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1.
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
2.
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
3.
Ioannis Kontoyiannis 《Journal of Theoretical Probability》1998,11(3):795-811
Let
be a discrete-valued stationary ergodic process distributed according to P and let x=(..., x
–1, x
0, x
1,...) denote a realization from X. We investigate the asymptotic behavior of the recurrence time R
n defined as the first time that the initial n-block
reappears in the past of x. We identify an associated random walk,
on the same probability space as X, and we prove a strong approximation theorem between log R
n and
. From this we deduce an almost sure invariance principle for log R
n. As a byproduct of our analysis we get unified proofs for several recent results that were previously established using methods from ergodic theory, the theory of Poisson approximation and the analysis of random trees. Similar results are proved for the waiting time W
n defined as the first time until the initial n-block from one realization first appears in an independent realization generated by the same (or by a different) process. 相似文献
4.
Let
be i.i.d. random variables and let, for each
and
. It is shown that
a.s. whenever the sequence of self-normalized sums S
n
/V
n is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if
相似文献
5.
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
6.
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
7.
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
8.
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
9.
We construct a metric space of set functions (
, d) such that a sequence {P
n} of Borel probability measures on a metric space (
, d*) satisfies the full Large Deviation Principle (LDP) with speed {a
n} and good rate function I if and only if the sequence
converges in (
, d) to the set function e
–I
. Weak convergence of probability measures is another special case of convergence in (
, d). Properties related to the LDP and to weak convergence are then characterized in terms of (
, d). 相似文献
10.
M. Yu. Zvagel'skii 《Journal of Mathematical Sciences》2001,104(4):1276-1282
We single out the obstruction for a closed
-null-homologous submanifold of codimension 2 to be the boundary of a submanifold of codimension 1. As an application, we calculate the groups
of cobordisms of embeddings of nonoriented n-manifolds in the Euclidean (n+2)-space for n=3 and 4. Namely, we show that
and
. A specific generator of the former group is explicitly given. Bibliography: 5 titles. 相似文献
11.
Asymptotic properties of partitions of the unit interval are studied through the entropy for random partition
where
are the order statistics of a random sample {X
i, i n}, X
0, n
–, X
n+1, n
+ and F(x) is a continuous distribution function. A characterization of continuous distributions based on
is obtained. Namely, a sequence of random observations {X
i, i1} comes from a continuous cumulative distribution function (cdf) F(x) if and only if
where = 0.577 is Euler's constant. If {X
i, i1} come from a density g(x) and F is a cdf with density f(x), some limit theorems for
are established, e.g.,
Statistical estimation as well as a goodness-of-fit test based on
are also discussed. 相似文献
0\} } {f(x)\log \frac{{f(x)}}{{g(x)}}dx + \gamma - 1{\text{ in probability}}}$$ " align="middle" vspace="20%" border="0"> |
12.
Kameswarrao S. Casukhela 《Journal of Theoretical Probability》1997,10(3):759-771
An infinite sequence of random variables X=(X
1, X
2,...) is said to be spreadable if all subsequences of X have the same distribution. Ryll-Nardzewski showed that X is spreadable iff it is exchangeable. This result has been generalized to various discrete parameter and higher dimensional settings. In this paper we show that a random measure on the tetrahedral space
is spreadable, iff it can be extended to an exchangeable random measure on
. The result is a continuous parameter version of a theorem by Kallenberg. 相似文献
13.
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
14.
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
15.
Gerold Alsmeyer 《Journal of Theoretical Probability》2002,15(2):259-283
It is proved that for each random walk (S
n
)
n0 on
d
there exists a smallest measurable subgroup
of
d
, called minimal subgroup of (S
n
)
n0, such that P(S
n
)=1 for all n1.
can be defined as the set of all x
d
for which the difference of the time averages n
–1
n
k=1
P(S
k
) and n
–1
n
k=1
P(S
k
+x) converges to 0 in total variation norm as n. The related subgroup
* consisting of all x
d
for which lim
n P(S
n
)–P(S
n
+x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S
n
)
n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on
d
. The main result shows that, up to regular linear transformations, the only subgroups of
d
admitting a quasi-invariant measure are those of the form
1×...×
k
×
l–k
×{0}
d–l
, 0kld, with
1,...,
k
being countable subgroups of
. The proof is based on a result recently proved by Kharazishvili(3) which states no uncountable proper subgroup of
admits a quasi-invariant measure. 相似文献
16.
Consider a double array
of i.i.d. random variables with mean and variance
and set
. Let
denote the empirical distribution function of Z1, n
,..., Z
N, n
and let be the standard normal distribution function. The main result establishes a functional law of the iterated logarithm for
, where n=n(N) as N. For the proof, some lemmas are derived which may be of independent interest. Some corollaries of the main result are also presented. 相似文献
17.
Aimé Lachal 《Journal of Theoretical Probability》2000,13(3):733-775
Let (B
t)
t0 be standard Brownian motion starting at y, X
t = x +
t
0
V(B
s) ds for x (a, b), with V(y) = y
if y0, V(y)=–K(–y)
if y0, where >0 and K is a given positive constant. Set
ab=inf{t>0: X
t(a, b)} and
0=inf{t>0: B
t=0}. In this paper we give several informations about the random variable
ab. We namely evaluate the moments of the random variables
, and also show how to calculate the expectations
. Then, we explicitly determine the probability laws of the random variables
as well as the probability
by means of special functions. 相似文献
18.
In this work we obtain an asymptotic estimate for the expected number of maxima of the random algebraic polynomial
, where a
j (j=0, 1,...,n–1) are independent, normally distributed random variables with mean and variance one. It is shown that for nonzero , the expected number of maxima is asymptotic to
log n, when n is large. 相似文献
19.
Milan Jasem 《Mathematica Slovaca》2007,57(2):107-118
In the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra
= (A, ⊕, −, ∼, 0, 1) there exists an internal direct decomposition
of
with
commutative such that
and
for each x ∈ A.
On the other hand, if
is an internal direct decomposition of a pseudo MV-algebra
= (A, ⊕, −, ∼, 0, 1) with
commutative, then the mapping g: A → A defined by
is an isometry in
and
.
相似文献
20.
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