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Re: Help please!

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  • Noel Vaillant
    The variables X1,..,Xn are i.i.d (independent, identically distributed) with cauchy distribution. If X is a random variable with cauchy distribution, then
    Message 1 of 2 , Dec 10, 2000
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      The variables X1,..,Xn are i.i.d (independent, identically
      distributed) with 'cauchy' distribution. If X is a random variable
      with cauchy distribution, then for all u in R:
      E[exp(iuX)]=exp(-|u|).
      So E[exp(iu(Sn/n))]=E[prod(exp(i(u/n)Xj))]=prod(E[i(u/n)Xj])
      =prod(exp(-|u|/n))=(exp(-|u|/n))^n=exp(-|u|)

      (where we have used the independence for E[prod.. = prod E[.. )

      so the characteristic function of Sn/n is also that of a cauchy
      distributed variable for all n.

      Regards. Noel
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