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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.

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  • $\begingroup$ " as Sn takes only positive values" Not true. $\endgroup$ – Did Dec 2 '18 at 21:32
  • $\begingroup$ 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 $\endgroup$ – ryszard eggink Dec 2 '18 at 22:06
  • $\begingroup$ The distribution functions $(F_n)$ of $(S_n)$ do not seem like they are tight. $\endgroup$ – parsiad Dec 3 '18 at 0:39

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