During examination of compound Poisson process, with log-normal distribution I came across to the following problem.
I have examined the following form $$L=\sum_{i=1}^{N}X_{i}$$ And $X_{i}\sim LogN(\mu,\sigma^{2})$.
And I have checked the conditions of Lyapunov version of the CLT for finite sum of conditional representation of $L$. And I found the conditions are not satisfied for the case $\delta=2k$.
Moreover I can derive form conditional representation of CPP, that the moment increase in order by $\approx e^{e}$. (I mean initial moments not central ones). Though all finite order moments exist and are computable, but increasing. (Like with Log-normal distribution itself but exponentially faster).
So here is my question.
$\textbf{Having disproved Lyapunov condition, can I state that my sequence}$ $\textbf{doesn't satisfy CLT?}$
As it is more likely to converge to heavy-tailed distibution.
And another question.
$\textbf{Are their any conditions, which can let me tell that the sequence doesn't}$ $\textbf{satisfy CLT?}$
Any ideas on the problem can be helpful.
Thank you very much.
You may want to look at Lindeberg's condition as well, which, as far as I know, is the most general version of the CLT that provides necessary and sufficient conditions for a sequence of random variables to converge in distribution to the standard normal.