# Mean and variance of the Gaussian resulting from Central Limit Theorem

Let $$\{x_i\}$$ be a set of iid random variables (not necessarily Gaussian distributed). The CLT states that $$\frac{1}{n}\sum_{i=1}^n x_i$$ is asymptotically normal.

What do we know about the mean and variance of this asymptotic Gaussian? I know that if $$x_i \sim \mathcal{N}(\mu,\sigma)$$ that the sample mean has distribution $$\mathcal{N}(\mu, \sqrt{\sigma}/n)$$. What would the approach be if $$x_i$$ comes from some arbitrary distribution (say, exponential)?

And in the multivariate case (i.e. $$x_i \in \mathbb{R}^n$$) or more generally the continuous case (i.e. $$x_i(t)$$ is a function over a continuous-valued $$t$$), we would also need to find covariance.

## 1 Answer

Let's assume that $$x$$ is distributed according to some distribution, not necessarily Gaussian, with mean $$\mu$$ and standard deviation $$\sigma$$. Then, the distribution for the mean of a sample of $$N$$ iid $${x_i}$$ converges to a Gaussian of mean $$\mu$$ and standard deviation $$\sigma/\sqrt{N}$$ if the following conditions hold:

• The $${x_i}$$ are independent, although the convergence will still occur if the correlation is not too strong
• The convergence to a Gaussian still occurs if the $${x_i}$$ have different pdf's, but their variances are of a similar order of magnitude
• The variances of the pdf's of the $$x_i$$ must be finite
• The theorem applies for $$N\to \inf$$. For finite $$N$$, the shape of the pdf of the sample mean will still be approximately Gaussian in its center (the tails will differ, though)

If each $$x_i$$ is a vector of dimension $$n$$, the situation is the same. The theorem can be applied to each dimension separately unless there are correlations between each vector's components.

In the continuous case, you say that $$t$$ is continuous, but $$x$$ is not necessarily continuous. Let's consider that $$x(t)$$ is piece-wise constant, that is, uniform over some intervals of $$t$$, but the value can differ from interval to interval. The width of each interval $$dt_i$$ can be in general a random variable too, where $$i$$ is the number of the interval. In this case, I expect that if the $$\{dt_i\}$$ fulfill the conditions mentioned above, and if there is no correlation between the $$dt_i$$ interval and the (uniform) value that $$x$$ takes in that interval, then the CLT should also apply.

• Fantastic, thank you! Could you share a source for the bullet points you gave? These are exactly what I'm looking for, and I'd like to know more. – 900edges Feb 24 at 14:59
• You can find more details about those bullet points and some other minor ones in e.g. "Critical Phenomena in Natural Sciences" by Didier Sornette – kastellane Feb 24 at 15:01