Sign in
Author

Conference

Journal

Organization

Year

DOI
Look for results that meet for the following criteria:
since
equal to
before
between
and
Search in all fields of study
Limit my searches in the following fields of study
Agriculture Science
Arts & Humanities
Biology
Chemistry
Computer Science
Economics & Business
Engineering
Environmental Sciences
Geosciences
Material Science
Mathematics
Medicine
Physics
Social Science
Multidisciplinary
Keywords
(7)
Absolute Continuity
Asymptotic Expansion
Large Deviation
Order Statistic
Random Variable
Weighted Sums
Independent Identically Distributed
Subscribe
Academic
Publications
Large Deviation Probabilities for Weighted Sums
Large Deviation Probabilities for Weighted Sums,10.1214/aoms/1177692474,The Annals of Mathematical Statistics
Edit
Large Deviation Probabilities for Weighted Sums
(
Citations: 12
)
BibTex

RIS

RefWorks
Download
Unknown
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 BahadurRanga 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. 12211234, 1972
DOI:
10.1214/aoms/1177692474
Cumulative
Annual
View Publication
The following links allow you to view full publications. These links are maintained by other sources not affiliated with Microsoft Academic Search.
(
projecteuclid.org
)
Citation Context
(2)
...random variables. Some extensions of Bahadur and Rao theorem were proposed in [
8
]...
Bernard Bercu
,
et 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(XkC) 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
Sort by:
Citations
(12)
Extreme value theory in some statistical analysis of genomic sequences
Lily Wang
,
Pranab K. Sen
Journal:
Extremes
, vol. 8, no. 4, pp. 295310, 2005
Grandes déviations pour des sommes pondérées de variables aléatoires i.i.d. appliquées à un problème géographique
Olivier Bonin
Journal:
Comptes Rendus De L Academie Des Sciences Serie Imathematique  C R ACAD SCI SER I MATH
, vol. 333, no. 4, pp. 369372, 2001
Sharp large deviations for Gaussian quadratic forms with applications
(
Citations: 13
)
Bernard Bercu
,
Fabrice Gamboa
,
Marc Lavielle
Journal:
Esaim: Probability and Statistics
, vol. 4, pp. 124, 2000
Probabilities of large deviations for random fields
(
Citations: 3
)
S. Čepulėnas
Journal:
Lithuanian Mathematical Journal  LITH MATH J
, vol. 25, no. 4, pp. 381390, 1985
Der zentrale grenzwertsatz und wahrseheinliehkeiten grober abweichungen f¨r summen
W. Wolf
Journal:
Statistics
, vol. 15, no. 3, pp. 389395, 1984