There is a sequence $(X_n)_{n≥1}$ of independent random variables, where for n ≥ 1 the distribution for $X_n$ is given $$P (X_n = 0) = \frac{1}{n}$$ $$P(X_n = 2n) = 1 −\frac{1}{n}$$ Examine if the sum below is convergent in distribution, if so, find the desired distribution
$$S_n=\frac{X_1 + X_2 + . . . + X_n}{n}-n$$ So here is the deal, the expected value: $E\frac{X_k}{n}=\frac{2k-2}{n}$, I was going to use Central Limit Theorem, then it would converge to N(0,2) -1, but the Linder erg condition is not satisfied.
Any hint is appricieted and also different approaches, as I must say I think I am in deep dark in here.