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Limit theorems for probabilities of large deviations

  • William Feller
Article

Summary

Let {X k,n ; k=1, ⋯, n} be a triangular array of independent variables with row sums S n . Suppose E(X k,n ) = 0 and E(S n 2 ) = 1 and that \(\psi _n (h) = \sum\limits_{k = 1}^n {\log E(e^{h X_{k,n} } )}\) exists for 0≦hɛ n . Under mild conditions we show that
$$P\{ S_n > z_n \} = \exp [ - r_n + 0(r_n )], r_n \to \infty$$
(1)
where the quantities z n and r n are related by the parametric equations
$$z_n = \psi '_n (h_n ), r_n = h_n \psi '_n (h_n ) - \psi _n (h_n ).$$
(2)
If the distributions of the X k,n behave reasonably well it is usually not difficult to obtain satisfactory asymptotic estimates for z n in terms of r n and vice versa. The principal application is to sequences X k . Then X k,n = X k /s n and S n = (X1+⋯.+X n )/s n . A familiar special case of (1) is given by
$$P\{ X_1 + \cdots + X_n > s_n z_n \} \sim [1 - \mathfrak{N}(z_n )] \exp [ - P_n (z_n )]$$
where \(\mathfrak{N}\) is the standard normal distribution and P n a certain power series. In this case rn = z n 2 but (2) may lead to radically different relationships between rn and zn.

Keywords

Normal Distribution Stochastic Process Probability Theory Power Series Limit Theorem 
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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References

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Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • William Feller
    • 1
  1. 1.Fine Hall Princeton UniversityPrincetonUSA

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