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1.
This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results
on equidistant systems of points in a Hilbert space to the case of the space L
2(Ω, F, ℙ) of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular
sequence of random variables (elements of which are taken from sets of equidistant elements of L
2(Ω, F, ℙ) to be orthogonal to some other sequence in L
2(Ω, F, ℙ). The result obtained is interesting from the point of view of the time series analysis, since it can be applied to a
class of sequences random variables that exhibit a monotonically increasing variance. An application to ergodic theorem is
also provided. 相似文献
2.
We prove a rank-dependent moderate deviation principle for U-empirical measures, where the underlying i.i.d. random variables take values in a measurable (not necessarily Polish) space
(S,𝒮). The result can be formulated on a suitable subset of all signed measures on (S
m
,𝒮⊗
m
). We endow this space with a topology, which is stronger than the usual τ-topology. A moderate deviation principle for Banach-space
valued U-statistics is obtained as a particular application. The advantage of our result is that we obtain in the degenerate case
moderate deviations in non-Gaussian situations with non-convex rate functions.
Received: 22 February 2000 / Revised version: 15 November 2002 /
Published online: 28 March 2003
Research partially supported by the Swiss National Foundation, Contract No. 21-298333.90.
Mathematics Subject Classification (2000): Primary 60F10; Secondary 62G20, 28A35
Key words or phrases: Rank-dependent moderate deviations – Empirical measures – Strong topology – U-statistics 相似文献
3.
This paper compares sequences of independent, mean zero random variables in a rearrangement-invariant space X on [0, 1] with
sequences of disjoint copies of individual terms in the corresponding rearrangement-invariant space Z
X
2
on [0, ∞). The principal results of the paper show that these sequences are equivalent in X and Z
X
2
, respectively, if and only if X possesses the (so-called) Kruglov property. We also apply our technique to complement well-known
results concerning the isomorphism between rearrangement-invariant spaces on [0, 1] and [0, ∞). Bibliography: 20 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 25–50. 相似文献
4.
Aurel I. Stan 《Journal of Theoretical Probability》2011,24(1):39-65
The paper is divided into two parts. In the first part we lay down the foundation for defining the joint annihilation–preservation–creation
decomposition of a finite family of not necessarily commutative random variables, and show that this decomposition is essentially
unique. In the second part we show that any two, not necessarily commutative, random variables X and Y for which the vector space spanned by the identity and their annihilation, preservation, and creation operators equipped
with the bracket given by the commutator forms a Lie algebra are equivalent up to an invertible linear transformation to two
independent Meixner random variables with mixed preservation operators. In particular, if X and Y commute, then they are equivalent up to an invertible linear transformation to two independent classic Meixner random variables.
To show this we start with a small technical condition called “non-degeneracy”. 相似文献
5.
The object of our investigations are isotropic convex bodies , centred at the origin and normed to volume one, in arbitrary dimensions. We show that a certain subset of these bodies –
specified by bounds on the second and fourth moments – is invariant under forming ‘expanded joinsrsquo;. Considering a body
K as above as a probability space and taking , we define random variables on K. It is known that for subclasses of isotropic convex bodies satisfying a ‘concentration of mass property’, the distributions
of these random variables are close to Gaussian distributions, for high dimensions n and ‘most’ directions . We show that this ‘central limit property’, which is known to hold with respect to convergence in law, is also true with
respect to -convergence and -convergence of the corresponding densities.
Received: 21 March 2001 / in final form: 17 October 2001 / Published online: 4 April 2002 相似文献
6.
We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules.
As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined
on L
ℱ
p
(ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L
p
(ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L
ℱ
p
(ℰ(E,H) the complete random normed module generated by L
p
(ℰ, H). 相似文献
7.
Kanter (Ann Probab 3(4):697–707, 1975) and Chambers et al. (J Am Stat Assoc 71(354):340–344, 1976) developed a method for
characterizing and simulating stable random variables, X, using nonlinear transformations involving two independent uniform random variables. Their method is scrutinized to provide
a characterization and then develop a method for simulating random variables with distribution P(X ≤ x| X > a), called here truncated stable random variables. Our characterization is rigorous when the characteristic exponent α ≠ 1. We extend our method to the case that α → 1. 相似文献
8.
We establish conditions under which there exists a function c(t) > 0 such that {fx1850-01}, where X(t) is a random process from an Orlicz space of random variables. We obtain estimates for the probabilities {fx1850-02}.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1647–1660, December, 2007. 相似文献
9.
D. A. Yarotskii 《Mathematical Notes》1999,66(3):372-383
A nonhomogeneous random walk on the grid ℤ1 with transition probabilities that differ from those of a certain homogeneous random walk only at a finite number of points
is considered. Trajectories of such a walk are proved to converge to trajectories of a certain generalized diffusion process
on the line. This result is a generalization of the well-known invariance principle for the sums of independent random variables
and Brownian motion.
Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 459–472, September, 1999. 相似文献
10.
We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with
probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real
axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces L
p
(Ω) and L
2 (Ω) in the norm of the space L
q
(ℝ).
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1340–1356, October, 2006. 相似文献
11.
Inequalities of Maximum of Partial Sums and Weak Convergence for a Class of Weak Dependent Random Variables 总被引:11,自引:0,他引:11
Jiang Feng WANG Feng Bin LU 《数学学报(英文版)》2006,22(3):693-700
In this paper, we establish a Rosenthal-type inequality of the maximum of partial sums for ρ^- -mixing random fields. As its applications we get the Hájeck -Rènyi inequality and weak convergence of sums of ρ^- -mixing sequence. These results extend related results for NA sequence and p^* -mixing random fields, 相似文献
12.
Kei Takeuchi Akimichi Takemura 《Annals of the Institute of Statistical Mathematics》1987,39(1):307-324
Summary Distribution of sum of vectors of 0–1 random variables is discussed generalizing the univariate results obtained in our previous
article Takeuchi and Takemura (1987,Ann. Inst. Statist. Math.,39, 85–102). As in our previous article no assumption is made on the independence of the 0–1 random variables. 相似文献
13.
In this paper, function spaces V∩l
A
p
(w) are considered in the context of their multiplicative structure. The space V is determined by conditions on the values
of a function in a disk (for example, CA,Lip
Aα). We denote by l
A
p
(w) the space of power series such that their Taylor coefficients are p-summable with weight w. For an analytic function Φ
acting in a space of this type, we prove the following alternative: either Φ″(z)≡0, or the space is a Banach algebra with
respect to pointwise multiplication. For a wide class of weights w, we establish the continuity of the identity embeddingmult(V∩l
A
p
(w))↪multl
A
p
. An estimate for the lp-multiplicative norm of random polynomials is found. This estimate can be considered as an extension of the known result by
Salem-Zygmund. Bibliography: 10 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 50–72.
Translated by S. Shimorin. 相似文献
14.
We obtain the almost sure convergence for sequences of H-valued random variables which are either associated or negatively associated.
Our results extend the results of Birkel (Stat. Probab. Lett. 7:17–20, 1989) and Matula (Stat. Probab. Lett. 15:209–213, 1992)
to a Hilbert space.
相似文献
15.
In this paper, we study sums of linear random fields defined on the lattice Z
2 with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and
mild sufficient conditions to obtain an approximation of order n
−p
are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence
in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum
and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes. 相似文献
16.
Jean-François Le Gall 《Inventiones Mathematicae》2007,169(3):621-670
We discuss scaling limits of large bipartite planar maps. If p≥2 is a fixed integer, we consider, for every integer n≥2, a random planar map M
n
which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of M
n
, equipped with the graph distance rescaled by the factor n
-1/4, converges in distribution as n→∞ towards a limiting random compact metric space, in the sense of the Gromov–Hausdorff distance. We prove that the topology
of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence
relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the
limit is almost surely equal to 4. 相似文献
17.
V. M. Korchevsky 《Vestnik St. Petersburg University: Mathematics》2011,44(4):268-271
New sufficient conditions for the applicability of the strong law of large numbers to a sequence of dependent random variables
X
1, X
2, …, with finite variances are established. No particular type of dependence between the random variables in the sequence
is assumed. The statement of the theorem involves the classical condition Σ
n
∞ (log2
n)2/n
2 < ∞, which appears in various theorems on the strong law of large numbers for sequences of random variables without the independence
condition. 相似文献
18.
19.
The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as
infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random
tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree.
The continuum random tree is a scaling limit as N→∞ of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous–Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of
all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling
limit as N→∞ of the Aldous–Broder chain. A key technical ingredient in this work is the use of a pointed Gromov–Hausdorff distance to
metrize the space of rooted compact real trees.
Berkeley Statistics Technical Report No. 654 (February 2004), revised October 2004. To appear in Probability Theory and Related Fields.
SNE supported in part by NSF grants DMS-0071468 and DMS-0405778, and a Miller Institute for Basic Research in Science research
professorship
JP supported in part by NSF grants DMS-0071448 and DMS-0405779
AW supported by a DFG Forchungsstipendium 相似文献
20.
A. K. Aleškevičienė 《Lithuanian Mathematical Journal》2006,46(2):129-145
Let X,X
1,X
2, … be independent identically distributed random variables, F(x) = P{X < x}, S
0 = 0, and S
n
=Σ
i=1
n
X
i
. We consider the random variables, ladder heights Z
+ and Z
− that are respectively the first positive sum and the first negative sum in the random walk {S
n
}, n = 0, 1, 2, …. We calculate the first three (four in the case EX = 0) moments of random variables Z
+ and Z
− in the qualitatively different cases EX > 0, EX < 0, and EX = 0.
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Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 159–179, April–June, 2006. 相似文献