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1.
Limit theorems for the ratio of the empirical distribution function to the true distribution function 总被引:4,自引:0,他引:4
Jon A. Wellner 《Probability Theory and Related Fields》1978,45(1):73-88
Summary We consider almost sure limit theorems for
and
where
n
is the empirical distribution function of a random sample ofn uniform (0, 1) random variables anda
n
0. It is shown that (1) ifna
n
/log2
n then both
and
converge to 1 a.s.; (2) ifna
n
/log2
n=d>0 (d>1) then
has an almost surely finite limit superior which is the solution of a certain transcendental equation; and (3) ifna
n
/log2
n0 then
and
have limit superior + almost surely. Similar results are established for the inverse function
n
–1
.Supported by the National Science Foundation under MCS 77-02255 相似文献
2.
Shiquan WU 《应用数学学报(英文版)》1996,12(4):377-383
Letn, s
1,s
2, ... ands
n
be positive integers. Assume
is an integer for eachi}. For
,
, and
, denotes
p
(a)={j|1jn,a
j
p},
, and
.
is called anI
t
p
-intersecting family if, for any a,b
,a
i
b
i
=min(a
i
,b
i
)p for at leastt i's.
is called a greedyI
t
P
-intersecting family if
is anI
t
p
-intersecting family andW
p
(A)W
p
(B+A
c
) for anyAS
p
(
) and any
with |B|=t–1.In this paper, we obtain a sharp upper bound of |
| for greedyI
t
p
-intersecting families in
for the case 2ps
i
(1in) ands
1>s
2>...>s
n
.This project is partially supported by the National Natural Science Foundation of China (No.19401008) and by Postdoctoral Science Foundation of China. 相似文献
3.
We find a regular deformation retraction
n,r
(K): Idem
n,r
(K) G
n,r
(K) from the manifold Idem
n,r
(K) of idempotent n × n matrices with rank r to the Grassmannian manifold G
n,r
(K) over K the reals, complex numbers or quaternions. Then we derive an injection
from the sets of homotopy classes of complex-valued polynomial to such a set of real-valued regular maps, where
denotes the Zariski closure in the affine space
n
of a subset
n
. Furthermore, we list complex-valued polynomial maps
2
2 of any Brouwer degree and deduce that the map ()2,1: Idem()2,1 G()2,1 yields an isomorphism
[
2
]
[
2,
2] of cyclic infinite homotopy groups. Finally, we show that every nonzero even Brouwer degree of the spheres
n
and
n
cannot be realized by a real-valued (resp. complex-valued) homogeneous polynomial map provided that n is even. 相似文献
4.
Dražen Adamović 《Algebras and Representation Theory》2004,7(4):457-469
Let
be the affine Lie algebra associated to the simple finite-dimensional Lie algebra
. We consider the tensor product of the loop
-module
associated to the irreducible finite-dimensional
-module V() and the irreducible highest weight
-module L
k,. Then L
k, can be viewed as an irreducible module for the vertex operator algebra M
k,0. Let A(L
k,) be the corresponding
-bimodule. We prove that if the
-module
is zero, then the
-module
is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras. 相似文献
5.
We introduce the notion ofweak subnormality, which generalizes subnormality in the sense that for the extension
ofT
we only require that
hold forf
; in this case we call
a partially normal extension ofT. After establishing some basic results about weak subnormality (including those dealing with the notion of minimal partially normal extension), we proceed to characterize weak subnormality for weighted shifts and to prove that 2-hyponormal weighted shifts are weakly subnormal. Let {
n
}
n=0
be a weight sequence and letW
denote the associated unilateral weighted shift on
. IfW
is 2-hyponormal thenW
is weakly subnormal. Moreover, there exists a partially normal extension
on
such that (i)
is hyponormal; (ii)
; and (iii)
. In particular, if is strictly increasing then
can be obtained as
whereW
is a weighted shift whose weight sequence {
n
·
n=0
is given by
In this case,
is a minimal partially normal extension ofW
. In addition, ifW
is 3-hyponormal then
can be chosen to be weakly subnormal. This allows us to shed new light on Stampfli's geometric construction of the minimal normal extension of a subnormal weighted shift. Our methods also yield two additional results: (i) the square of a weakly subnormal operator whose minimal partially normal extension is always hyponormal, and (ii) a 2-hyponormal operator with rank-one self-commutator is necessarily subnormal. Finally, we investigate the connections of weak subnormality and 2-hyponormality with Agler's model theory.Supported by NSF research grant DMS-9800931.Supported by the Brain Korea 21 Project from the Korean Ministry of Education. 相似文献
6.
Let be a positive number, and letE
n,n
(x
;[0,1]) denote the error of best uniform rational approximation from
n,n
tox
on the interval [0,1]. We rigorously determined the numbers {E
n,n
(x
;[0,1])}
n
=1/30
for six values of in the interval (0, 1), where these numbers were calculated with a precision of at least 200 significant digits. For each of these six values of , Richardson's extrapolation was applied to the products
to obtain estimates of
相似文献
7.
Summary.
Let
We say that
preserves the distance d 0 if
for each
implies
Let A
n
denote the set of all positive numbers
d such that any map
that preserves unit distance preserves also distance
d.
Let D
n
denote the set of all positive numbers
d with the property: if
and
then there exists a finite set
S
xy
with
such that any map
that preserves unit distance preserves also the distance between
x and y.
Obviously,
We prove:
(1)
(2)
for n 2
D
n
is a
dense subset of
(2) implies that each mapping
f
from
to
(n 2)
preserving unit distance preserves all distances,
if f is continuous with respect to the product topologies
on
and
相似文献
8.
Let (X
t
) be a one dimensional diffusion corresponding to the operator
, starting from x>0 and T
0 be the hitting time of 0. Consider the family of positive solutions of the equation
with (0, ), where
. We show that the distribution of the h-process induced by any such is
, for a suitable sequence of stopping times (S
M
: M0) related to which converges to with M. We also give analytical conditions for
, where
is the smallest point of increase of the spectral measure associated to
. 相似文献
9.
Niels Jakob Laustsen 《K-Theory》2001,22(3):241-249
Let
Figiel's reflexive Banach space which is not isomorphic to its Cartesian square. We show that the K
0group of the algebra
of continuous, linear operators on
contain a subgroup isomorphic to the group c
00(
) of sequences
rational numbers with z
n=0 eventually. 相似文献
10.
A. I. Budkin 《Algebra and Logic》2000,39(6):363-369
Let
be a class of all groups G for which the normal closure (x)
G
of every element x belongs to a class
.
is a Levi class generated by
. Let
and
0 be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that
and
, and so
and
. It is shown that quasivarieties
and
are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that
is closed under free products if so is
. 相似文献
11.
Alain Rivière 《Geometriae Dedicata》2001,85(1-3):217-235
For a separable Hilbert space E whose dimension is 2 and for an open subset of E, not empty and different from E, let
be the set of all points of which have at least two projections on the close set E\, and let
be the set of all the centres of the open balls contained in and which are maximal for inclusion. We show that the Hausdorff dimension dimH(
) of
may be any real value s such that 0sdim E; we also show that can be chosen so that
is everywhere dense in and so that we have dimH(
)=1.Associons à un ouvert d'un espace de Hilbert séparable E de dimension 2, non vide et distinct de E, l'ensemble
des points de admettant plusieurs projections sur le fermé E\, et l'ensemble
des centres des boules ouvertes inclues dans et maximales pour l'inclusion. Nous montrons d'une part que la dimension de Hausdorff dimH(
) de
peut prendre toute valeur réelle s telle que 0sdim E, et d'autre part qu'on peut choisir de sorte que
soit dense dans et qu'on ait dimH(
)=1. 相似文献
12.
The one-step prediction problem is studied in the context ofP
n-weakly stationary stochastic processes
, where
is an orthogonal polynomial sequence defining a polynomial hypergroup on
. This kind of stochastic processes appears when estimating the mean of classical weakly stationary processes. In particular, it is investigated whether these processes are asymptoticP
n-deterministic, i.e. the prediction mean-squared error tends to zero. Sufficient conditions on the covariance function or the spectral measure are given for
being asymptoticP
n-deterministic. For Jacobi polynomialsP
n(x) the problem of
being asymptoticP
n-deterministic is completely solved. 相似文献
13.
We consider a random instance I of k-SAT with n variables and m clauses, where k=k(n) satisfies k—log2
n. Let m
0=2
k
nln2 and let =(n)>0 be such that n. We prove that
14.
Summary Let
denote the extended Weyl algebra,
, the Weyl algebra. It is well known that every element of
of the formA=B
k
*
B
k
is positive. We prove that the converse implication also holds: Every positive elementA in
has a quadratic sum factorization for some finite set of elements (B
k
) in
. The corresponding result is not true for the subalgebra
. We identify states on
which do not extend to states on
. It follows from a result of Powers (and Arveson) that such states on
cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on
, and this property is not in general shared by representations generated by states defined only on the subalgebra
.Work supported in part by the NSF 相似文献
15.
16.
Let S = x
1,...,x
n} be a finite subset of a partially ordered set P. Let f be an incidence function of P. Let
denote the n × n matrix having f evaluated at the meet
of x
i and x
j as its i, j-entry and
denote the n × n matrix having f evaluated at the join
of x
i and x
j as its i, j-entry. The set S is said to be meet-closed if
for all 1 i, j n. In this paper we get explicit combinatorial formulas for the determinants of matrices
and
on any meet-closed set S. We also obtain necessary and sufficient conditions for the matrices
and
on any meet-closed set S to be nonsingular. Finally, we give some number-theoretic applications. 相似文献
17.
Suppose that
(j) is the lag-j autocorrelation of the squared residuals computed from a realization of length n under the assumption that the observations follow a GARCH(1,1) model. We study the asymptotic distribution of the statistics of the form
, where the j are nonnegative summable weights and the matrix
, can be estimated from the data. We show that, under weak assumptions on model errors, the statistic Q
n converges in distribution to
, where the N
i are iid standard normal. We discuss choices of the weights j for which the distribution of Q is tabulated. Our results lead to and provide a rigorous justification for Portmanteau goodness-of-fit tests for GARCH(1,1) specification. 相似文献
18.
Hansmartin Zeuner 《Journal of Theoretical Probability》1994,7(2):225-245
Let (X
n:n) be i.i.d. with finite variance and values in a hypergroupK:=+ or and
j=1
n
X
j
be the randomized sum of these random variables. It is shown that the processes
converge in distribution to a Gaussian process in the caseK=+, that the processes
converge towards a Bessel process on + in the case of polynomial growth of the hypergroupK=+ or , and that in the case of exponential growth
converges towards a Brownian motion asn. 相似文献
19.
The boundedness below of 2×2 upper triangular operator matrices 总被引:2,自引:0,他引:2
Wen
and
are given we denote byM
C
an operator acting on the Hilbert space
of the form
20.
Let X,
,X
1,...,X
n
be i.i.d. random variables taking values in a measurable space (
). Consider U-statistics of degree two
|