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1.
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
2.
R. J. Tomkins 《Journal of Theoretical Probability》1996,9(4):841-851
Forr1 and eachnr, letM
nr
be therth largest ofX
1,X
2, ...,X
n
, where {X
n
,n1} is an i.i.d. sequence. Necessary and sufficient conditions are presented for the convergence of
for all >0 and some –1, where {a
n
} is a real sequence. Furthermore, it is shown that this series converges for all >–1, allr1 and all >0 if it converges for some >–1, somer1 and all >0. 相似文献
3.
V. A. Egorov 《Journal of Mathematical Sciences》1990,52(2):2878-2883
Let {Xn}
n=1
be a sequence of independent, symmetric random variables and let {Xin}
i=1
n
be the absolute order statistics. The rate of growth of
and X2,n is investigated for n.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 25–31, 1988. 相似文献
4.
5.
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
6.
Péter Major 《Probability Theory and Related Fields》1988,77(1):117-128
Summary In this paper we prove the following statement. Given a random walk
,n=1, 2, ... where
1,
2 ... are i.i.d. random variables,
let (n) denote the number of points visited exactly once by this random walk up to timen. We show that there exists some constantC, 0 <C < , such that
with probability 1. The proof applies some arguments analogous to the techniques of the large deviation theory.Research supported by the Hungarian National Foundation for Scientific Research, Grant No # 819/1 相似文献
7.
Gerold Wagner 《Monatshefte für Mathematik》1981,92(3):239-245
Letf be a periodic function on with period 1, piecewise continuously differentiable, satisfying
. For an arbitrary sequence = (
i
) in [0,1) put
and
. If
then
n
(f,) >c· logn holds for some positive constantc (depending onf only) and almost alln. In a certain sense the converse is also true: there is a class of functionsf with
such that
n
(f,) =o (logn).Support has been received from Netherlands Organization for the Advancement of Pure Research (Z. W. O.). 相似文献
8.
Jonathan Eckstein Peter L. Hammer Ying Liu Mikhail Nediak Bruno Simeone 《Computational Optimization and Applications》2002,23(3):285-298
Given two finite sets of points X
+ and X
– in
n
, the maximum box problem consists of finding an interval (box) B = {x : l x u} such that B X
– = , and the cardinality of B X
+ is maximized. A simple generalization can be obtained by instead maximizing a weighted sum of the elements of B X
+. While polynomial for any fixed n, the maximum box problem is
-hard in general. We construct an efficient branch-and-bound algorithm for this problem and apply it to a standard problem in data analysis. We test this method on nine data sets, seven of which are drawn from the UCI standard machine learning repository. 相似文献
9.
Let X,X
n
;n1 be a sequence of real-valued i.i.d. random variables with E(X)=0. Assume B(u) is positive, strictly increasing and regularly-varying at infinity with index 1/2<1. Set b
n
=B(n),n1. If
and
for some [0,), then it is shown that
and
for every real triangular array (a
n,k
;1kn,n1) and every array of bounded real-valued i.i.d. random variables W,W
n,k
;1kn,n1`` independent of {X,X
n
;n1}, where (W)=(E(W–E(W))2)1/2. An analogous law of the iterated logarithm for the unweighted sums
n
k=1
X
k
;n1} is also given, along with some illustrative examples. 相似文献
10.
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
11.
Oleg T. Izhboldin 《K-Theory》2001,22(3):199-229
Let F be a field of characteristic different from 2 and be a quadratic form over F. Let X be an arbitrary projective homogeneous generic splitting variety of . For example, we can take X to be equal to the variety X,m of totally isotropic m-dimensional subspaces of V, where V is the quadratic space corresponding to and <
dim V. In this paper, we study the groups CH2(X) and H3(F(X)/F) = ker(H
3(F) H
3(F(X))). One of the main results of this paper claims that the group Tors CH2(X) is always zero or isomorphic to
. In many cases we prove that Tors CH2(X) = 0 and compute the group H
3(F(X)/F) completely. As an application of the main results, we give a criterion of motivic equivalence of eight-dimensional forms except for the case where the Schur indices of their Clifford algebras equal 4. 相似文献
12.
Nous montrons que toute fonction séparément finement surharmonique sur un ouvert de la topologie produit
n_1×s×
n_k des topologies fines des espaces R
n
1,. . ., R
n
k,
n_1×s×
n_k-localement bornée inférieurement est finement surharmonique dans . On en déduit que toute fonction séparément finement harmonique,
n_1×s×
n_k-localement bornée sur est finement harmonique dans .Separately Finely Superharmonic Functions
Abstract.We prove that every separately finely surperharmonic function on an open set in R
n
1×s×R
n
k for the product
n_1×s×
n_k of the fine topologies on the spaces R
n
1,. . ., R
n
k,
n_1×s×
n-klocally lower bounded, is finely superharmonic in . We then deduce that every separateltly finely harmonic function
n_1×s×
n
k-locally bounded in is finely harmonic. 相似文献
13.
A. Spătaru 《Journal of Theoretical Probability》2004,17(4):943-965
Let
be i.i.d. random variables, and set S
n
=
k
n
X
k
. We exhibit a method able to provide exact loglog rates. The typical result is that
whenever EX=0,EX
2=2 and E[X
2(log+ | X |)
r-1] < . To get this and other related precise asymptotics, we derive some general estimates concerning the Dirichlet divisor problem, of interest in their own right. 相似文献
14.
L. V. Rozovskii 《Journal of Mathematical Sciences》1984,27(6):3279-3288
For independent, identically distributed vectors X and Borel sets An, one investigates the accuracy of the approximation of the probability Pn(An)= P{n1/2(X1+...+Xn)An} by the distribution. One obtains criteria in order to have the relations, uniformly with respect to all sequences {An} such that,
\bar \Lambda \left. {\left. {\left( {\sqrt n } \right)} \right\}} \right)$$
" align="middle" border="0">
, where
are functions satisfying certain conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 157–166, 1983. 相似文献
15.
Abdelhafed Elkhadiri 《Bulletin of the Brazilian Mathematical Society》2000,31(1):45-71
Let be an open subset of
n
and
be a subalgebra of the algebra of analytic functions on . We suppose that
satisfies some weak conditions of noetherianity such that we can construct a finite stratification for each ideal of
. We also suppose that
satifies global £ojasiewicz's inequalities. We prove the following: Let
andf C
on
flat on ; if for eacha the Taylor's serie off ata, T
a
f, is in the ideal generated byT
a
f
1,...,T
a
f
p
in the ring of formal power series, then there exist
1,...,
p
,C
on
flat on such that
. This result extends the classic Hormander's theorem of division (for a polynomial) or the £ojasiewicz-Malgrange theorem in the local analytic case.Reherches menées dans le cadre du Programme d'Appui à la Recherche Scientifique (PARS MI 33) 相似文献
16.
F. H. Simons 《Probability Theory and Related Fields》1970,15(3):177-179
Summary In [1], an example was given of a measure-preserving dissipative transformation T in a -finite measure space (X, , ), such that T is conservative in the measure space (X, , ) where
. Here we shall show that for this transformation we actually have R
={ØX}[]. 相似文献
17.
On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results 总被引:1,自引:0,他引:1
Gerold Alsmeyer 《Journal of Theoretical Probability》2003,16(1):217-247
A result by Elton(6) states that an iterated function system
of i.i.d. random Lipschitz maps F
1,F
2,... on a locally compact, complete separable metric space
converges weakly to its unique stationary distribution if the pertinent Liapunov exponent is a.s. negative and
for some
. Diaconis and Freedman(5) showed the convergence rate be geometric in the Prokhorov metric if
for some p>0, where L
1 denotes the Lipschitz constant of F
1. The same and also polynomial rates have been recently obtained in Alsmeyer and Fuh(1) by different methods. In this article, necessary and sufficient conditions are given for the positive Harris recurrence of (M
n
)
n0 on some absorbing subset
. If
and the support of has nonempty interior, we further show that the same respective moment conditions ensuring the weak convergence rate results mentioned above now lead to polynomial, respectively geometric rate results for the convergence to in total variation or f-norm
f
, f(x)=1+d(x,x
0)
for some (0,p]. The results are applied to various examples that have been discussed in the literature, including the Beta walk, multivariate ARMA models and matrix recursions. 相似文献
18.
Ishay Weissman 《Probability Theory and Related Fields》1976,37(1):35-41
Let be an Euclidean space; Y
n
, Z, U random vectors in ; h
n
, g
n
affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that gn
and
where Z is nonsingular. The behaviour of
n
= h
n
g
n
–1
as n is discussed first. The results are used then to prove that if
for all t(0, ), where h
n
þ and Z
1 is nonsingular and nonsymmetric with respect to þ then
H,
for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown. 相似文献
19.
Let X
1, ..., Xn be an i.i.d. sequence of random variables, from an unknown distribution F, and X
1
W
, ... X
n
W
be a sample from
, the weighted empirical distribution function of X
1, ..., Xn. We define the order statistics X
1,n
W
... X
n,n
W
of X
1
W
, ..., X
n
W
. Under suitable assumptions on weights, we study the influence of the maxima in the construction of limit theorems. We choose a resample size m(n) and we derive conditions on m(n) for the in probability and with probability 1 consistency of X
m(n),m(n)
W
. The presence of weights has an influence on the resample size and requires the use of new tools. When X
n,n is in the domain of attraction of an extreme value distribution, m(n) , and
, as n , all our results hold. 相似文献
20.
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