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On the convergence of convolutions of distributions with regularly varying tails

  • Thomas Höglund
Article

Summary

Let x1,..., xn be a sequence of independent random variables with the common distribution F. Suppose E xk=0 and that F belongs to the domain of attraction of the normal distribution. Under conditions which do not involve the existence of any particular moment we show that
$$P\{ x_1 + \cdots + x_n \underline \leqslant xa_n \} = \Phi (x) - \frac{n}{{a_n^3 }}\int\limits_{ - a_n }^{a_n } {|y|^3 F(dy)(\omega \Phi (x) + o(1))} $$
uniformly in x, provided the norming constants a1, a2,... are properly chosen. Here Φ is the standard normal distribution and Ω a certain operator (depending on F).

The local counterparts are also treated.

Keywords

Normal Distribution Stochastic Process Convolution Probability Theory Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • Thomas Höglund
    • 1
  1. 1.Stockholms UniversitetStockholm

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