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Law of Large Numbers - Rate of convergence

  1. Feb 9, 2010 #1
    What is the rate of convergence of the law of large numbers?

    lim_{n \rightarrow \infty} \frac{1}{n} \sum Z_n = \mu

    1. can we say that the sum converges to [tex]\mu[/tex] as [tex]n^\alpha[/tex] for some [tex]\alpha\in \Re[/tex]?

    2. If so, what is the value of [tex]\alpha[/tex]?

  2. jcsd
  3. Feb 9, 2010 #2


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    It depends very much on the distribution functions of the random variables involved.
  4. Feb 24, 2010 #3


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    This paper has some results without proofs.
  5. Feb 25, 2010 #4


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    Theorems generically known as "Laws of the iterated logarithm" will give some answers. You can find discussions in probability texts (Chung, for example). A very good discussion is in the book "Approximation Theorems of Mathematical Statistics".
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