Limit Theorems on Large Deviations

Saulis, L.; Statulevičius, V.

doi:10.1007/978-3-662-04172-7_5

Cited by 28 publications

(38 citation statements)

References 50 publications

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“…In the following theorem, we obtain a one term sharp large deviation expansion similar to Cramér [13], Bahadur-Rao [2], Saulis and Statulevicius [41] and Sakhanenko [43].…”

Section: And R(t) By (27) In Particular For Allmentioning

confidence: 70%

“…Cramér [13] (see also Theorem 3.1 of Saulis and Statulevicius [41] for more general results) proved that, for all 0 x = o( √ n),…”

Section: Comparison With the Expansions Of Cramér And Bahadur-raomentioning

confidence: 97%

“…When Bernstein's condition fails, we refer to Theorem 3.1 of Saulis and Statulevicius [41], where explicit and asymptotic expansions have been established via the Cramér series (cf. Petrov [34] for details).…”

Section: And R(t) By (27) In Particular For Allmentioning

confidence: 99%

“…Various precise inequalities and asymptotic results have been established by Hoeffding [25], Nagaev [32], Saulis and Statulevicius [41], Chaganty and Sethuraman [12] and Petrov [35] under different backgrounds. Assume that (ξ i ) i=1,...,n satisfies Bernstein's condition…”

Section: Introductionmentioning

confidence: 99%

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Sharp large deviation results for sums of independent random variables

2015

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We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein's condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds improve the inequalities of Bennett and Hoeffding by adding missing factors in the spirit of Talagrand (1995). We also complete Talagrand's inequality by giving a lower bound of the same form, leading to an equality. As a consequence, we obtain large deviation expansions similar to those of Cramér (1938), Bahadur-Rao (1960) and Sakhanenko (1991). We also show that our bound can be used to improve a recent inequality of Pinelis (2014). KeywordsBernstein's inequality, sharp large deviations, Cramér large deviations, expansion of BahadurRao, sums of independent random variables, Bennett's inequality, Hoeffding's inequality MSC(2010) 60F10, 60F05, 60E15, 60G50 Citation: Fan X Q, Grama I, Liu Q S. Sharp large deviation results for sums of independent random variables.

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“…In the following theorem, we obtain a one term sharp large deviation expansion similar to Cramér [13], Bahadur-Rao [2], Saulis and Statulevicius [41] and Sakhanenko [43].…”

Section: And R(t) By (27) In Particular For Allmentioning

confidence: 70%

“…Cramér [13] (see also Theorem 3.1 of Saulis and Statulevicius [41] for more general results) proved that, for all 0 x = o( √ n),…”

Section: Comparison With the Expansions Of Cramér And Bahadur-raomentioning

confidence: 97%

Section: And R(t) By (27) In Particular For Allmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sharp large deviation results for sums of independent random variables

2015

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“…It is worth noting that in all the above-mentioned works, as well as in [1], [2], [13], [15], [23], [28], [31], [32], the order of deviations is bounded by O(n 1/2 …”

Section: Introduction Let ξ ξmentioning

confidence: 99%

Abelian Theorems, Limit Properties of Conjugate Distributions, and Large Deviations for Sums of Independent Random Vectors

Zaigraev¹,

Nagaev$^dag$²

2004

Theory Probab. Appl.

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Abstract.A class of multidimensional absolutely continuous distributions is considered. Each distribution has a moment generating function, which is finite in a bounded convex set S and generates a family of the so-called conjugate distributions. We focus our attention on the limit distributions for this family when the conjugate parameter tends to the boundary of S. As in the one-dimensional case, each limit distribution is obtained as a corollary of the Abel-type theorem. The results obtained are utilized for establishing a local limit theorem for large deviations of arbitrarily high order.

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The necessity of the Linnik condition in a theorem on probabilities of large deviations

Amosova¹

1999

J Math Sci

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Limit Theorems on Large Deviations

Cited by 28 publications

References 50 publications

Sharp large deviation results for sums of independent random variables

Sharp large deviation results for sums of independent random variables

Abelian Theorems, Limit Properties of Conjugate Distributions, and Large Deviations for Sums of Independent Random Vectors

The necessity of the Linnik condition in a theorem on probabilities of large deviations

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