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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.
N. Guillotin-Plantard 《Journal of Theoretical Probability》2001,14(1):241-260
In this paper, we study a
d
-random walk
on nearest neighbours with transition probabilities generated by a dynamical system
. We prove, at first, that under some hypotheses,
verifies a local limit theorem. Then, we study these walks in a random scenery
, a sequence of independent, identically distributed and centred random variables and show that for certain dynamic random walks,
satisfies a strong law of large numbers. 相似文献
3.
Let u(x) xR
q
be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p
x, (·) be the density of an R
q
valued canonical normal random variable with mean x and variance and let {G
x, ; (x, )R
q
×[0,1 ]} be the mean zero Gaussian process with covariance
A finite positive measure on R
q
is said to be in
with respect to u, if
When
, a multiple Wick product chaos
is defined to be the limit in L
2, as 0, of
where
,
denotes the Wick product of the m
j
normal random variables
.Consider also the associated decoupled chaos processes
,
defined as the limit in L
2, as 0, of
where
are independent copies of G
x,.Define
Note that a neighborhood of the diagonals of
in
is excluded, except those points on the diagonal which originate in the same Wick product in (i). Set
One of the main results of this paper is:
Theorem A. If
is continuous on (R
q
)
r
for all
then
is continuous on
.When u satisfies some regularity conditions simple sufficient conditions are obtained for the continuity of
on (R
q
)
r
. Also several variants of (i) are considered and related to different types of decoupled processes. These results have applications in the study of intersections of Lévy process and continuous additive functionals of several Lévy processes. 相似文献
4.
Let
be sequences of real numbers which are symmetric in k. Let
be independent sequences of independent normal random variables with mean zero and variance one. For each fixed choice of
we consider
Let
Several examples are given in which the condition
is either a sufficient, a necessary, or a necessary and sufficient condition for {Q(x), x[0, 2]
n
} to have a continuous version. 相似文献
5.
We demonstrate how a well studied combinatorial optimizationproblem may be used as a new cryptographic primitive. The problemin question is that of finding a "large" clique in a randomgraph. While the largest clique in a random graph with nvertices and edge probability p is very likely tobe of size about
, it is widely conjecturedthat no polynomial-time algorithm exists which finds a cliqueof size
with significantprobability for any constant > 0. We presenta very simple method of exploiting this conjecture by hidinglarge cliques in random graphs. In particular, we show that ifthe conjecture is true, then when a large clique—of size,say,
is randomlyinserted (hidden) in a random graph, finding a clique ofsize
remains hard.Our analysis also covers the case of high edge probabilitieswhich allows us to insert cliques of size up to
. Our result suggests several cryptographicapplications, such as a simple one-way function. 相似文献
6.
For an array {V
nk
,k1,n1} of rowwise independent random elements in a real separable Banach space
with almost surely convergent row sums
, we provide criteria for S
n
–A
n
to be stochastically bounded or for the weak law of large numbers
to hold where {A
n
,n1} is a (nonrandom) sequence in
. 相似文献
7.
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
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.
Let
be a continuous semimartingale and let
be a continuous function of bounded variation. Setting
and
suppose that a continuous function
is given such that F is C1,2 on
and F is
on
. Then the following change-of-variable formula holds:
where
is the local time of X at the curve b given by
and
refers to the integration with respect to
. A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping. 相似文献
10.
Florence Merlevède Magda Peligrad Sergey Utev 《Journal of Theoretical Probability》1997,10(3):681-693
In this paper we study the behavior of sums of a linear process
associated to a strictly stationary sequence
with values in a real separable Hilbert space and
are linear operators from H to H. One of the results is that
satisfies the CLT provided
are i.i.d. centered having finite second moments and
. We shall provide an example which shows that the condition on the operators is essentially sharp. Extensions of this result are given for sequences of weak dependent random variables
under minimal conditions. 相似文献
11.
Let * be the convolution on M(
+) associated with a second order singular differential operator L on ]0, +[. If is a probability measure on
+ with suitable moment conditions, we study how to normalize the measures *
n
; n
} (resp.
) in order to get vague convergence if n+ (resp. x+). The results depend on the asymptotic drift of the operator L and on a precise study of the asymptotic behaviour of its eigenfunctions. 相似文献
12.
13.
A compound Poisson process is of the form
where Z, Z
1, Z
2, are arbitrary i.i.d. random variables and N
is an independent Poisson random variable with parameter . This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate
. The truncation level introduced depends only on and Z and not on the overall exceedance level a. 相似文献
14.
Mohan Namasivayam 《Annali di Matematica Pura ed Applicata》1986,144(1):157-172
Summary
Integral representations of junctions in the anisotropic Sobolev spaces
on unbounded domains are used in the study of the embeddings of these spaces into Lebesgue spaces. Estimates of entropy numbers of the embedding
are obtained, where k, p and q satisfy certain conditions and where is a certain type of quasibounded domain, 相似文献
15.
It is well known that (i) for every irrational number the Kronecker
sequence m (m = 1,...,M) is equidistributed modulo one in the
limit
, and (ii) closed horocycles of length
become equidistributed
in the unit tangent bundle
of a hyperbolic surface
of finite area, as
. In the present paper both equidistribution
problems are studied simultaneously: we prove that for any constant
the Kronecker sequence embedded in
along a long closed
horocycle becomes equidistributed in
for almost all , provided
that
. This equidistribution result holds in fact under
explicit diophantine conditions on (e.g. for = 2) provided that
,
with additional assumptions on the Fourier coefficients
of certain automorphic forms. Finally, we show that for
, our
equidistribution theorem implies a recent result of Rudnick and Sarnak
on the uniformity of the pair correlation density of the sequence
n2 modulo one. 相似文献
16.
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
17.
A. A. Dosiev 《Functional Analysis and Its Applications》2003,37(1):61-64
This note deals with homological characteristics of algebras of holomorphic functions of noncommuting variables generated by a finite-dimensional nilpotent Lie algebra
. It is proved that the embedding
of the universal enveloping algebra
of
into its Arens–Michael hull
is an absolute localization in the sense of Taylor provided that
相似文献
18.
Peter Hellekalek 《Monatshefte für Mathematik》1980,90(4):291-295
Ifp2 is an integer, then every nonnegative integerk is represented by an expression of the form
with integersa
i
(k), 0a
i
(k)p–1,i=0.1,...,s. The radical-inverse function to the basep,
p
(k), is defined by
. The sequence
is uniformly distributed modulo 1 (it may be called a one-dimensional Halton sequence). In the casep=2 it is the van der Corput sequence. The set of all numbers (0, 1] such that the local discrepancy
is bounded inn is determined. 相似文献
19.
Matthew Harris 《Journal of Theoretical Probability》2001,14(2):299-317
We consider a Poisson point process on
with intensity , and at each Poisson point we place a two sided mirror of random length and orientation. The length and orientation of a mirror is taken from a fixed distribution, and is independent of the lengths and orientations of the other mirrors. We ask if light shone from the origin will remain in a bounded region. We find that there exists a
with 0 <
< for which, if
<
, light leaving the origin in all but a countable number of directions will travel arbitrariliy far from the origin with positive probability. Also, if
>
, light from the origin will almost surely remain in a bounded region. 相似文献
20.
Let {S
n}
n0 be a random walk on the line. We give criteria for the existence of a nonrandom sequence n
i for which
respectively
We thereby obtain conditions for to be a strong limit point of {S
n} or {S
n
/n}. The first of these properties is shown to be equivalent to
for some sequence a
i , where T(a) is the exit time from the interval [–a,a]. We also obtain a general equivalence between
and
for an increasing function fand suitable sequences n
i and a
i. These sorts of properties are of interest in sequential analysis. Known conditions for
and
(divergence through the whole sequence n) are also simplified. 相似文献