My book is trying to prove this version of the central limit theorem:
The proof starts like this:
I am totally confused about this proof. I have two questions:
Question 1: Why must there exist $N(0,\sigma^2)$ distributed random variables $\overline{Y_1}, \overline{Y_2},...,\overline{Y_n}$ with the desired condition of independence ? And what is $\sigma$ ?
Question 2: I know that weak convergence implies convergence in distribution. However, the statement:
" we have to prove $E[f(S _n)] − E[f(\overline{S _n})] → 0$ for any $f ∈ C_b(R)$. It is enough to verify it for smooth f of compact support."....(1)
does not seem to be justifiable by the equivalence of weak convergence and convergence of distribution, does it ? What is the justification for statement $(1)$ ?
I would really appreciate ii if someone can point me to the right direction and give even short answers or hints to answer my questions.
Thank you a lot.
This is possible by enlarging the probability space (that will not change the convergence in distribution).
Assume that we know that the convergence $$\tag{*}\mathbb E\left[f\left(S_n^*\right)\right] -\mathbb E\left[f\left(\overline{S_n}/\sqrt n\right)\right]\to 0$$ holds for each $f$ smooth with compact support. We want to prove it for each continuous and bounded $f$. Using for a suitable $R$ a smooth function with compact support $\eta_R$ such that $\eta_R=1$ on $[-R,R]$, and tightness of the sequences $\left(S_n^*\right)_{n\geqslant 1}$ and $\left(\overline{S_n}/\sqrt n\right)_{n\geqslant 1}$, we get the convergence (*) for each smooth bounded function $f$. In order to prove it for any continuous bounded function, we use convolution by a mollifier.