Summary
LetX 1,X 2,h. be i.i.d. random variables in the domain of attraction of a stable lawG, and denote Sn = X1 + Xn, Ln(ω, A) =n−1 \(L_n (\omega ,A) = n^{ - 1} \sum\limits_{j = 0}^{n - 1} {\chi _A ({{S_j } \mathord{\left/ {\vphantom {{S_j } {a(n)}}} \right. \kern-\nulldelimiterspace} {a(n)}})}\) satisfiesa(n) −1 S n →G. Large deviation probability estimates of Donsker-Varadhan type are obtained forL n (ω, ·), 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 ofX 1. The techniques used are those developed by Donsker and Varadhan in their theory of large deviations.
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This work was partially supported by the National Science Foundation.
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Jain, N.C. A Donsker-Varadhan type of invariance principle. Z. Wahrscheinlichkeitstheorie verw Gebiete 59, 117–138 (1982). https://doi.org/10.1007/BF00575529
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DOI: https://doi.org/10.1007/BF00575529