Probabilities of Large Deviations

  • Valentin V. Petrov
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 82)

Abstract

We shall consider a sequence of independent random variables {X n } having a common distribution with finite positive variance σ2 and a mean value which we can take to be zero, without loss of generality. We shall write \({S_n} = \sum\limits_{j = 1}^n {{X_j}} , {F_n}\left( x \right) = P\left( {\frac{{{s_n}}}{{\sigma \sqrt n }} < x} \right)\) . We have F n (x) → Φ(x) uniformly in x. Therefore when x = O(1) we have
$$\frac{{1 - {F_n}\left( x \right)}}{{1 - \Phi \left( x \right)}} \to 1, \frac{{{F_n}\left( { - x} \right)}}{{\Phi \left( { - x} \right)}} \to 1$$
(1.1)
.

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1975

Authors and Affiliations

  • Valentin V. Petrov
    • 1
  1. 1.Leningard UniversityLeningradUSSR

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