Lindeberg-Feller central limit theorem.

roposition

(Lindeberg-Feller CLT). Let is a family of r.v. such that

1. ,

2. ,

3. .

Then for the following two conditions A,B to hold

A. converges in distribution to ,

B. is holospoudic (see the definition ( Holospoudic ))

it is necessary and sufficient that for each $\eta>0$ we have

(Lindeberg-Feller condition)

Proof

(Sufficiency). We follow the proof of the proposition ( Lyapunov CLT ). As before, we show that MATH because then the statement would follow from the proposition ( Convergence of p.m. and ch.f. 2 ).

To prove that we verify conditions of the proposition ( Convergence lemma for family of complex numbers ) pointwise in for : MATH We substitute the Taylor decomposition of $e^{itx}$ around in the following form: MATH for some fixed $\eta$ and numbers , . Hence, MATH Using the condition 3 of the proposition we rewrite the above as MATH The first term tends to zero according to the condition MATH According to the condition MATH we have MATH hence MATH We use the proposition ( Lyapunov inequality ) MATH for , and MATH to conclude MATH or MATH This shows that MATH because the $\eta$ can be arbitrarily small. The verification of the rest is similar.