# Convergence to normal distribution.

Let us suppose we have an infinite number if Bernoulli variables that can take one of two values: $$-1,1$$, with equal probability. Then their sum converges to a normal probability distribution.

What about variables that can take three values, $$-1,0,1$$ with equal probability? Does the probability distribution of their sum also converge to the normal distribution?

• Bernoulli variables are $0$ or $1$ but this does not affect your question Sep 26 at 0:37

Firstly, you said when you have Bernoulli random variables that take values $$-1, 1,$$ then their sum converges to a normal distribution (presumably you mean by the Central Limit Theorem). This is slightly false; actually, the normalized sum will converge. In particular, if your variables are $$X_1, X_2, \dots,$$ then $$\frac{1}{\sqrt{n}} \sum_{i=1}^n X_i \stackrel{d}{\to} \mathcal{N}(0, \text{Var}(X)).$$
The Central Limit Theorem in general states that if you have a sequence of independent random variables $$X_1, X_2, \dots$$ that have the same distribution, so that $$\sigma^2 := \text{Var}(X_i) < \infty$$ and $$\mu := \mathbb{E}[X_i]$$ is the expected value, then $$\frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - \mu) \stackrel{d}{\to} \mathcal{N}(0, \sigma^2).$$