ and variance $V$.
The problem I'm having is that it seems like the distribution of the average should converge to something like a discrete variable with a PMF like $1$ at $M$ and $0$ everywhere else. This is because the strong LLN says the average must be $M$, as $n$ approaches infinity. Instead, the normal distribution given by the CLT seems to say that there's a chance of the average not being $M$, as $n$ approaches infinity, which seems to contradict the strong LLN.
Where's the flaw in my reasoning?
Best Answer
The problem is that you left out the scaling. The average does converge to a constant: what converges to a normal distribution is $$\left(\text{average} - M\right)\sqrt{n}$$