# Moment convergence, but only some moments

I have an integer valued random variable $$X_n$$ (that takes values in $$\{0,1\dots,n\}$$) that is a sum of $$n$$ (dependent) indicator functions. I believe that it converges to some random variable $$X$$, and I have shown that $$E(X_n^k) \rightarrow E(X^k)$$ for a $$k$$ that is relatively small $$(k/n \text{ close to 0})$$, and large $$k$$. $$(k/n \text{ close to 1})$$. Is there anything I can do with this information without computing other moments which are tough to compute?

• Can you say more about the random variable, in which sense it converges to what, and what are you looking to prove? – R_B Apr 7 at 1:31