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Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} \frac{X_1+...+X_n}{\sqrt{n}} = \infty)=1$. But I don't really understand this reasoning? I suspect that for almost sure convergence we would need to have that the sequence $\frac{X_1+...+X_n}{\sqrt{n}}$ is almost surely bounded, but I am not quite sure about this.

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    $\begingroup$ The limit of a sequence $(y_n)_{n\geq 1}$ exists in $\mathbb{R}$ if and only if $\limsup y_n=\liminf y_n<\infty$. $\endgroup$ – Stefan Hansen Jun 27 '14 at 14:15
  • $\begingroup$ Why $P(\limsup\frac{X_1+...+X_n}{\sqrt{n}}=\infty)=1$? $\endgroup$ – Connor Apr 18 '17 at 4:23
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  • A convergent sequence is bounded.
  • If $(x_n)_{n\geqslant 1}$ is a sequence of real numbers such that $\limsup_{n\to \infty}x_n= +\infty$, then this sequence is unbounded.
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