Let $(X_{n,i})_{1\le i\le n}$ be a triangular array of independent random variables, satisfying the uniform infinitesimality condition $$\lim_{n\rightarrow\infty}\max_{1\le i\le n}P(|X_{n,i}>\delta|)=0$$ for all $\delta>0$. Assume that the distribution of $S_n=\sum_{i=1}^n X_{n,i}$ converges weakly to a limit $\mu$. Prove that $\mu$ is Gauss iff $\lim_{n\rightarrow\infty}\sum_{i=1}^nP(|X_{n,i}>\delta|)=0$ for all $\delta>0$.
Hint: use Levy-Khintchine representation theorem
This answer can be found in the following website. Convergence to Gaussian with infinitesimal condition with relation to Levy Triple.. Actually Petite Etincelle provided an counterexample. But I don't know whether it is true at the moment. I will check it later. Uniform infinitesimality Condition and convergence in distribution to Gaussian distribution