# Limit theorems for probabilities of large deviations

• William Feller
Article

## Summary

Let {Xk,n; k=1, ⋯, n} be a triangular array of independent variables with row sums Sn. Suppose E(Xk,n) = 0 and E(Sn2) = 1 and that $$\psi _n (h) = \sum\limits_{k = 1}^n {\log E(e^{h X_{k,n} } )}$$ exists for 0≦hɛn. 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 zn and rn 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 Xk,n behave reasonably well it is usually not difficult to obtain satisfactory asymptotic estimates for zn in terms of rn and vice versa. The principal application is to sequences Xk. Then Xk,n= Xk/sn and Sn = (X1+⋯.+Xn)/sn. 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 Pn a certain power series. In this case rn = zn2but (2) may lead to radically different relationships between rn and zn.

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