Is there any proof for the CLT not using characteristic functions, a simpler method?

Maybe Tikhomirov or Stein's methods?

Something self-contained you can explain to a university student (first year of mathematics or physics) and takes less than one page?

  • 2
    $\begingroup$ I sketched out such an elementary approach at stats.stackexchange.com/a/3904/919. Arguably, using cumulant generation functions is the simplest possible method: your "simpler" is probably intended to read "more elementary." $\endgroup$
    – whuber
    Oct 4, 2016 at 17:45
  • 2
    $\begingroup$ Under more restrictive conditions than when using characteristic functions you can use moment generating functions instead (indeed the first CLT I saw was of this form) -- but the exposition is quite similar. $\endgroup$
    – Glen_b
    Oct 5, 2016 at 1:38
  • $\begingroup$ @Glen_b I also thought it could be easier with moments. Anyway I'll leave open the question in case anybody else posts a different demonstration. $\endgroup$
    – skan
    Oct 5, 2016 at 19:16
  • $\begingroup$ As a proof, it's not actually any easier (the proof with cfs can be written in the same form as the proof with mgfs), but may be preferable for students who may not have any background with functions involving $i$. That is, you may save introducing new concepts, but if they have those concepts already a proof of the corresponding statement with cfs isn't actually harder to do (though it is more general). Whether this is better depends on the students you're dealing with. $\endgroup$
    – Glen_b
    Oct 5, 2016 at 21:20
  • $\begingroup$ I recall my first year graduate statistics professor provided a visual "proof" of the CLT by showing sampling distributions of the mean with $n=10, 100, 1000$ under a variety of probability models. Normal, of course, showed no tendency, but exponential, bernoulli, and various heavy-tailed distributions all "rounded off" to the familiar shape visually per each increase in $n$. $\endgroup$
    – AdamO
    Nov 27, 2017 at 23:07

2 Answers 2


You can prove it with Stein's method, however it's debatable if the proof is elementary. The plus side of Stein's method is you get a slightly weaker form of Berry Esseen bounds essentially for free. Also, Stein's method is nothing short of black magic! You can find an exposition of the proof in section 6 of this link. You'll find other proofs of the CLT in the link as well.

Here's a brief outline:

1) Prove, using simple integration by parts and the normal distribution density, that $Ef'(A)-Xf(A)=0$ for all continuously differentiable iff $A$ is $N(0,1)$ distributed. It's easier to show $A$ normal implies the result and a bit harder to show the converse, but perhaps it can be taken on faith.

2) More generally, if $Ef(X_n)-X_nf(X_n)\rightarrow 0$ for every continuously differentiable $f$ with $f,f'$ bounded, then $X_n$ converges to $N(0,1)$ in distribution. The proof here is again by integration by parts, with some tricks. Specifically, we need to know that convergence in distribution is equivalent to $Eg(X_n)\rightarrow E g(A)$ for all bounded continuous functions $g$. Fixing $g$, this is used to reformulate:


where one solves for $f$ using basic ODE theory, and then shows $f$ is nice. Thus if we can find such a nice $f$, by assumption the r.h.s. goes to 0, and therefore so does the left side.

3) Finally, prove the central limit theorem for $Y_n:=\frac{X_1+\cdots+X_n}{\sqrt{n}}$ where $X_i$ are iid with mean 0 and variance 1. This again exploits the trick in step 2, where for every $g$ we find an $f$ such that:



Here's how I would do it if I were in high school.

Take any probability distribution with density $f(x)$, get its mean and variance $\mu_x,\sigma_x^2$. Next, approximate it with the random variable $z$ which has the following form: $$z=\mu_x-\sigma_x+2\sigma_x\xi,$$ where $\xi$ is Bernoulli random variable with parameter $p=1/2$. You can see that $\mu_z=\mu_x$ and $\sigma_z^2=\sigma_x^2$.

Now we can look at the sum $$S_n=\sum_{i=1}^n z_i$$ $$=n(\mu_x-\sigma_x)+2\sigma_x\sum_{i=1}^n\xi_i$$

You can recognize the Binomial distribution here: $\eta=\sum_{i=1}^n\xi_i$, where $\eta\sim B(n,1/2)$. You don't need characteristic function to see that it converges to normal distribution's shape.

So, in some regard you could say that the Bernoulli is the least precise approximation for any distribution, and even it converges to normal.

For instance, you can show that the moments match normal. Let's define look at the variable: $y=(S_n/n-\mu_x)\sqrt n$

$$y=\sigma_x(-1+2\eta/n)\sqrt n$$

Let's see what's the mean and variance: $$\mu_y=\sigma_x(-1+2(n/2)/n)\sqrt n=0$$ $$Var[y]=\sigma_x^2Var[2\eta/n] n=4\sigma_x^2/nn(1/4)=\sigma_x^2$$

The skewness and excess kurtosis converge to zero with $n\to\infty$, it's easy to show by plugging the known formulae for Binomial.

  • $\begingroup$ Interesting. Is it possible to turn this idea in a complete proof? $\endgroup$
    – Elvis
    Oct 4, 2016 at 19:31
  • $\begingroup$ @Elvis, I was trying to think like myself many years ago, and I wasn't into proofs so much. One thing I thought of is to represent the continuous distribution as a combination of bernoullis, but not sure if it's possible $\endgroup$
    – Aksakal
    Oct 4, 2016 at 19:53
  • $\begingroup$ What you wrote above could be much better. No need to approximate closely the distribution: a rough approximation by a variable taking two different values would do the job. $\endgroup$
    – Elvis
    Oct 4, 2016 at 20:10
  • $\begingroup$ That is, if it is possible to derive some bound on the accuracy of the normal approximation. Like, the normal approximation is at least as good for the original distribution as it is for the scaled Bernoulli. Or more probably something weaker but still allowing to conclude. $\endgroup$
    – Elvis
    Oct 4, 2016 at 20:18

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