# Convergence in distribution of sum of random, independent variables.

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.

• " as Sn takes only positive values" Not true. – Did Dec 2 '18 at 21:32
• You are right obviousely, but I guess the limit still holds as n goes to infinity, but I have no idea how to prove it – ryszard eggink Dec 2 '18 at 22:06
• The distribution functions $(F_n)$ of $(S_n)$ do not seem like they are tight. – parsiad Dec 3 '18 at 0:39