LIMIT THEOREMS FOR PROBABILITIES OF LARGE DEVIATIONS

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 {Mathematical expression} exists for 0≦h≦e{open}n. Under mild conditions we show that {Mathematical expression} where the quantities zn and rn are related by the parametric equations {Mathematical expression} 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 {Mathematical expression} where {Mathematical expression} 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. © 1969 Springer-Verlag.