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Is it possible to prove that the Cesàro mean of a converging sequence is the limit of the sequence through probabilities using the weak law of large numbers ? Has it ever been done ?

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The weak law of large numbers usually concerns the case where the initial sequence is divergent. So there is no chance to apply the LLN.

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  • $\begingroup$ I don't really get it. What do you mean by saying it usually concerns that case ? In what way does it entail we can't apply it to this problem ? $\endgroup$ – Minipipo1 Jan 26 '17 at 12:07
  • $\begingroup$ @Minipipo1, because LLN usually speaks about a sequence of independent random variables. Such sequence can converge only to a deterministic limit. This virtually removes stochasticity from the LLN, almost transforming it to a deterministic result. $\endgroup$ – zhoraster Jan 26 '17 at 16:39

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