This is a follow-up to this question.
Suppose I have a sequence of independently distributed positive random variables: $X_1\sim A_1,X_2\sim A_2, \ldots, X_n\sim A_n$, where $A_i$'s have support over $[0,\infty)$.
Furthermore, suppose that, for each $X_i$, variance $\sigma_{lb}^2<\sigma_i^2<\sigma_{ub}^2$, that is variances of $X_i$'s are bounded from above and below.
With $s_n^2=\sum_{i=1}^n\sigma_i^2$ and $s_n=\sqrt{s_n^2}$, Lindeberg's Condition is defined as:
$\lim_{n\rightarrow\infty}\frac{1}{s_n^2}\sum_{i=1}^n\int_{\{|x-\mu_i|>\epsilon s_n\}}(x-\mu_i)^2f_i(x)dx=0$
From the answer to my previous question, it does not hold in the case where just the variance is bounded.
Suppose we define two more possible restrictions on all $A_i$'s, in addition to the variance bound:
- $\mu_{lb} < \mu_i < \mu_{ub}$, that is, bound the means of all distributions from above and below;
- $\mathbf{P}(a\leq X_i\leq b)>0$ for all intervals $[a,b]\subseteq [0,\infty)$
Is there a subset (not necessarily proper) of the three above-defined restrictions on $A_i$'s for which Lindeberg condition holds?