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1.
V. Bentkus 《Lithuanian Mathematical Journal》2008,48(2):137-157
Let S = X
1 + ⋯ + X
n
be a sum of independent random variables such that 0 ⩽ X
k
⩽ 1 for all k. Write p = E S/n and q = 1 − p. Let 0 < t < q. In this paper, we extend the Hoeffding inequality [16, Theorem 1]
, to the case where X
k
are unbounded positive random variables. Our inequalities reduce to the Hoeffding inequality if 0 ⩽ X
k
⩽ 1. Our conditions are X
k
⩾ 0 and E S < ∞. We also provide improvements comparable with the inequalities of Bentkus [5]. The independence of X
k
can be replaced by supermartingale-type assumptions. Our methods can be extended to prove counterparts of other inequalities
of Hoeffding [16] and Bentkus [5].
The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No T-25/08. 相似文献
2.
We provide precise bounds for tail probabilities, say {M
n
x}, of sums M
n
of bounded i.i.d. random variables. The bounds are expressed through tail probabilities of sums of i.i.d. Bernoulli random variables. In other words, we show that the tails are sub-Bernoullian. Sub-Bernoullian tails are dominated by Gaussian tails. Possible extensions of the methods are discussed. 相似文献
3.
V. Bentkus 《Lithuanian Mathematical Journal》2008,48(3):237-255
Let S
n = X
1 + ⋯ + X
n be a sum of independent random variables such that 0 ⩽ X
k ⩽ 1 for all k. Write {ie237-01} and q = 1 − p. Let 0 < t < q. In our recent paper [3], we extended the inequality of Hoeffding ([6], Theorem 1) {fx237-01} to the case where X
k are unbounded positive random variables. It was assumed that the means {ie237-02} of individual summands are known. In this
addendum, we prove that the inequality still holds if only an upper bound for the mean {ie237-03} is known and that the i.i.d.
case where {ie237-04} dominates the general non-i.i.d. case. Furthermore, we provide upper bounds expressed in terms of certain
compound Poisson distributions. Such bounds can be more convenient in applications. Our inequalities reduce to the related
Hoeffding inequalities if 0 ⩽ X
k ⩽ 1. Our conditions are X
k ⩾ 0 and {ie237-05}. In particular, X
k can have fat tails. We provide as well improvements comparable with the inequalities in Bentkus [2]. The independence of
X
k can be replaced by super-martingale type assumptions. Our methods can be extended to prove counterparts of other inequalities
in Hoeffding [6] and Bentkus
The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No T-25/08. 相似文献
4.
Let X be a random variable with survival function G(x)=ℙ{X≥x}. Let α > 0. Consider a transform, say T
α, of survival functions G↦T
α
G defined by
. In this paper, we examine the properties of the transform as well as provide explicit expressions for T
α
G in some special cases. Our motivation to study the properties of the transform comes from the theory of inequalities for
tail probabilities of sums of independent random variables, martingales, super-martingales, etc. The transform is a commonly
used tool in this field, and up to date it has led to most advanced or new-type inequalities. For statistical applications,
such as construction of upper confidence bounds, particularly, in problems related to the auditing mathematics, computable
and as precise as possible inequalities are required. This is the main motivation to write this paper. We examine general
properties of the transform and consider special cases of normal, exponential, Bernoulli, binomial, uniform, and Poisson tails.
These types and other survival functions appear in known applications. Until now only the values of α = 1, 2, 3 were actually used, and we concentrate our attention to these three special cases.
This research was supported by the Technology Foundation STW, applied science division of NWO and the technology program of
the Ministry of Economic Affairs.
Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 1, pp. 3–54, January–March, 2006. 相似文献
5.
Let M
n =X1+...+Xn be a martingale with bounded differences Xm=Mm-Mm-1 such that {|Xm| m}=1 with some nonnegative m. Write 2=
1
2
+ ... +
n
2
. We prove the inequalities {M
nx}c(1-(x/)), {M
n x} 1- c(1- (-x/)) with a constant
. The result yields sharp inequalities in some models related to the measure concentration phenomena. 相似文献
6.
An Inequality for Tail Probabilities of Martingales with Differences Bounded from One Side 总被引:4,自引:0,他引:4
V. Bentkus 《Journal of Theoretical Probability》2003,16(1):161-173
Let M
n
=X
1+ +X
n
be a martingale with differences X
k
=M
k
–M
k–1bounded from above such that
with some non-random positive
k
.Let the conditional variance
2
k
=E(X
2
k
|X
1,...,X
k–1) satisfy
2
k
s
2
k
with probability one, where s
2
k
are some non-random numbers. Write
2
k
=max{
2
k
,s
2
k
} and
2=
2
1+ +
2
n
. We prove the inequality
with a constant
. 相似文献
7.
Let M
n
= X
1 + ⋯ + X
n
be a sum of independent random variables such that X
k
⩽ 1,
and EX
k
2
= σ
k
2
for all k. Hoeffding [15, Theorem 3] proved that
with
. Bentkus [5] improved Hoeffding’s inequalities using binomial tails as upper bounds. Let
and
stand for the skewness and kurtosis of X
k
. In this paper we prove (improved) counterparts of the Hoeffding inequality replacing σ
2 by certain functions of γ
1, ..., γ
n
(respectively ϰ1, ..., ϰ1). Our bounds extend to a general setting where X
k
are martingale differences, and they can combine the knowledge of skewness and/or kurtosis and/or variances of X
k
. Up to factors bounded by e
2/2 the bounds are final. All our results are new since no inequalities incorporating skewness or kurtosis control are known
so far.
The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No T-15/07. 相似文献
8.
A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables 总被引:19,自引:0,他引:19
Qi-Man Shao 《Journal of Theoretical Probability》2000,13(2):343-356
Let {X
i, 1in} be a negatively associated sequence, and let {X*
i
, 1in} be a sequence of independent random variables such that X*
i
and X
i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef(
n
i=1
X
i)Ef(
n
i=1
X*
i
) for any convex function f on R
1 and that Ef(max1kn
n
i=k
X
i)Ef(max1kn
k
i=1
X*
i
) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population. 相似文献
9.
Periodica Mathematica Hungarica - Let X 1,X 2,... be a sequence of independent and identically distributed random variables, and put % MATHTYPE!MTEF!2!1!+-%... 相似文献
10.
Mark Veraar 《Potential Analysis》2009,30(4):341-370
In this paper we extend certain correlation inequalities for vector-valued Gaussian random variables due to Kolmogorov and Rozanov. The inequalities are applied to sequences of Gaussian random variables and Gaussian processes. For sequences of Gaussian random variables satisfying a correlation assumption, we prove a Borel-Cantelli lemma, maximal inequalities and several laws of large numbers. This extends results of Be?ka and Ciesielski and of Hytönen and the author. In the second part of the paper we consider a certain class of vector-valued Gaussian processes which are α-Hölder continuous in p-th moment. For these processes we obtain Besov regularity of the paths of order α. We also obtain estimates for the moments in the Besov norm. In particular, the results are applied to vector-valued fractional Brownian motions. These results extend earlier work of Ciesielski, Kerkyacharian and Roynette and of Hytönen and the author. 相似文献