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Large Deviation Probabilities for Weighted Sums

Large Deviation Probabilities for Weighted Sums,10.1214/aoms/1177692474,The Annals of Mathematical Statistics

Large Deviation Probabilities for Weighted Sums   (Citations: 12)
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Taking as our point of departure the methods and results in a 1960 paper of Bahadur and Ranga Rao, we derive asymptotic representations of large deviation probabilities for weighted sums of independent, identically distributed random variables. The main theorem generalizes the Bahadur-Ranga Rao result in the absolutely continuous case. The method of proof closely parallels that of the 1960 paper, a major component of which was the use of Cramer's 1923 theorem on asymptotic expansions. For our result, we need an extension of Cramer's theorem to triangular arrays, and that extension is also developed in the paper. We then show that the main theorem implies a logarithmic result which generalizes a 1952 theorem of Chernoff and is of more precision but less generality than a 1969 result of Feller. Finally, we note that in the exponential case the theorem can be used to estimate large deviation probabilities for linear combinations of exponential order statistics.
Journal: The Annals of Mathematical Statistics , vol. 43, no. 1972, pp. 1221-1234, 1972
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    • ...random variables. Some extensions of Bahadur and Rao theorem were proposed in [8]...

    Bernard Bercuet al. Sharp large deviations for Gaussian quadratic forms with applications

    • ...All these results, including the one in the present article, represent the asymptotic behavior less precisely than, for example, Cram& [4], Bahadur and Ranga Rao [1], and Book [2]...
    • ...The theorems in [2] on weighted sums, while more precise than those here, apply at present only in the absolutely continuous case...
    • ...As in Section 1 of [2], we define random variables Y,k = a,k(Xk--C) and observe...
    • ...It turns out that, under Conditions I and II, there exists a sequence {h.' 1 _-< n < oo } of h's to appear in the definition of the associated random variables such that E(S.)=0 and the variances Var(S.) are uniformly bounded away from 0 and oe. In particular, we have the following two results, whose proofs can be found in Section 2 of [2] :...

    Unknown. A large deviation theorem for weighted sums

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