I am trying to find a counterexample or prove the following:
$\dfrac{Var\left(X_{n}\right)}{\left[EX_{n}\right]^{2}}\rightarrow0 , then \dfrac{X_{n}}{EX_{n}}\rightarrow1$ in probability. Assuming $EX_n\rightarrow\infty$. Note that $X_n$ doesn't have any special restrictions.
I can show this holds no problem using L2 convergence implies convergence in probability. However, I don't know if the converse holds. I tried to give some counterexamples but they all seem to fail. I tried the form $X_n=n^k1_{[0,1-\frac{1}{n})}$ or something similar. I hope someone can point to me if I am on the right direction.
I think I have figured it out. The converse doesn't hold. A counterexample is pretty intuitive. My idea was right, but one just need to tune the parameters a little to get satisfy all the conditions.