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the central limit theorem states that the sums Sn scaled by the factor $1/\sqrt{n}$ converge in distribution to a standard normal distribution. Combined with Kolmogorov's zero-one law, this implies that these quantities converge neither in probability nor almost surely: $$ \frac{S_n}{\sqrt n} \ \stackrel{p}{\nrightarrow}\ \forall, \qquad \frac{S_n}{\sqrt n} \ \stackrel{a.s.}{\nrightarrow}\ \forall, \qquad \text{as}\ \ n\to\infty. $$

How do Kolmogorov 0-1 law and CLT imply normalized sample mean doesn't converge in probability nor a.s.?



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