What impact if any does the choice of the parent distributio

What impact (if any) does the choice of the parent distribution has on the speed of convergence to the Central Limit Theorem?

If there is an impact, order the uniform/normal/rayleigh distributions according to the speed of convergence.

Solution

We have Z=(S_n-)/, then Z=(summation over i =1 to n(X_i-₁)/). Since X_i\'s are independent, we get M_Z(t)=[1+(t²/2n)+O(n^-3/2)]ⁿ. For every fixed t, the terms O(n^-3/2)0 as ninfinity. Therefore as ninfinity, we get lim(ninfinity)M_Z(t)= lim(ninfinity)[1+(t²/2n)+O(n^-3/2)]ⁿ=exp[t²/2], which is the M.G.F of standard normal variate. Hence by Central limit theorem, X₁,X₂,.....,X_n are independently and identically distributed random variables with E(X_i)=_1, V(X_i)= ²₁, then the sum S_n=X₁+X₂+....+X_n is asymptotically normal with mean =n₁ and variance ²=n²₁. Lim(ninifinity)P[a<=[(S_n-n₁)/₁n]<=b]=phi(b)-phi(a).