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1.
Leonid Bogachev 《Journal of Theoretical Probability》2006,19(4):849-873
We are concerned with the limit distribution of l
t
-norms (of order t) of samples of i.i.d. positive random variables, as N→∞, t→∞. The problem was first considered by Schlather [(2001), Ann. Probab. 29, 862–881], but the case where {X
i
} belong to the domain of attraction of Gumbel’s double exponential law (in the sense of extreme value theory) has largely
remained open (even for an exponential distribution). In this paper, it is assumed that the log-tail distribution function
is regularly varying at infinity with index . We proceed from studying the limit distribution of the sums , which is of interest in its own right. A proper growth scale of N relative to t appears to be of the form (). We show that there are two critical points, α1 = 1 and α2 = 2, below which the law of large numbers and the central limit theorem, respectively, break down. For α < 2, under a slightly
stronger condition of normalized regular variation of h, we prove that the limit laws for S
N
(t) are stable, with characteristic exponent and skewness parameter . A complete picture of the limit laws for the norms R
N
(t) = S
N
(t)1/t
is then derived. In particular, our results corroborate a conjecture in Schlather [(2001), Ann. Probab. 29, 862–881] regarding the “endpoints” , α→ 0.
相似文献
2.
Henri Heinich 《Journal of Theoretical Probability》2006,19(2):509-534
In this paper, we generalize the Kantorovich functional to K?the-spaces for a cost or a profit function. We examine the convergence
of probabilities with respect to this functional for some K?the-spaces. We study the Monge problem: Let
be a K?the-space, P and Q two Borel probabilities defined on a Polish space M and a cost function
. A K?the functional
is defined by
(P, Q) = inf
where
is the law of X. If c is a profit function, we note
. (P, Q) = sup
Under some conditions, we show the existence of a Monge function, φ, such that
, or
.
相似文献
3.
In this paper, we introduce the class of
-stopping lines which generalize the planar stopping lines in Merzbach [(1980), Stochastic Process. Appl. 10, 49–63] by replacing the positive quadrant of the plane by a collection
of compact subsets of a fixed topological space. Our notion of stopping line also compliments and generalizes the stopping
sets defined in Ivanoff and Merzbach [(1995), Stochastic Process. Appl. 57, 83–98].
相似文献
4.
Let (, ) be a separable Banach space and let be a class of probability measures on , and let denote the symmetrization of . We provide two sufficient conditions (one in terms of certain quantiles and the other in terms of certain moments of relative to μ and , ) for the “uniform comparison” of the μ and measure of the complements of the closed balls of centered at zero, for every . As a corollary to these “tail comparison inequalities,” we show that three classical results (the Lévy-type Inequalities,
the Kwapień-Contraction Inequality, and a part of the It?–Nisio Theorem) that are valid for the symmetric (but not for the
general non-symmetric) independent -valued random vectors do indeed hold for the independent random vectors whose laws belong to any which satisfies one of the two noted conditions and which is closed under convolution. We further point out that these three
results (respectively, the tail comparison inequalities) are valid for the centered log-concave, as well as, for the strictly
α-stable (or the more general strictly (r, α) -semistable) α ≠ 1 random vectors (respectively, probability measures). We also present several examples which we believe
form a valuable part of the paper.
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5.
Aimé Lachal 《Journal of Theoretical Probability》2006,19(4):757-771
Let (B
t
)
t≥ 0 be standard Brownian motion starting at y and set X
t
= for , with V(y) = y
γ if y≥ 0, V(y) = −K(−y)γ if y≤ 0, where γ and K are some given positive constants. Set . In this paper, we provide some formulas for the probability distribution of the random variable as well as for the probability (or b)}. The formulas corresponding to the particular cases x = a or b are explicitly expressed by means of hypergeometric functions.
相似文献
6.
Patrizia Berti Luca Pratelli Pietro Rigo 《Probability Theory and Related Fields》2007,137(3-4):277-288
Let $(\Omega,\mathcal{A},P)Let
be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and
an arbitrary map, n = 1,2,.... If μ is tight and X
n
converges in distribution to μ (in Hoffmann–J?rgensen’s sense), then X∼μ for some S-valued random variable X on
. If, in addition, the X
n
are measurable and tight, there are S-valued random variables
and X, defined on
, such that
, X∼μ, and
a.s. for some subsequence (n
k
). Further,
a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X
n
= x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to
for some H⊂Ω with P
*(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space
can be taken
, for some H⊂ (0,1) with outer Lebesgue measure 1, where
is the Borel σ-field on (0,1) and m
H
the only extension of Lebesgue measure such that m
H
(H) = 1. In order to prove the previous results, it is also shown that, if X
n
converges in distribution to a separable limit, then X
n
k
converges stably for some subsequence (n
k
).
相似文献
7.
Kyle Siegrist 《Journal of Theoretical Probability》2006,19(1):204-220
Let (S,·) be a positive semigroup and T a sub-semigroup of S. In many natural cases, an element can be factored uniquely as x=yz, where and where z is in an associated “quotient space” S/T. If X has an exponential distribution on S, we show that Y and Z are independent and that Y has an exponential distribution on T. We prove a converse when the sub-semigroup is for . Specifically, we show that if Y
t
and Z
t
are independent and Y
t
has an exponential distribution on S
t
for each , then X has an exponential distribution on S. When applied to ([0,∞), +) and , these results unify and extend known results on the quotient and remainder when X is divided by t. 相似文献
8.
Michele Baldini 《Journal of Theoretical Probability》2007,20(1):65-86
Given a one-dimensional positive recurrent diffusion governed by the Stratonovich SDE , we show that the associated stochastic flow of diffeomorphisms focuses as fast as , where is the finite stationary measure. Moreover, if the drift is reversed and the diffeomorphism is inverted, then the path function
so produced tends, independently of its starting point, to a single (random) point whose distribution is . Applications to stationary solutions of X
t
, asymptotic behavior of solutions of SPDEs and random attractors are offered.
This paper was written while the author was visiting Northwestern University and the opinions expressed in it are those of
the author alone and do not necessarily reflect the views of Merrill Lynch, its subsidiaries or affiliates. 相似文献
9.
Sanatan Rai 《Journal of Theoretical Probability》2007,20(2):119-132
Consider n points, x
1,... , x
n
, distributed uniformly in [0, 1]
d
. Form a graph by connecting two points x
i
and x
j
if . This gives a random geometric graph, , which is connected for appropriate r(n). We show that the spectral measure of the transition matrix of the simple random walk on is concentrated, and in fact converges to that of the graph on the deterministic grid.
相似文献
10.
Richard Nickl 《Journal of Theoretical Probability》2009,22(1):38-56
Let μ
n
be a sequence of random finite signed measures on the locally compact group G equal to either
or ℝ
d
. We give weak conditions on the sequence μ
n
and on functions K such that the convolution product μ
n
*K, and its derivatives, converge in law, in probability, or almost surely in the Banach spaces
or L
p
(G). Examples for sequences μ
n
covered are the empirical process (possibly arising from dependent data) and also random signed measures
where
is some (nonparametric) estimator for the measure ℙ, including the usual kernel and wavelet based density estimators with
MISE-optimal bandwidths. As a statistical application, we apply the results to study convolutions of density estimators.
相似文献
11.
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
12.
Martin J. B. Appel Michael J. Klass Ralph P. Russo 《Journal of Theoretical Probability》1999,12(1):27-47
Let U
1, U
2,... be a sequence of i.i.d. random mappings taking values in a space S and let h be a symmetric function on S×S with global maximum
Let {x
n} be any nondecreasing real sequence converging to
Then p=P(H
n>x
n, infinitely often) is either zero or one, where H
n=max{h(U
i, U
j), 1 ijn}. This paper provides a nonrandom series criterion which is necessary and sufficient to determine the value of p. In addition, various sufficient conditions are presented which may be easier to apply. A number of metric space applications are given. 相似文献
13.
Pavel Drábek Peter Takáč 《Calculus of Variations and Partial Differential Equations》2007,29(1):31-58
An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ1, where −λ1 is the first eigenvalue of the Dirichlet p-Laplacian Δ
p
on , λ1 > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ
p
associated with −λ1. 相似文献
14.
Let (Ω,ℬ,P) be a probability space,
a sub-σ-field, and μ a regular conditional distribution for P given
. For various, classically interesting, choices of
(including tail and symmetric), we prove the following 0–1 law: There is a set
such that P(A
0)=1 and μ(ω)(A)∈{0,1} for all
and ω∈A
0. If ℬ is countably generated (and certain regular conditional distributions exist), the result applies whatever P is.
相似文献
15.
Consider the one-dimensional catalytic super-Brownian motion X (called the reactant) in the catalytic medium
which is an autonomous classical super-Brownian motion. We characterize
both in terms of a martingale problem and (in dimension one) as solution of a certain stochastic partial differential equation.
The focus of this paper is for dimension one the analysis of the longtime behavior via a mass-time-space rescaling. When scaling
time by a factor of K, space is scaled by K
η and mass by K
−η. We show that for every parameter value η ≥ 0 the rescaled processes converge as K→ ∞ in path space. While the catalyst’s limiting process exhibits a phase transition at η = 1, the reactant’s limit is always the same degenerate
process.
相似文献
16.
We consider perturbed empirical distribution functions
, where {Ginn, n1} is a sequence of continuous distribution functions converging weakly to the distribution function of unit mass at 0, and {X
i, i1} is a non-stationary sequence of absolutely regular random variables. We derive the almost sure representation and the law of the iterated logarithm for the statistic
whereU
n
is aU-statistic based onX
1,...,X
n
. The results obtained extend or generalize the results of Nadaraya,(7) Winter,(16) Puri and Ralescu,(9,10) Oodaira and Yoshihara,(8) and Yoshihara,(19) among others.Research supported by the Office of Naval Research Contract N00014-91-J-1020. 相似文献
17.
Let A be a chain ring that is a faithful algebra over a commutative chain ring R, such that is a separable, normal, algebraic field extension of and is countably generated over . It has been recently proved by Alkhamees and Singh that A has a coefficient ring R
0, and there exists a pair (θ, σ) with θ ∈ A, σ an R-automorphism of R
0 such that J (A) = θ A = Aθ, and θa = σ (a) θ, a ∈ R
0. The question of the extension of certain R-automorphisms of R
0 to R-automorphisms of A is investigated.
Mathematics Subject Classification (2000) Primary 16H05, 16W20, secondary 13F20, 13J15 相似文献
18.
Let $A=-(\nabla-i\vec{a})^2+VLet be a magnetic Schr?dinger operator acting on L
2(R
n
), n≥1, where and 0≤V∈L
1
loc. Following [1], we define, by means of the area integral function, a Hardy space H
1
A
associated with A. We show that Riesz transforms (∂/∂x
k
-i
a
k
)A
-1/2 associated with A, , are bounded from the Hardy space H
1
A
into L
1. By interpolation, the Riesz transforms are bounded on L
p
for all 1<p≤2. 相似文献
19.
Let V
n
be an open manifold of non-negative sectional curvature with a soul Σ of co-dimension two. The universal cover of the unit normal bundle N of the soul in such a manifold is isometric to the direct product M
n-2 × R. In the study of the metric structure of V
n
an important role plays the vector field X which belongs to the projection of the vertical planes distribution of the Riemannian submersion on the factor M in this metric splitting . The case n = 4 was considered in [Gromoll, D., Tapp, K.: Geom. Dedicata 99, 127–136 (2003)] where the authors prove that X is a Killing vector field while the manifold V
4 is isometric to the quotient of by the flow along the corresponding Killing field. Following an approach of [Gromoll, D., Tapp, K.: Geom. Dedicata 99, 127–136 (2003)] we consider the next case n = 5 and obtain the same result under the assumption that the set of zeros of X is not empty. Under this assumption we prove that both M
3 and Σ3 admit an open-book decomposition with a bending which is a closed geodesic and pages which are totally geodesic two-spheres,
the vector field X is Killing, while the whole manifold V
5 is isometric to the quotient of by the flow along corresponding Killing field.
Supported by the Faculty of Natural Sciences of the Hogskolan i Kalmar (Sweden). 相似文献