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1.
M. V. Myronyuk 《Ukrainian Mathematical Journal》2008,60(9):1437-1447
According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables
with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak
for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators.
In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another
characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms,
it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second
part of the paper, we prove an analog of the Heyde theorem in a Banach space.
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1234–1242, September, 2008. 相似文献
2.
The well-known Skitovich-Darmois theorem asserts that a Gaussian distribution is characterized by the independence of two
linear forms of independent random variables. The similar result was proved by Heyde, where instead of the independence, the
symmetry of the conditional distribution of one linear form given another was considered. In this article we prove that the
Heyde theorem on a locally compact Abelian group X remains true if and only if X contains no elements of order two. We describe also all distributions on the two-dimensional torus which are characterized by the symmetry of the conditional distribution of one linear form given another. In so doing we
assume that the coefficients of the forms are topological automorphisms of X and the characteristic functions of the considering random variables do not vanish. 相似文献
3.
According to the well-known Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups X containing an element of order 2. We prove that if X contains an element of order 2, this leads to the fact that a wide class of non-Gaussian distributions on X is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. While coefficients of linear forms are topological automorphisms of a group.
相似文献4.
G. M. Feldman 《Doklady Mathematics》2013,87(2):198-201
We prove some analogues of the well-known Skitovich-Darmois and Heyde characterization theorems for a second countable locally compact Abelian group X under the assumption that the distributions of the random variables have continuous positive densities with respect to a Haar measure on X and the coefficients in the linear forms considered are topological automorphisms of the group X. 相似文献
5.
G. M. Feldman 《Doklady Mathematics》2017,95(2):147-150
According to the well-known Heyde theorem Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in n independent random variables given another. For n = 2 we prove analogs of this theorem in the case when random variables take values in a locally compact Abelian group X, and coefficients of the linear forms are topological automorphisms of the group X. 相似文献
6.
7.
Gennadiy Feldman 《Mathematische Nachrichten》2013,286(4):340-348
We prove some analogues of the well‐known Skitovich–Darmois and Heyde characterization theorems for a second countable locally compact Abelian group X under the assumption that the distributions of the random variables have continuous positive densities with respect to a Haar measure on X and the coefficients in the linear forms considered are topological automorphisms of X. 相似文献
8.
M. V. Myronyuk 《Ukrainian Mathematical Journal》2004,56(10):1602-1618
We prove theorems that generalize the Skitovich-Darmois theorem to the case where independent random variables ξj, j = 1, 2, ..., n, n ≥ 2, take values in a locally compact Abelian group and the coefficients αj and βj of the linear forms L
1 = α1ξ1 + ... + αnξn and L
2 = β1ξ1 + ... + βnξn are automorphisms of this group.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1342 – 1356, October, 2004. 相似文献
9.
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
10.
Let X be a locally compact Abelian group, \(\alpha _{j}, \beta _j\) be topological automorphisms of X. Let \(\xi _1, \xi _2\) be independent random variables with values in X and distributions \(\mu _j\) with non-vanishing characteristic functions. It is known that if X contains no subgroup topologically isomorphic to the circle group \(\mathbb {T}\), then the independence of the linear forms \(L_1=\alpha _1\xi _1+\alpha _2\xi _2\) and \(L_2=\beta _1\xi _1+\beta _2\xi _2\) implies that \(\mu _j\) are Gaussian distributions. We prove that if X contains no subgroup topologically isomorphic to \(\mathbb {T}^2\), then the independence of \(L_1\) and \(L_2\) implies that \(\mu _j\) are either Gaussian distributions or convolutions of Gaussian distributions and signed measures supported in a subgroup of X generated by an element of order 2. The proof is based on solving the Skitovich–Darmois functional equation on some locally compact Abelian groups. 相似文献
11.
We introduce a notion of derived Azumaya algebras over ring and schemes generalizing the notion of Azumaya algebras of Grothendieck (Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses. Dix Exposés sur la Cohomologie des Schémas, pp. 46–66, North-Holland, Amsterdam, 1968). We prove that any such algebra B on a scheme X provides a class ϕ(B) in . We prove that for X a quasi-compact and quasi-separated scheme ϕ defines a bijective correspondence, and in particular that any class in , torsion or not, can be represented by a derived Azumaya algebra on X. Our result is a consequence of a more general theorem about the existence of compact generators in twisted derived categories, with coefficients in any local system of reasonable dg-categories, generalizing the well known existence of compact generators in derived categories of quasi-coherent sheaves of Bondal and Van Den Bergh (Mosc. Math. J. 3(1):1–36, 2003). A huge part of this paper concerns the treatment of twisted derived categories, as well as the proof that the existence of compact generator locally for the fppf topology implies the existence of a global compact generator. We present explicit examples of derived Azumaya algebras that are not represented by classical Azumaya algebras, as well as applications of our main result to the localization for twisted algebraic K-theory and to the stability of saturated dg-categories by direct push-forwards along smooth and proper maps. 相似文献
12.
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
13.
Povilas Banys 《Lithuanian Mathematical Journal》2011,51(3):303-309
In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations
were proved. In this paper, we present the central limit theorem for linear random fields with martingale-differences innovations
satisfying the central limit theorem from [J. Dedecker, A central limit theorem for stationary random fields, Probab. Theory Relat. Fields, 110(3):397–426, 1998] and arranged in lexicographical order. 相似文献
14.
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
15.
Alexander R. Pruss 《Probability Theory and Related Fields》1998,111(3):323-332
Summary. A sequence of random variables X
1,X
2,X
3,… is said to be N-tuplewise independent if X
i
1,X
i
2,…,X
i
N
are independent whenever (i
1,i
2,…,i
N
) is an N-tuple of distinct positive integers. For any fixed N∈ℤ+, we construct a sequence of bounded identically distributed N-tuplewise independent random variables which fail to satisfy the central limit theorem.
Received: 17 May 1996 / In revised form: 28 January 1998 相似文献
16.
William J. McGraw 《Mathematische Annalen》2003,326(1):105-122
In a recent paper [Duke Math. J., 97, 219–233], Borcherds asks whether or not the spaces of vector valued modular forms associated to the Weil representation
have bases of modular forms whose Fourier expansions have only integer coefficients. We give an affirmative answer to Borcherds'
question. This strengthens and simplifies Borcherds' main theorem which is a generalization of a theorem of Gross, Kohnen,
and Zagier [Math. Ann., 278, 497–562].
Received: 27 September 2001 / Revised version: 22 July 2002 /
Published online: 28 March 2003
Mathematics Subject Classification (1991): 11F30; 11F27 相似文献
17.
TieXin Guo 《中国科学A辑(英文版)》2008,51(9):1651-1663
Let (Ω,A,μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S,X) a random normed space over K with base (ω,A,μ). Denote the support of (S,X) by E, namely E is the essential supremum of the set {A ∈ A: there exists an element p in S such that X
p
(ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit
ball S
*(1) = {f ∈ S
*: X
*
f
⩽ 1} of the random conjugate space (S
*,X
*) of (S,X) is compact under the random weak star topology on (S
*,X
*) iff E∩A=: {E∩A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {A
n
: n ∈ N} of at most countably many μ-atoms from E ∩ A such that E = ∪
n=1∞
A
n
and for each element F in E ∩ A, there is an H in the σ-algebra generated by {A
n
: n ∈ N} satisfying μ(FΔH) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding
classical case. Further, Banach-Bourbaki-Kakutani-Šmulian (briefly, BBKS) theorem in a complete random normed module is established
as follows: If (S,X) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S: X
p
⩽ 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive and E ∩ A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary
complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary
almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that
the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they
possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James
theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous
classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent
in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another
in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely
simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of
random metric theory. 相似文献
18.
We prove limit theorems for row sums of a rowwise independent infinitesimal array of random variables with values in a locally compact Abelian group. First we give a proof of Gaiser's theorem [4, Satz 1.3.6], since it does not have an easy access and it is not complete. This theorem gives sufficient conditions for convergence of the row sums, but the limit measure cannot have a nondegenerate idempotent factor. Then we prove necessary and sufficient conditions for convergence of the row sums, where the limit measure can be also a nondegenerate Haar measure on a compact subgroup. Finally, we investigate special cases: the torus group, the group of p ‐adic integers and the p ‐adic solenoid. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
19.
Hong Jun LI 《数学学报(英文版)》2006,22(4):971-988
Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo's parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction. 相似文献
20.
Osamu Hiwatashi Masaru Nagisa Hiroaki Yoshida 《Probability Theory and Related Fields》1999,113(1):115-133
In usual probability theory, various characterizations of the Gaussian law have been obtained. For instance, independence
of the sample mean and the sample variance of independently identically distributed random variables characterizes the Gaussian
law and the property of remaining independent under rotations characterizes the Gaussian random variables. In this paper,
we consider the free analogue of such a kind of characterizations replacing independence by freeness. We show that freeness
of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables,
which covers the case for the sample mean and the sample variance, characterizes the semicircle law. Moreover we give the
alternative proof for Nica's result that the property of remaining free under rotations characterizes a semicircular system.
Our proof is more direct and straightforward one.
Received: 12 February 1997 / Revised version: 16 June 1998 相似文献