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 2 n ) = 1 and that \(\psi _n (h) = \sum\limits_{k = 1}^n {\log E(e^{h X_{k,n} } )}\) exists for 0≦h≦ɛ n . Under mild conditions we show that
where the quantities z n and r n are related by the parametric equations
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 = (X 1+⋯.+X n )/s n . A familiar special case of (1) is given by
where \(\mathfrak{N}\) is the standard normal distribution and P n a certain power series. In this case rn = z 2n but (2) may lead to radically different relationships between rn and zn.
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Work connected with a Project for Research in Probability at Princeton University sponsored by the U. S. Army Research Office.
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Feller, W. Limit theorems for probabilities of large deviations. Z. Wahrscheinlichkeitstheorie verw Gebiete 14, 1–20 (1969). https://doi.org/10.1007/BF00534113
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DOI: https://doi.org/10.1007/BF00534113