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1.
Define
, where
is a symmetric U-type statistic, H
k() is the Hermite polynomial of degree k, and {X, X
n, n1} are independent identically distributed binary random variables with Pr(X{–1, 1}})=1. We show that
according as EX=0 or EX0, respectively. 相似文献
2.
Let X
1, X
2,... denote an i.i.d. sequence of real valued random variables which ly in the domain of attraction of a stable law Q with index 0<1. under=" a=" von=" mises=" condition=" we=" show=" that=" the=" sum=" of=" order=" statistics=">1.>
相似文献
3.
Associated random variables and martingale inequalities 总被引:9,自引:0,他引:9
Summary Many of the classical submartingale inequalities, including Doob's maximal inequality and upcrossing inequality, are valid for sequences S
j such that the (S
j+1
-S
j's are associated (positive mean) random variables, and for more general demisubmartingales. The demisubmartingale maximal inequality is used to prove weak convergence to the two-parameter Wiener process of the partial sum processes constructed from a stationary two-parameter sequence of associated random variables X
ijwith
.Alfred P. Sloan Research Fellow. Research supported in part by NSF Grants MCS 77-20683 and MCS 80-19384 相似文献
4.
Pavle Mladenovic´ 《Extremes》1999,2(4):405-419
Let X
n1
*
, ... X
nn
*
be a sequence of n independent random variables which have a geometric distribution with the parameter p
n = 1/n, and M
n
*
= \max\{X
n1
*
, ... X
nn
*
}. Let Z
1, Z2, Z3, ... be a sequence of independent random variables with the uniform distribution over the set N
n = {1, 2, ... n}. For each j N
n let us denote X
nj = min{k : Zk = j}, M
n = max{Xn1, ... Xnn}, and let S
n be the 2nd largest among X
n1, Xn2, ... Xnn. Using the methodology of verifying D(un) and D'(un) mixing conditions we prove herein that the maximum M
n has the same type I limiting distribution as the maximum M
n
*
and estimate the rate of convergence. The limiting bivariate distribution of (Sn, Mn) is also obtained. Let
n, n Nn,
,
and T
n = min{M(An), M(Bn)}. We determine herein the limiting distribution of random variable T
n in the case
n ,
n/n > 0, as n . 相似文献
5.
On outstanding values in a sequence of random variables 总被引:3,自引:0,他引:3
Dr. Mahabanoo N. Tata 《Probability Theory and Related Fields》1969,12(1):9-20
Summary A sequence {X
n} of independent and identically distributed (i.i.d.) random variables is considered. Outstanding values in the sequence are those that strictly exceed values preceding them. Let L
n be the index of the n-th outstanding value. Limit theorems are given for the sequences
and {L
n} and {
n} where
n=Ln–Ln–1. A characterization of the exponential distribution in terms of the sequence
is also given. 相似文献
6.
Vladislav Kargin 《Probability Theory and Related Fields》2007,139(3-4):397-413
Let X
i
denote free identically-distributed random variables. This paper investigates how the norm of products behaves as n approaches infinity. In addition, for positive X
i
it studies the asymptotic behavior of the norm of where denotes the symmetric product of two positive operators: . It is proved that if EX
i
= 1, then is between and c
2
n for certain constant c
1 and c
2. For it is proved that the limit of exists and equals Finally, if π is a cyclic representation of the algebra generated by X
i
, and if ξ is a cyclic vector, then for all n. These results are significantly different from analogous results for commuting random variables. 相似文献
7.
F. Morits 《Mathematical Notes》1975,17(2):127-133
Let {Xi}
–
be a sequence of random variables, E(Xi) 0. If 1, estimates for the -th moments
can be derived from known estimates
of the -th moment. Here we generalized the Men'shov-Rademacher inequality for =2 for orthonormal Xi, to the case 1 and dependent random variables. The Men'shov-Payley inequality >2 for orthonormal Xi) is generalized for >2 to general random variables. A theorem is also proved that contains both the Erdös -Stechkin theorem and Serfling's theorem withv > 2 for dependent random variables.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 219–230, February, 1975.This article was written while the author was working in the V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR. 相似文献
8.
A. V. Sudakov 《Journal of Mathematical Sciences》2002,109(6):2219-2224
It is shown that, for the Kantorovich metric
on probability measures, the integral
9.
The Asymptotic Distributions of Sums of Records 总被引:3,自引:0,他引:3
LetX
1,X
2... be a sequence of i.i.d. random variables and let
be the associated (upper) record sequence. Resnick (1973) identified the class of all possible limit distributions for
. Here we focus on sums of records,
. We describe three cases in which T
n can be normalized to have a non-trivial limiting distribution. The problem of identifying all possible limit laws for normalized sums of records remains open. 相似文献
10.
11.
Lucien Chevalier 《Probability Theory and Related Fields》1979,49(3):249-255
Summary We prove the following extension of classical Burkholder-Davis-Gundy inequalities: let (X
n
)
nN
be a martingale; for p1, in order that
and
belong to L
p, it is sufficient that Inf(X
*, S(X)) belong to L
p. For «regular» martingales this result holds for p>0. 相似文献
12.
H. Fiedler 《Numerische Mathematik》1987,51(5):571-581
Summary Interpolatory quadrature formulae consist in replacing
by
wherep
f
denotes the interpolating polynomial off with respect to a certain knot setX. The remainder
may in many cases be written as
wherem=n resp. (n+1) forn even and odd, respectively. We determine the asymptotic behaviour of the Peano kernelP
X
(t) forn for the quadrature formulae of Filippi, Polya and Clenshaw-Curtis. 相似文献
13.
Asymptotic properties of partitions of the unit interval are studied through the entropy for random partition
14.
Helena Ferreira 《Extremes》2000,3(4):385-392
Let
be a sequence of identically distributed variables. We study the asymptotic distribution of
, where Y
[r:n] denotes the concomitant of the rth order statistic X
r:n
, corresponding to
, and
is held fixed while
. Conditions are given for the
and
to have the same asymptotic behavior as that we would apply if
were i.i.d. The result is illustrated with a simple linear regression model
, where
is a stationary sequence with extremal index
. 相似文献
15.
Joseph Rosenblatt 《Mathematische Annalen》1977,230(3):245-272
For a mean zero norm one sequence (f
n
)L
2[0, 1], the sequence (f
n
{nx+y}) is an orthonormal sequence inL
2([0, 1]2); so if
, then
converges for a.e. (x, y)[0, 1]2 and has a maximal function inL
2([0, 1]2). But for a mean zerofL
2[0, 1], it is harder to give necessary and sufficient conditions for theL
2-norm convergence or a.e. convergence of
. Ifc
n
0 and
, then this series will not converge inL
2-norm on a denseG
subset of the mean zero functions inL
2[0, 1]. Also, there are mean zerofL[0, 1] such that
never converges and there is a mean zero continuous functionf with
a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c
n
| = 0(n
–) for >1/2, then
converges a.e. and unconditionally inL
2[0, 1]. In addition, for any mean zerof of bounded variation, the series
has its maximal function in allL
p[0, 1] with 1p<. Finally, if (f
n
)L
[0, 1] is a uniformly bounded mean zero sequence, then
is a necessary and sufficient condition for
to converge for a.e.y and a.e. (x
n
)[0, 1]. Moreover, iffL
[0, 1] is mean zero and
, then for a.e. (x
n
)[0, 1],
converges for a.e.y and in allL
p
[0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one. 相似文献
16.
Kameswarrao S. Casukhela 《Journal of Theoretical Probability》1997,10(3):759-771
An infinite sequence of random variables X=(X
1, X
2,...) is said to be spreadable if all subsequences of X have the same distribution. Ryll-Nardzewski showed that X is spreadable iff it is exchangeable. This result has been generalized to various discrete parameter and higher dimensional settings. In this paper we show that a random measure on the tetrahedral space
is spreadable, iff it can be extended to an exchangeable random measure on
. The result is a continuous parameter version of a theorem by Kallenberg. 相似文献
17.
Summary Let {X
ij; i>0, j>0} be a double sequence of i.i.d. random variables taking values in the d-dimensional integer lattice E
d
. Also let
. Then the range of random walk {S
mn: m>0, n>0} up to time (m, n), denoted by R
mn
, is the cardinality of the set {S
pq: 0
18.
Lin Zhengyan 《数学学报(英文版)》1989,5(2):185-192
Consider the weighted sums
of a sequence {X
n} of independent random variables or random elements inD [0,1]. For convergence ofS
n in probability and with probability one, in [2],[3] etc., the following stronger condition is required: {X
n} is uniformly bounded by a random variableX,i.e.P(¦X
n¦x)P(¦X¦x) for allx>0. Our paper aims at trying to drop this restriction.The Project supported by National Natural Science Foundation of China 相似文献
19.
O. L. Vinogradov 《Journal of Mathematical Sciences》2003,114(5):1608-1627
Let
be the space of 2-periodic functions whose (r – 1)th-order derivative is absolutely continuous on any segment and rth-order derivative belongs to L
p, S
2n,m
is the space of 2-periodic splines of order m of minimal defect over the uniform partition
. In this paper, we construct linear operators
such that
20.
We obtain precise large deviations for heavy-tailed random sums
, of independent random variables.
are nonnegative integer-valued random variables independent of r.v. (X
i
)i
N with distribution functions F
i. We assume that the average of right tails of distribution functions F
i is equivalent to some distribution function with regularly varying tail. An example with the Pareto law as the limit function is given. 相似文献
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