On the convergence of convolutions of distributions with regularly varying tails

  • Thomas Höglund


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.


Normal Distribution Stochastic Process Convolution Probability Theory Mathematical Biology 
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Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • Thomas Höglund
    • 1
  1. 1.Stockholms UniversitetStockholm

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