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1.
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
2.
Let X
1, X
2, ... be i.i.d. positive random variables, and let
n
be the initial rank of X
n
(that is, the rank of X
n among X
1, ..., X
n). Those observations whose initial rank is k are collected into a point process N
k on +, called the k-record process. The fact that {itNk; k=1, 2, ... are independent and identically distributed point processes is the main result of the paper. The proof, based on martingales, is very rapid. We also show that given N
1, ..., N
k, the lifetimes in rank k of all observations of initial rank at most k are independent geometric random variables.These results are generalised to continuous time, where the analogue of the i.i.d. sequence is a time-space Poisson process. Initially, we think of this Poisson process as having values in +, but subsequently we extend to Poisson processes with values in more general Polish spaces (for example, Brownian excursion space) where ranking is performed using real-valued attributes. 相似文献
3.
Arunava Mukherjea 《Probability Theory and Related Fields》1991,87(3):389-401
Summary In this paper we present a necessary and sufficient condition for tightness of products of i.i.d. finite dimensional random nonnegative matrices. We give an example illustrating the use of our theorem and treat completely the case of 2×2 matrices. We also describe stationary solutions of the linear equationy
n=Xnyn–1, n>0, in (R
d
)+, whereX
1,X
2,... are i.i.d.d×d nonnegative matrices. 相似文献
4.
David Mejzler 《Israel Journal of Mathematics》1987,57(1):1-27
LetX
1, ...,X
n be independent random variables, letF
i be the distribution function ofX
i (1≦i≦n) and letX
1n
≦... ≦X
nn be the corresponding order statistics. We consider the statisticsX
kn, wherek=k(n),k/n → 1 andn−k → ∞. Under some additional restrictions concerning the behaviour of the sequences {a
n>0,b
n,k(n),F
n} we characterize the class of all distribution functionsH such that Prob{(X
kn
−b
n
)/a
n
<x)}→H.
Dedicated to the Memory of N. V. Smirnov (1900–1966) 相似文献
5.
Fuqing Gao 《Journal of Mathematical Analysis and Applications》2003,285(1):1-7
Let X0,X1,… be i.i.d. random variables with E(X0)=0, E(X20)=1 and E(exp{tX0})<∞ for any |t|<t0. We prove that the weighted sums V(n)=∑j=0∞aj(n)Xj, n?1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös-Rényi-Shepp laws for the weighted sums. 相似文献
6.
We give an example of two distinct stationary processes {X
n} and {X′
n} on {0, 1} for whichP[X0=1|X−1=a−1,X−2=a−2, …]=P[X′0=1|X′−1=a−1,X′−2=a−2, …] for all {a
i},i=−1, −2, …, even though these probabilities are bounded away from 0 and 1, and are continuous in {a
i}.
Supported in part by NSF Grant DMS 89-01545.
Supported in part by the US Army Research Office. 相似文献
7.
André Adler 《Central European Journal of Mathematics》2006,4(1):1-4
Consider independent and identically distributed random variables {X
nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X
n(i) ≤ X
n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables
R
ij = X
n(j)/X
n(i). 相似文献
8.
9.
R. Shimizu 《Annals of the Institute of Statistical Mathematics》1979,31(1):367-372
Summary A necessary and sufficient condition is given under which the minimum ofX
k
/a
k
,k=1, 2, ...,n has the same distribution asX
1, whereX's are i.i.d. positive random variables anda's are given positive constants.
The Institute of Statistical Mathematics 相似文献
10.
P. Vértesi 《Constructive Approximation》1999,15(3):355-367
For a wide class of Freud-type weights of form w = exp(-Q) we investigate the behavior of the corresponding weighted Lebesgue function λ
n
(w,X,x) , where X = { x
kn
}
(-∞,∞) is an interpolatory matrix. We prove that for arbitrary
X
(-∞,∞) and ɛ > 0 , fixed,
λ
n
(w, X, x)
≥ c ɛ log n,
x ∈
[-a
n
, a
n
]\H
n
,
n ≥ 1,
where a
n
is the MRS number and |H
n
| ≤ 2
ɛ
a
n
. The result corresponds to the behavior of the ``ordinary' Lebesgue function in [-1,1] .
Other exponential weights are considered in our subsequent paper.
October 28, 1996. Date revised: April 7, 1997. Date accepted: March 18, 1998. 相似文献
11.
Let X, X1 , X2 , ··· be a sequence of nondegenerate i.i.d. random variables with zero means, which is in the domain of attraction of the normal law. Let {a ni , 1≤i≤n, n≥1} be an array of real numbers with some suitable conditions. In this paper, we show that a central limit theorem for self-normalized weighted sums holds. We also deduce a version of ASCLT for self-normalized weighted sums. 相似文献
12.
A. I. Martikainen 《Journal of Mathematical Sciences》2006,133(3):1308-1313
Let {Xi, Yi}i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y1 = y) = 0 for all y. Put Mn(j) = max0≤k≤n-j (Xk+1 + ... Xk+j)Ik,j, where Ik,k+j = I{Yk+1 < ⋯ < Yk+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let Ln be the largest index l ≤ n for which Ik,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,”
i.e., for Mn(Ln) has been recently derived for the case where X1 has a finite moment of order 3 + ε, ε > 0. Assuming that X1 has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(Ln - a) is equivalent to EX
1
3+a
I{X1 > 0} < ∞. We derive also some new results for the a.s. asymptotics of Ln. Bibliography: 5 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 179–189. 相似文献
13.
Péter Kevei 《Periodica Mathematica Hungarica》2010,60(1):25-36
Let X1,X2, ... be iid random variables, and let a
n
= (a
1,n, ..., a
n,n
) be an arbitrary sequence of weights. We investigate the asymptotic distribution of the linear combination $
S_{a_n }
$
S_{a_n }
= a
1,n
X
1 + ... + a
n,n
X
n
under the natural negligibility condition lim
n→∞
max{|a
k,n
|: k = 1, ..., n} = 0. We prove that if $
S_{a_n }
$
S_{a_n }
is asymptotically normal for a weight sequence a
n
, in which the components are of the same magnitude, then the common distribution belongs to $
\mathbb{D}
$
\mathbb{D}
(2). 相似文献
14.
Lucia Alessandrini Giovanni Bassanelli Marco Leoni 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2002,72(1):255-268
We study here K?hler-type properties of 1-convex manifolds, using the duality between forms and compactly supported currents,
and some properties of the Aeppli groups of (q-convex manifolds. We prove that, when the exceptional setS of the l-convex manifoldX has dimensionk, X is p-K?hler for everyp > k, and isk-K?hler if and only if “the fundamental class” ofS does not vanish. There are classical examples whereX is notk-K?hler even with a smoothS, but we prove that this cannot happen if2k ≥n = dimX, nor for suitable neighborhoods of S; in particular,X is always balanced (i.e.,(n - 1)-Kahler).
Partially supported by MIUR research funds. 相似文献
15.
Dimitra Kosta 《Proceedings of the Steklov Institute of Mathematics》2009,264(1):102-109
Let X be a complete intersection of two hypersurfaces F
n
and F
k
in ℙ5 of degree n and k, respectively, with n ≥ k, such that the singularities of X are nodal and F
k
is smooth. We prove that if the threefold X has at most (n + k − 2)(n − 1) − 1 singular points, then it is factorial. 相似文献
16.
A strong law for weighted sums of i.i.d. random variables 总被引:4,自引:0,他引:4
Jack Cuzick 《Journal of Theoretical Probability》1995,8(3):625-641
A strong law is proved for weighted sumsS
n=a
in
X
i whereX
i are i.i.d. and {a
in} is an array of constants. When sup(n
–1|a
in
|
q
)1/q
<, 1<q andX
i are mean zero, we showE|X|
p
<,p
l+q
–1=1 impliesS
n
/n
0. Whenq= this reduces to a result of Choi and Sung who showed that when the {a
in} are uniformly bounded,EX=0 andE|X|< impliesS
n
/n
0. The result is also true whenq=1 under the additional assumption that lim sup |a
in
|n
–1 logn=0. Extensions to more general normalizing sequences are also given. In particular we show that when the {a
in} are uniformly bounded,E|X|1/< impliesS
n
/n
0 for >1, but this is not true in general for 1/2<<1, even when theX
i are symmetric. In that case the additional assumption that (x
1/ log1/–1
x)P(|X|x)0 asx provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a
in}. 相似文献
17.
André Adler 《Journal of Theoretical Probability》1992,5(4):707-726
Let {X,X
n,nZ
+
d
} be a sequence of independent and identically distributed random variables and {a
n
,n Z
+
d
} be a sequence of constants. We examine the almost sure limiting behavior of weighted partial sums of the form |n|N
a
n
X
n
. Suppose further that eitherEX=0 orE|X|=. In most situations these normalized partial sums fail to have a limit, no matter which normalizing sequence we choose. Thus, the investigation lends itself to the study of the limit inferior and limit superior of these sequences. On the way to proving results of this type we first establish several weak laws. These weak laws prove to be of great value in establishing generalized laws of the iterated logarithm. 相似文献
18.
LU Chuanrong QIU Jin & XU Jianjun School of Mathematics Statistics Zhejiang University of Finance Economics Hangzhou China Department of Mathematics Zhejiang University Hangzhou China 《中国科学A辑(英文版)》2006,49(12):1788-1799
Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc. 相似文献
19.
V. Anantharam 《Queueing Systems》1989,5(4):345-367
LetW
k
denote the waiting time of customerk, k 0, in an initially empty GI/G/1 queue. Fixa> 0. We prove weak limit theorems describing the behaviour ofW
k
/n, 0kn, given Wn >na. LetX have the distribution of the difference between the service and interarrival distributions. We consider queues for which Cramer type conditions hold forX, and queues for whichX has regularly varying positive tail.The results can also be interpreted as conditional limit theorems, conditional on large maxima in the partial sums of random walks with negative drift.Research supported by the NSF under Grant NCR 8710840 and under the PYI Award NCR 8857731. 相似文献
20.
Xiaotao Sun 《Inventiones Mathematicae》2008,173(2):427-447
Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→X
1 be the relative Frobenius morphism. For any vector bundle W on X, we prove that instability of F
*
W is bounded by instability of W⊗Tℓ(Ω1
X
) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F
*
W being stable whenever W is stable.
Dedicated to Professor Zhexian Wan on the occasion of his 80th birthday. 相似文献