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From WIkipedia

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.?

Thanks!

  • 1
    I wrote up the argument here: http://math.stackexchange.com/questions/210131/how-should-i-understand-the-sigma-algebra-in-kolmogorovs-zero-one-law/210152#2101522012-10-18

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