Skip to main content
Cross Validated

You are not logged in. Your edit will be placed in a queue until it is peer reviewed.

We welcome edits that make the post easier to understand and more valuable for readers. Because community members review edits, please try to make the post substantially better than how you found it, for example, by fixing grammar or adding additional resources and hyperlinks.

5
  • 1
    $\begingroup$ This comes down to a series of assertions beginning with "as you increase the number of random variables that enter your sum, the distribution of resulting values across trials will grow increasingly peaked in the middle." How do you demonstrate that? How do you show there aren't multiple peaks when the original distribution is not uniform? What can you demonstrate intuitively about how the spread of the distribution grows? Why does the same limiting distribution appear in the limit, no matter what distribution you start with? $\endgroup$ Commented Mar 17, 2022 at 15:31
  • $\begingroup$ @whuber My goal here was intuition, as OP requested. The logic can be evaluated numerically. If a particular value arises with probability 1/6 in a single roll, then the probability of getting that same value twice will be 1/6*1/6, etc. As there are relatively fewer combinations of values that yield sums in the tails, the tails will arise with decreasing probability as die are added to the set. The same logic holds with a loaded die, i.e., any distribution (you can see this numerically in a simulation): github.com/gburtch/simple_CLT_sim/blob/main/r_simulation.R. $\endgroup$ Commented Mar 18, 2022 at 16:55
  • $\begingroup$ Here is a Python gist producing a similar plot. $\endgroup$ Commented Oct 16, 2022 at 5:36
  • 2
    $\begingroup$ I don't think this is fully satisfying though, as it doesn't explain why the CLT couldn't say that $\bar{X}_n$ converges to, say, a triangle distribution with decaying tails. $\endgroup$ Commented Apr 18 at 15:17
  • $\begingroup$ I suspect this won’t be a fully satisfying answer, but it should be clear that this simulation setup isn’t going to converge to a triangular distribution. We’re summing i.i.d. random variables—so each step involves convolving the base distribution with itself. That process inherently smooths the distribution and spreads out the support. You can get a triangle from summing two uniforms, but as soon as you add more variables, the distribution starts pulling away from that shape. The sum doesn't stay bounded. $\endgroup$ Commented Apr 19 at 16:17