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1.
Péter Major 《Probability Theory and Related Fields》2006,134(3):489-537
Let a sequence of iid. random variables ξ
1, . . . ,ξ
n
be given on a space
with distribution μ together with a nice class
of functions f(x
1, . . . ,x
k
) of k variables on the product space
For all f ∈
we consider the random integral J
n,k
(f) of the function f with respect to the k-fold product of the normalized signed measure
where μ
n
denotes the empirical measure defined by the random variables ξ
1, . . . ,ξ
n
and investigate the probabilities
for all x>0. We show that for nice classes of functions, for instance if
is a Vapnik–Červonenkis class, an almost as good bound can be given for these probabilities as in the case when only the
random integral of one function is considered. A similar result holds for degenerate U-statistics, too.
Supported by the OTKA foundation Nr. 037886 相似文献
2.
Yu Zhang 《Probability Theory and Related Fields》2006,136(2):298-320
We consider the first passage percolation model on Z
d
for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p. We denote by T(0, ν) the passage time from the origin to ν for ν ∈ R
d
and
It is well known that if p < p
c
, there exists a compact shape B
d
⊂ R
d
such that for all
> 0,
t
B
d
(1 −
) ⊂ B(t) ⊂ tB
d
(1 +
) and G(t)(1 −
) ⊂ B(t) ⊂ G(t)(1 +
) eventually w.p.1. We denote the fluctuations of B(t) from tB
d
and G(t) by
In this paper, we show that for all d ≥ 2 with a high probability, the fluctuations F(B(t), G(t)) and F(B(t), tB
d
) diverge with a rate of at least C log t for some constant C. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the
paths themselves. This linearity is also independently interesting.
Research supported by NSF grant DMS-0405150 相似文献
3.
If E and F are real Banach lattices and there is an algebra and order isomorphism Φ:(E) →(F) between their respective ordered Banach algebras of regular operators then there is a linear order isomorphism U:E→F such that Φ(T) =UTU−1 for all T ∈ (E). 相似文献
4.
For a random closed set obtained by exponential transformation of the closed range of a subordinator, a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of . We focus on the number of parts Kn of the composition when is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for Kn and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous
work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at 0+.
Research supported in part by N.S.F. Grant DMS-0071448 相似文献
5.
We investigate the large N behavior of the time the simple random walk on the discrete cylinder
needs to disconnect the discrete cylinder. We show that when d≥2, this time is roughly of order N
2
d
and comparable to the cover time of the slice
, but substantially larger than the cover timer of the base by the projection of the walk. Further we show that by the time
disconnection occurs, a massive ``clogging' typically takes place in the truncated cylinders of height
. These mechanisms are in contrast with what happens when d=1. 相似文献
6.
Wen-Bin Zhang 《Mathematische Zeitschrift》2005,251(2):359-391
Halász’s general mean-value theorem for multiplicative functions on ℕ is classical in probabilistic number theory. We extend this theorem to functions f, defined on a set of generalized integers associated with a set of generalized primes in Beurling’s sense, which satisfies Halász’s conditions, in particular,Assume that the distribution function N(x) of satisfieswith γ>γ0, where ρ1<ρ2<···<ρm are constants with ρm≥1 and A1,···,Am are real constants with Am>0. Also, assume that the Chebyshev function ψ(x) of satisfieswith M>M0. Then the asymptoticimplieswhere τ is a positive constant with τ≥1 and L(u) is a slowly oscillating function with |L(u)|=1. 相似文献
7.
Davar Khoshnevisan David A. Levin Pedro J. Méndez-Hernández 《Probability Theory and Related Fields》2006,134(3):383-416
Consider a sequence
of i.i.d. random variables. Associate to each X
i
(0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy and restart the clock. In this way, we obtain i.i.d. stationary processes {X
i
(t)}
t
≥0 (i=1,2,···) whose invariant distribution is the law ν of X
1(0).
Benjamini et al. (2003) introduced the dynamical walk S
n
(t)=X
1(t)+···+X
n
(t), and proved among other things that the LIL holds for n↦S
n
(t) for all t. In other words, the LIL is dynamically stable. Subsequently (2004b), we showed that in the case that the X
i
(0)'s are standard normal, the classical integral test is not dynamically stable.
Presently, we study the set of times t when n↦S
n
(t) exceeds a given envelope infinitely often. Our analysis is made possible thanks to a connection to the Kolmogorov ɛ-entropy.
When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some
of which we relate.
We prove also that the infinite-dimensional process
converges weakly in
to the Ornstein–Uhlenbeck process in
For this we assume only that the increments have mean zero and variance one.
In addition, we extend a result of Benjamini et al. (2003) by proving that if the X
i
(0)'s are lattice, mean-zero variance-one, and possess (2+ɛ) finite absolute moments for some ɛ>0, then the recurrence of
the origin is dynamically stable. To prove this we derive a gambler's ruin estimate that is valid for all lattice random walks
that have mean zero and finite variance. We believe the latter may be of independent interest.
The research of D. Kh. is partially supported by a grant from the NSF. 相似文献
8.
In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict
positivity fails.
Specifically, we treat a ``symmetric' polynomial q(x, h) in noncommuting variables, {x1, . . . , } and {h1, . . . , } for which q(X,H) is positive semidefinite whenever
are tuples of selfadjoint matrices with ||Xj|| ≤ 1 but Hj unconstrained. The representation we obtain is a Gram representation in the variables h
where Pq is a symmetric matrix whose entries are noncommutative polynomials only in x and V is a ``vector' whose entries are polynomials in both x and h. We show that one can choose Pq such that the matrix Pq(X) is positive semidefinite for all ||Xj|| ≤ 1. The representation covers sum of square results ([Am. Math. (to appear); Linear Algebra Appl. 326 (2001), 193–203; Non commutative Sums of Squares, preprint]) when gx = 0. Also it allows for arbitrary degree in h, rather than degree two, in the main result of [Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically
to appear IOET July 2003] when restricted to x-domains of the type ||Xj|| ≤ 1.
Partially supported by NSF, DARPA and Ford Motor Co.
Partially supported by NSF grant DMS-0140112
Partially supported by NSF grant DMS-0100367 相似文献
9.
S. Fourati 《Probability Theory and Related Fields》2006,135(2):201-215
We proove a new duality relation between stable Lévy processes with index and those with index . This duality appears to be the trajectorial version of the duality of Zolotarev which concerns one dimensional stable laws.
We give an application of this result to the behaviour of the paths at small and large times of the process ``conditioned
to stay positive'.
Mots-clefs: Lévy process Stable process Fluctuation identities Ladder process 相似文献
10.
Let be a C*-algebra, a subalgebra of its center and Φ: → a tracial faithful conditional expectation. We define the positive projective space as the quotient
where G+ is the space of positive invertible elements of , and if there exists g invertible in such that a′ = |g|2a. When is abelian, this space is a set of representatives for probability densities equivalent to a given one. The aim of this paper
is to endow ℙ+ with differentiable structure, a linear connection and a Finsler metric. This is done in a way that given any pair of elements
in ℙ+, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric.
The metric space ℙ+ with the given geodesic distance is non positively curved. 相似文献
11.
Let {Xn} be a stationary and ergodic time series taking values from a finite or countably infinite set Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times n along which we will be able to estimate the conditional probability P(=x|X0,...,) from data segment (X0,...,) in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (among others, all stationary and ergodic Markov chains are included in this class) then almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then n is upperbounded by a polynomial, eventually almost surely.Mathematics Subject Classification (2000): 62G05, 60G25, 60G10 相似文献
12.
We study a certain class of von Neumann algebras generated by selfadjoint elements ωi=ai+ai+, where ai, ai+ satisfy the general commutation relations:We assume that operator T for which the constants are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality were shown. In this paper we prove that T-Ornstein-Uhlenbeck semigroup Ut=ΓT(e−t), t>0 arising from the second quantization procedure is hyper- and ultracontractive. The optimal bounds for hypercontractivity are also discussed.This paper was partially supported by KBN grant no 2P03A00732 and also by RTN grant HPRN-CT-2002-00279. 相似文献
13.
Anna Erschler 《Probability Theory and Related Fields》2006,136(4):560-586
We prove an isoperimetric inequality for wreath products of Markov chains with variable fibers. We use isoperimetric inequalities
for wreath products to estimate the return probability of random walks on infinite groups and graphs, drift of random loops,
the expected value E(exp(−tR
n
)), where R
n
is the number of distinct sites, visited up to the moment n, and, more generally,
(where L
z,n
is the number of visits of z up to the moment n and F(x, y) is some non-negative function). 相似文献
14.
Amílcar Pacheco 《manuscripta mathematica》2005,118(3):361-381
Let be a smooth projective curve defined over a number field k, A/k() an abelian variety and (τ, B) the k()/k-trace of A. We estimate how the rank of A(k())/τB(k) varies when we take a finite geometrically abelian cover defined over k.
This work was partially supported by CNPq research grant 304424/2003-0, Pronex 41.96.0830.00 and CNPq Edital Universal 470099/2003-8.
I would like to thank Douglas Ulmer for comments on how to treat the case of arbitrary ramification, but the conductor prime
to the ramification locus, in the case of elliptic fibrations. I would also like to thank Marc Hindry for comments on the
inequality comparing the conductors of A and A'. Finally, I also thank the referee for his comments and criticisms. 相似文献
15.
Summary We introduce a class of n×n structured matrices which includes three well-known classes of generalized companion matrices: tridiagonal plus rank-one matrices (comrade matrices), diagonal plus rank-one matrices and arrowhead matrices. Relying on the structure properties of , we show that if A then A=RQ , where A=QR is the QR decomposition of A. This allows one to implement the QR iteration for computing the eigenvalues and the eigenvectors of any A with O(n) arithmetic operations per iteration and with O(n) memory storage. This iteration, applied to generalized companion matrices, provides new O(n2) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations. Numerical experiments confirm the effectiveness and the robustness of our approach.The results of this paper were presented at the Workshop on Nonlinear Approximations in Numerical Analysis, June 22 – 25, 2003, Moscow, Russia, at the Workshop on Operator Theory and Applications (IWOTA), June 24 – 27, 2003, Cagliari, Italy, at the Workshop on Numerical Linear Algebra at Universidad Carlos III in Leganes, June 16 – 17, 2003, Leganes, Spain, at the SIAM Conference on Applied Linear Algebra, July 15 – 19, 2003, Williamsburg, VA and in the Technical Report [8]. This work was partially supported by MIUR, grant number 2002014121, and by GNCS-INDAM. This work was supported by NSF Grant CCR 9732206 and PSC CUNY Awards 66406-0033 and 65393-0034. 相似文献
16.
We study limiting distributions of exponential sums as t→∞, N→∞, where (Xi) are i.i.d. random variables. Two cases are considered: (A) ess sup Xi = 0 and (B) ess sup Xi = ∞. We assume that the function h(x)= -log P{Xi>x} (case B) or h(x) = -log P {Xi>-1/x} (case A) is regularly varying at ∞ with index 1 < ϱ <∞ (case B) or 0 < ϱ < ∞ (case A). The appropriate growth scale of N relative to t is of the form , where the rate function H0(t) is a certain asymptotic version of the function (case B) or (case A). We have found two critical points, λ1<λ2, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < λ < λ2, under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable, with characteristic exponent α = α (ϱ, λ) ∈ (0,2) and skewness parameter β ≡ 1.Research supported in part by the DFG grants 436 RUS 113/534 and 436 RUS 113/722.Mathematics Subject Classification (2000): 60G50, 60F05, 60E07 相似文献
17.
Given γ ∈ (−1,1), we present a dyadic growth condition on the finite dimensional distributions of operator semigroups on C0(E which - for γ>0 and Feller semigroups - assures that the corresponding Feller process has paths in local Hölder spaces and in weighted Besov spaces of order γ. We show that, for operator semigroups satisfying Gaussian kernel estimates of order m>1, condition holds for all and even for all in the case of Feller semigroups. Such Gaussian kernel estimates are typical for Feller semigroups on fractals of walk dimension m and for semigroups generated by elliptic operators on ℝD of order m≥D. 相似文献
18.
In the bootstrap percolation on the n-dimensional hypercube, in the initial position each of the 2n sites is occupied with probability p and empty with probability 1−p, independently of the state of the other sites. Every occupied site remains occupied for ever, while an empty site becomes
occupied if at least two of its neighbours are occupied. If at the end of the process every site is occupied, we say that
the (initial) position spans the hypercube. We shall show that there are constants c1,c2>0 such that for the probability of spanning tends to 1 as n→∞, while for the probability tends to 0. Furthermore, we shall show that for each n the transition has a sharp threshold function.
J. Balogh: work was done while at The University of Memphis, USA
Research supported in part by NSF grant DMS0302804
Research supported in part by NSF grant ITR 0225610 and DARPA grant F33615-01-C-1900 相似文献
19.
We consider a real random walk Sn=X1+...+Xn attracted (without centering) to the normal law: this means that for a suitable norming sequence an we have the weak convergence Sn/an⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit
Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S1>0,...,Sn>0) and let Sn+ denote the random variable Sn conditioned on : it is known that Sn+/an ↠ ϕ+(x) dx, where ϕ+(x):=x exp (−x2/2)1(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence.
We also consider the particular case when X1 has an absolutely continuous law: in this case the uniform convergence of the density of Sn+/an towards ϕ+(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our
main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical
proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation
Theory for random walks. 相似文献
20.
T. Bodineau 《Probability Theory and Related Fields》2006,135(2):153-168
We prove that all the translation invariant Gibbs states of the Ising model are a linear combination of the pure phases for any . This implies that the average magnetization is continuous for . Furthermore, combined with previous results on the slab percolation threshold [B2] this shows the validity of Pisztora's
coarse graining [Pi] up to the critical temperature.
We would like to thank C. Pfister and Y. Velenik for very useful comments. 相似文献