共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
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
3.
Complete moment and integral convergence for sums of negatively associated random variables 总被引:2,自引:0,他引:2
For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE(max1≤k≤n|Sk|^1/q-∈bn^1/qp)^+〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 (log n) ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence. 相似文献
4.
In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f
1(z), f
2(z), …, f
n
(z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ
n
and
$
\begin{gathered}
\frac{{D^{tk + 1} + f_p \left( 0 \right)\left( {z^{tk + 1} } \right)}}
{{\left( {tk + 1} \right)!}} = \sum\limits_{l_1 ,l_2 ,...,l_{tk + 1} = 1}^n {\left| {apl_1 l_2 ...l_{tk + 1} } \right|e^{i\tfrac{{\theta pl_1 + \theta pl_2 + ... + \theta pl_{tk + 1} }}
{{tk + 1}}} zl_1 zl_2 ...zl_{tk + 1} ,} \hfill \\
p = 1,2,...,n. \hfill \\
\end{gathered}
$
\begin{gathered}
\frac{{D^{tk + 1} + f_p \left( 0 \right)\left( {z^{tk + 1} } \right)}}
{{\left( {tk + 1} \right)!}} = \sum\limits_{l_1 ,l_2 ,...,l_{tk + 1} = 1}^n {\left| {apl_1 l_2 ...l_{tk + 1} } \right|e^{i\tfrac{{\theta pl_1 + \theta pl_2 + ... + \theta pl_{tk + 1} }}
{{tk + 1}}} zl_1 zl_2 ...zl_{tk + 1} ,} \hfill \\
p = 1,2,...,n. \hfill \\
\end{gathered}
相似文献
5.
Zong-mao Cheng Xiu-yun Wang Zheng-yan Lin 《应用数学学报(英文版)》2006,22(1):81-90
In this paper, we introduce a class of Gaussian processes Y={Y(t):t∈R^N},the so called hifractional Brownian motion with the indcxes H=(H1,…,HN)and α. We consider the (N, d, H, α) Gaussian random field x(t) = (x1 (t),..., xd(t)),where X1 (t),…, Xd(t) are independent copies of Y(t), At first we show the existence and join continuity of the local times of X = {X(t), t ∈ R+^N}, then we consider the HSlder conditions for the local times. 相似文献
6.
Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables 总被引:1,自引:0,他引:1
Ye JIANG Li Xin ZHANG 《数学学报(英文版)》2006,22(3):781-792
Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1+X2+…+Xn,Mn=max k≤n│Sk│,n≥1.Let an=O(1/loglogn).In this paper,we prove that,for b〉-1,lim ε→0 →^2(b+1)∑n=1^∞ (loglogn)^b/nlogn n^1/2 E{Mn-σ(ε+an)√2nloglogn}+σ2^-b/(b+1)(2b+3)E│N│^2b+3∑k=0^∞ (-1)k/(2k+1)^2b+3 holds if and only if EX=0 and EX^2=σ^2〈∞. 相似文献
7.
Let X, X1 , X2 , . . . be i.i.d. random variables, and set Sn = X1 +···+Xn , Mn = maxk≤n |Sk|, n ≥1. Let an = o( (n)(1/2)/logn). By using the strong approximation, we prove that, if EX = 0, VarX = σ2 0 and E|X| 2+ε ∞ for some ε 0, then for any r 1, lim ε1/(r-1)(1/2) [ε-2-(r-1)]∞∑n=1 nr-2 P{Mn ≤εσ (π2n/(8log n))(1/2) + an } = 4/π . We also show that the widest a n is o( n(1/2)/logn). 相似文献
8.
Zamira Abdikalikova Ryskul Oinarov Lars-Erik Persson 《Czechoslovak Mathematical Journal》2011,61(1):7-26
We consider a new Sobolev type function space called the space with multiweighted derivatives $
W_{p,\bar \alpha }^n
$
W_{p,\bar \alpha }^n
, where $
\bar \alpha
$
\bar \alpha
= (α
0, α
1,…, α
n
), α
i
∈ ℝ, i = 0, 1,…, n, and $
\left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|}
$
\left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|}
,
|