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For questions about the central limit theorem, which states: "Given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed." (Wikipedia)

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aquiring different results for a limit using CLT and LLN

I'm having some trouble with convergence in distribution and convergence in probability, mainly because I'm getting different results that seem to contradict each other using the Central limit theorem …
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