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Suppose that [itex]X_i\sim N(0,1)[/itex] are iid, then by the Strong Law of Large Numbers the sample mean [itex]\bar{X}_n=\frac{1}{n}\sum_{i=1}^n X_i[/itex] converges almost surely to the mean, which in this case is 0.

Recall that by definition, [itex]\bar{X}_n \to 0[/itex] almost surely means that [itex]P(\lim_{n\to\infty}|\bar{X_n}-0|=0)=1[/itex]. Since for the limit to converge, the limsup (and liminf) must also converge to the same value, we have [itex]P(\limsup_{n\to\infty}|\bar{X_n}-0|=0)=1[/itex].

But [itex]P(\limsup_{n\to\infty}|\bar{X_n}|=0)=P(|\bar{X_n}|=0\text{ i.o.})[/itex], where i.o. means infinitely often (this is the definition of i.o.). Now, [itex]\sum_n P(|\bar{X_n}|=0)=0<\infty[/itex], since [itex]\bar{X}_n[/itex] is a continuous random variable and the probability it takes any particular value is 0. But by the Borel-Cantelli Lemma this would imply that [itex]P(|\bar{X_n}|=0\text{ i.o.})=0[/itex], not 1, as the Strong Law of Large Numbers says.

I've been trying all day to find what's wrong with my argument. We can replace the normal distribution with any continuous distribution with finite mean. Can anyone help me?

Recall that by definition, [itex]\bar{X}_n \to 0[/itex] almost surely means that [itex]P(\lim_{n\to\infty}|\bar{X_n}-0|=0)=1[/itex]. Since for the limit to converge, the limsup (and liminf) must also converge to the same value, we have [itex]P(\limsup_{n\to\infty}|\bar{X_n}-0|=0)=1[/itex].

But [itex]P(\limsup_{n\to\infty}|\bar{X_n}|=0)=P(|\bar{X_n}|=0\text{ i.o.})[/itex], where i.o. means infinitely often (this is the definition of i.o.). Now, [itex]\sum_n P(|\bar{X_n}|=0)=0<\infty[/itex], since [itex]\bar{X}_n[/itex] is a continuous random variable and the probability it takes any particular value is 0. But by the Borel-Cantelli Lemma this would imply that [itex]P(|\bar{X_n}|=0\text{ i.o.})=0[/itex], not 1, as the Strong Law of Large Numbers says.

I've been trying all day to find what's wrong with my argument. We can replace the normal distribution with any continuous distribution with finite mean. Can anyone help me?

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