A probability exercise related to Central Limit Thm

This exercise appears in K.L.Chung's A Course in Probability Theory, Chapter 7.

Ex.7.1-4

Let ${X_j}$ be independent r.v.'s such that $max_{1<=j<=n} \frac{|X_j|}{b_n} \to 0$ in pr. and $(S_n - a_n)/b_n$ converges to a nondegenerate d.f. Then $b_n \to \infty$, $b_{n+1}/b_n \to 1$, and $(a_{n+1} - a_n)/b_n \to 0$.

I found it difficult, and I do not have any idea why this is put in the exercise of CLT.Anyone helps me solve this? Thanks.

edited Jun 11 2010 at 22:36

François G. Dorais♦
22.5k●2●48●113

asked Jun 11 2010 at 13:56

AlgSoul
13●2

	I'm guessing you at least mean that $b_n\rightarrow\infty$? Also, could you edit and put your math within dollar signs so that we can read it more easily? – Noah Stein Jun 11 2010 at 14:44
	Probably it is in a chapter on the CLT because it is sort of like a converse for a CLT. That is to say, if your centering and scaling sequences $a_n$ and $b_n$ are such that a CLT-like statement holds about your sequence of independent random variables $X_j$, then the sequences $a_n$ and $b_n$ behave asymptotically as the corresponding sequences in the CLT do. – Noah Stein Jun 11 2010 at 14:54