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Sharp large deviation results for sums of independent random variables 被引量:1
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作者 FAN XieQuan GRAMA Ion LIU QuanSheng 《Science China Mathematics》 SCIE CSCD 2015年第9期1939-1958,共20页
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 certai... 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 Cram′er(1938),Bahadur-Rao(1960) and Sakhanenko(1991). We also show that our bound can be used to improve a recent inequality of Pinelis(2014). 展开更多
关键词 Bernstein’s inequality sharp large deviations Cramér large deviations expansion of BahadurRao sums of independent random variabl
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On Random Taylor Series
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作者 Ding Xiaoqing College of Mathematical Sciences, Wuhan University, Wuhan 430072, China 《Wuhan University Journal of Natural Sciences》 EI CAS 1998年第3期3-6,共4页
This paper deals with random Taylor series whose coefficients consist of independent random variables {X n} with the property: αE 1/2 {|X n| 2}≤E{|X n|}<∞, E{X n}=0 (n ) for some positive cons... This paper deals with random Taylor series whose coefficients consist of independent random variables {X n} with the property: αE 1/2 {|X n| 2}≤E{|X n|}<∞, E{X n}=0 (n ) for some positive constant α. The convergence, growth, and value distribution of the series are investigated. 展开更多
关键词 random Taylor series independent random variables CONVERGENCE GROWTH value distribution
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A LIL for independent non-identically distributed random variables in Banach space and its applications 被引量:2
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作者 LIU WeiDong FU KeAng ZHANG LiXin 《Science China Mathematics》 SCIE 2008年第2期219-232,共14页
In this paper,we prove a general law of the iterated logarithm (LIL) for independent non-identically distributed B-valued random variables.As an interesting application,we obtain the law of the iterated logarithm for ... In this paper,we prove a general law of the iterated logarithm (LIL) for independent non-identically distributed B-valued random variables.As an interesting application,we obtain the law of the iterated logarithm for the empirical covariance of Hilbertian autoregressive processes. 展开更多
关键词 law of the iterated logarithm independent random variable autoregressive Hilbertian processes covariance operator 60F15
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Sharp large deviations for sums of bounded from above random variables
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作者 FAN XieQuan 《Science China Mathematics》 SCIE CSCD 2017年第12期2465-2480,共16页
We show large deviation expansions for sums of independent and bounded from above random variables. Our moderate deviation expansions are similar to those of Cram′er(1938), Bahadur and Ranga Rao(1960), and Sakhanenko... We show large deviation expansions for sums of independent and bounded from above random variables. Our moderate deviation expansions are similar to those of Cram′er(1938), Bahadur and Ranga Rao(1960), and Sakhanenko(1991). In particular, our results extend Talagrand's inequality from bounded random variables to random variables having finite(2 + δ)-th moments, where δ∈(0, 1]. As a consequence,we obtain an improvement of Hoeffding's inequality. Applications to linear regression, self-normalized large deviations and t-statistic are also discussed. 展开更多
关键词 sharp large deviations Cram′er large deviations Talagrand’s inequality Hoeffding’s inequality sums of independent random variables
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NOTES ON ERDOS' CONJECTURE
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作者 孔繁超 唐启鹤 《Acta Mathematica Scientia》 SCIE CSCD 2000年第4期533-541,共9页
Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The aut... Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent, it is both sufficient and necessary that there exist three numbers X0 and X1 < X2 such that limsup(FN(X2) - FN(X1)) > 0 holds, and Lo = N→ ∞ lim FN(X0) exists. Moreover, the authors point out that they can also obtain the same result in the weakened case of lim inf P(Xn = 0) > 0. 展开更多
关键词 Erdos' conjecture additive arithmetic function sums of independent random variables essential convergence weak convergence
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Extended Rama Distribution:Properties and Applications
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作者 Khaldoon M.Alhyasat Kamarulzaman Ibrahim +1 位作者 Amer Al-Omari Mohd Aftar Abu Bakar 《Computer Systems Science & Engineering》 SCIE EI 2021年第10期55-67,共13页
In this paper,the Rama distribution(RD)is considered,and a new model called extended Rama distribution(ERD)is suggested.The new model involves the sum of two independent Rama distributed random variables.The probabili... In this paper,the Rama distribution(RD)is considered,and a new model called extended Rama distribution(ERD)is suggested.The new model involves the sum of two independent Rama distributed random variables.The probability density function(pdf)and cumulative distribution function(cdf)are obtained and analyzed.It is found that the new model is skewed to the right.Several mathematical and statistical properties are derived and proved.The properties studied include moments,coefficient of variation,coefficient of skewness,coefficient of kurtosis and moment generating function.Some simulations are undertaken to illustrate the behavior of these properties.In addition,the reliability analysis of the distribution is investigated through the hazard rate function,reversed hazard rate function and odds function.The parameter of the distribution is estimated based on the maximum likelihood method.The distributions of order statistics for ERD are also presented.The performance of the suggested model is compared with several other lifetime distributions based on some goodness of fit tests on a real dataset.It turns out that the suggested model is more flexible than its competitors considered in this study,for modeling real lifetime data. 展开更多
关键词 Coefficient of kurtosis coefficient of skewness order statistics two independent Rama random variables Maximum likelihood estimation Rama distribution
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On the asymptotic independence of the sum and maximum of normal random variables
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作者 XIE ShengrongDepartment of Mathematics, Southwest-China Normal University, Chongqing 630715, China 《Chinese Science Bulletin》 SCIE EI CAS 1997年第21期1846-1846,共1页
RECENTLY, a number of papers have been published concerning the asymptotic independentproperties of V X<sub>i</sub> and sum from 1 X<sub>i</sub> of weakly dependent stationary sequence {X<su... RECENTLY, a number of papers have been published concerning the asymptotic independentproperties of V X<sub>i</sub> and sum from 1 X<sub>i</sub> of weakly dependent stationary sequence {X<sub>i</sub>}.In this letter, let {X<sub>i</sub>} be a standard normal sequence of random variables with zero meanand unit variance and write r<sub>ij</sub>=cov(X<sub>i</sub>, X<sub>j</sub>). 展开更多
关键词 On the asymptotic independence of the sum and maximum of normal random variables
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On necessary and sufficient conditions for the self-normalized central limit theorem In Honor of Professor Chuanrong Lu on His 85th Birthday 被引量:1
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作者 Qiman Shao 《Science China Mathematics》 SCIE CSCD 2018年第10期1741-1748,共8页
Let X_1, X_2,... be a sequence of independent random variables and S_n=sum X_1 from i=1 to n and V_n^2=sum X_1~2 from i=1 to n . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S... Let X_1, X_2,... be a sequence of independent random variables and S_n=sum X_1 from i=1 to n and V_n^2=sum X_1~2 from i=1 to n . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n/V_n converges to a standard normal distribution if and only if max1≤i≤n|X_i|/V_n → 0 in probability and the mean of X_1 is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max1≤i≤n|X_i|/V_n → 0 in probability, then these sufficient conditions are necessary. 展开更多
关键词 central limit theorem SELF-NORMALIZED independent random variables
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