A Donsker-Varadhan type of invariance principle

Summary LetX₁,X₂,h. be i.i.d. random variables in the domain of attraction of a stable lawG, and denote S_n = X₁ + X_n, L_n(, A) =n_–1 satisfiesa(n)^–1S_nG. Large deviation probability estimates of Donsker-Varadhan type are obtained forL_n(, ·), and these are then used to study the behavior of small values of (S_n/a(n). These latter results are analogues of Strassen's results which described the behavior of large values of (S_n/a(n)) when the limit law was Gaussian. The limiting constants are seen to depend only on the limit lawG and not on the distribution ofX₁. The techniques used are those developed by Donsker and Varadhan in their theory of large deviations.This work was partially supported by the National Science Foundation.