Large Deviation Probabilities for Some Classes of Distributions Satisfying the Cramer Condition

Rozovsky, L. V.

doi:10.1007/s10958-005-0207-y

Cited by 4 publications

(3 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is obvious that a precise lower bound of Ee λSn allows to improve Nagaev's bound (24) by equality (22).…”

Section: Resultsmentioning

confidence: 99%

“…The study of sharp large deviation probabilities has a long history. Many interesting asymptotic expansions have been established in Cramér [9], Bahadur and Ranga Rao [2], Petrov [26] and Rozovky [24,25]. Various exponential upper bounds have been obtained by Prohorov [30], Nagaev [18], Petrov [27] and Talagrand [33], see also McDiarmid [17], Nagaev [20,21] and [11,12] for martingales.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Sharp large deviation probabilities for sums of independent bounded random variables

Fan,

Grama,

Liu

2012

Preprint

View full text Add to dashboard Cite

We obtain some optimal inequalities on tail probabilities for sums of independent bounded random variables. Our main result completes an upper bound on tail probabilities due to Talagrand by giving a one-term asymptotic expansion for large deviations. This result can also be regarded as sharp large deviations of types of Cramér and Bahadur-Ranga Rao. Keywords sharp large deviations • Talagrand's inequality • large deviations • Bahadur-Rao theorem • sums of independent random variables • random walks Mathematics Subject Classification (2000) 60F10 • 60E15 • 60G50 • 62E20 X. Fan ( ) •

show abstract

“…It is obvious that a precise lower bound of Ee λSn allows to improve Nagaev's bound (24) by equality (22).…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sharp large deviation probabilities for sums of independent bounded random variables

Fan,

Grama,

Liu

2012

Preprint

View full text Add to dashboard Cite

show abstract

“…where c y , σ 1y and t y depend on y and the distribution of ξ 1 ; see also Bercu [8,9], Rozovky [31] and Györfi, Harremöes and Tusnády [24] for more general results. Our bound (2.13) implies that, for y 0 small enough,…”

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

confidence: 99%

Sharp large deviation results for sums of independent random variables

2015

View full text Add to dashboard Cite

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.

show abstract