A non-trivial Example that Lindeberg holds but Lyapunov fails


In STAT 643, we learned Lyapunov condition implies Lindeberg condition, but not the other way around. In homework, we were required to give an example that Lindeberg condition holds but Lyapunov condition fails. One easy trivial example is give a R.V. without only moments of the order of less or equal than 2. I spent a lot of time trying to find a nontrivial one but Lindeberg and Lyapunov just live and die together. Come on, Why you two guys stay together all the time?

I didn’t search out any nontrivial example. There must be some other guys found it but I just couldn’t find it. Finally I got it myself when I couldn’t sleep last night.