Limit theorems for probabilities of large deviations

  • William Feller


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
$$P\{ S_n > z_n \} = \exp [ - r_n + 0(r_n )], r_n \to \infty$$
where the quantities z n and r n are related by the parametric equations
$$z_n = \psi '_n (h_n ), r_n = h_n \psi '_n (h_n ) - \psi _n (h_n ).$$
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.


Normal Distribution Stochastic Process Probability Theory Power Series Limit Theorem 
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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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