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1.
For a supercritical branching process (Zn) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population Wn=Zn/E[Zn|ξ] to its limit W: we show a central limit theorem for WWn with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distributed. Similar results are also shown for Wn+kWn for each fixed kN.  相似文献   

2.
Let (Zn) be a supercritical branching process with immigration in a random environment. Firstly, we prove that under a simple log moment condition on the offspring and immigration distributions, the naturally normalized population size Wn converges almost surely to a finite random variable W. Secondly, we show criterions for the non-degeneracy and for the existence of moments of the limit random variable W. Finally, we establish a central limit theorem, a large deviation principle and a moderate deviation principle about log Zn.  相似文献   

3.
Let (Z n ) be a supercritical branching process in an independent and identically distributed random environment ζ = (ζ 0, ζ 1,…), and let W be the limit of the normalized population size Z n / $\mathbb{E}$ (Z n |ζ). We show a necessary and sufficient condition for the existence of weighted moments of W of the form $\mathbb{E}$ , where α ≥ 1 and ? is a positive function slowly varying at ∞.  相似文献   

4.
I. Biswas 《Topology》2006,45(2):403-419
Let X be a nonsingular algebraic curve of genus g?3, and let Mξ denote the moduli space of stable vector bundles of rank n?2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g=3 then n?4 and if g=4 then n?3, and suppose further that n0, d0 are integers such that n0?1 and nd0+n0d>nn0(2g-2). Let E be a semistable vector bundle over X of rank n0 and degree d0. The generalised Picard bundle Wξ(E) is by definition the vector bundle over Mξ defined by the direct image where Uξ is a universal vector bundle over X×Mξ. We obtain an inversion formula allowing us to recover E from Wξ(E) and show that the space of infinitesimal deformations of Wξ(E) is isomorphic to H1(X,End(E)). This construction gives a locally complete family of vector bundles over Mξ parametrised by the moduli space M(n0,d0) of stable bundles of rank n0 and degree d0 over X. If (n0,d0)=1 and Wξ(E) is stable for all EM(n0,d0), the construction determines an isomorphism from M(n0,d0) to a connected component M0 of a moduli space of stable sheaves over Mξ. This applies in particular when n0=1, in which case M0 is isomorphic to the Jacobian J of X as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over J, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles.  相似文献   

5.
In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σn?N1rP(n)?s ξn, where PR+ [X1,…,Xr] and ξn = ξ1n1ξrnr, with ξiC, such that |ξi| = 1 and ξi ≠ 1, 1 ≦ ir. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over Q by the ξi and the coefficients of P. If, there exists a number field K, containing the ξi, 1 ≦ ir, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a B-adic function ZB(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k).  相似文献   

6.
Let Z = {Z0, Z1, Z2,…} be a martingale, with difference sequence X0 = Z0, Xi = Zi ? Zi ? 1, i ≥ 1. The principal purpose of this paper is to prove that the best constant in the inequality λP(supi |Xi| ≥ λ) ≤ C supiE |Zi|, for λ > 0, is C = (log 2)?1. If Z is finite of length n, it is proved that the best constant is Cn = [n(21n ? 1)]?1. The analogous best constant Cn(z) when Z0z is also determined. For these finite cases, examples of martingales attaining equality are constructed. The results follow from an explicit determination of the quantity Gn(z, E) = supzP(maxi=1,…,n |Xi| ≥ 1), the supremum being taken over all martingales Z with Z0z and E|Zn| = E. The expression for Gn(z,E) is derived by induction, using methods from the theory of moments.  相似文献   

7.
Given an antisymmetric kernel K (K(z, z′) = ?K(z′, z)) and i.i.d. random variates Zn, n?1, such that EK2(Z1, Z2)<∞, set An = ∑1?i?j?nK(Zi,Zj), n?1. If the Zn's are two-dimensional and K is the determinant function, An is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the An's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.  相似文献   

8.
Let S(m|n,r)Z be a Z-form of a Schur superalgebra S(m|n,r) generated by elements ξi,j. We solve a problem of Muir and describe a Z-form of a simple S(m|n,r)-module Dλ,Q over the field Q of rational numbers, under the action of S(m|n,r)Z. This Z-form is the Z-span of modified bideterminants [T?:Ti] defined in this work. We also prove that each [T?:Ti] is a Z-linear combination of modified bideterminants corresponding to (m|n)-semistandard tableaux Ti.  相似文献   

9.
Let β > 1 be a Pisot number andg be a positive Hölder continuous function with period one and g(0) = 1. The multiperiodic functionG(ξ)=Π n=0 g(ξ/βn) is studied and the asymptotic behaviour ofI G(T) = ∫ 0 T G(ξ)dξ investigated. We prove that the limit of logI(T)/ logT exists asT tends to infinity. We also provide a method to calculate this limit for the caseg(ξ) = cos2 2πξ, corresponding to the Fourier transform of the Bernoulli convolution associated to the golden number (or some of its generalizations).  相似文献   

10.
As is well known, for a supercritical Galton-Watson process Z n whose offspring distribution has mean m > 1, the ratio W n := Z n /m n has almost surely a limit, say W. We study the tail behaviour of the distributions of W n and W in the case where Z 1 has a heavy-tailed distribution, that is, $\mathbb{E}e^{\lambda {\rm Z}_1 } = \infty $ for every λ > 0. We show how different types of distributions of Z 1 lead to different asymptotic behaviour of the tail of W n and W. We describe the most likely way in which large values of the process occur.  相似文献   

11.
We consider the random variable Zn,α=Y1+2αY2+?+nαYn, with αR and Y1,Y2,… independent and exponentially distributed random variables with mean one. The distribution function of Zn,α is in terms of a series with alternating signs, causing great numerical difficulties. Using an extended version of the saddle point method, we derive a uniform asymptotic expansion for P(Zn,α<x) that remains valid inside (α≥−1/2) and outside (α<−1/2) the domain of attraction of the central limit theorem. We discuss several special cases, including α=1, for which we sharpen some of the results in Kingman and Volkov (2003).  相似文献   

12.
Let ξt, t ? 0, be a d-dimensional Brownian motion. The asymptotic behaviour of the random field ??∫t0?(ξs) ds is investigated, where ? belongs to a Sobolev space of periodic functions. Particularly a central limit theorem and a law of iterated logarithm are proved leading to a so-called universal law of iterated logarithm.  相似文献   

13.
Let {W(t): t ≥ 0} be μ-Brownian motion in a real separable Banach space B, and let aT be a nondecreasing function of T for which (i) 0 < aTT (T ≥ 0), (ii) aTT is nonincreasing. We establish a Strassen limit theorem for the net {ξT: T ≥ 3}, where
ξT =W(T + taT) ? W(T){2aT[log(TaT) + log log T]}12, 0 ? t ? 1
  相似文献   

14.
By solving a free analog of the Monge-Ampère equation, we prove a non-commutative analog of Brenier’s monotone transport theorem: if an n-tuple of self-adjoint non-commutative random variables Z 1,…,Z n satisfies a regularity condition (its conjugate variables ξ 1,…,ξ n should be analytic in Z 1,…,Z n and ξ j should be close to Z j in a certain analytic norm), then there exist invertible non-commutative functions F j of an n-tuple of semicircular variables S 1,…,S n , so that Z j =F j (S 1,…,S n ). Moreover, F j can be chosen to be monotone, in the sense that and g is a non-commutative function with a positive definite Hessian. In particular, we can deduce that C ?(Z 1,…,Z n )?C ?(S 1,…,S n ) and \(W^{*}(Z_{1},\dots,Z_{n})\cong L(\mathbb{F}(n))\) . Thus our condition is a useful way to recognize when an n-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors \(\varGamma_{q}(\mathbb{R}^{n})\) are isomorphic (for sufficiently small q, with bound depending on n) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport.  相似文献   

15.
Let (Xm,n)(m,n)∈Z2 be a Cp-valued wide sense stationary process. We study the prediction theory of such processes according to different total orders on Z2. In the case of a “rational order”, we give the spectral distribution of the resulting evanescent component and prove that for two different rational orders, the resulting evanescent components are mutually orthogonal.  相似文献   

16.
We introduce new families of orthogonal polynomials HD,n, motivated by the non-equilibrium evolution of a quantum Brownian particle (qBp). The HD,n’s generalize non-trivially the standard Hermite polynomials, employed for classical Brownian motion. We treat several models (labelled by D) for a non-equilibrium qBp, by means of the Wigner function W, in the presence of a “heat bath” at thermal equilibrium, with and without ab initio friction. For long times (for a suitable class of initial conditions), the non-equilibrium Wigner function W should approach, in some sense, the (time-independent) equilibrium Wigner function Weq,D, which describes the thermal equilibrium of the qBp with the “heat bath” and plays a central role. Weq,D is chosen to be the weight function which orthogonalizes the HD,n’s. New results on Weq,D and on the HD,n’s are reported. We justify the key role of the HD,n’s as follows. Using the HD,n’s, moments Weq,D,n and Wn are introduced for Weq,D and W, respectively. At equilibrium, all moments Weq,D,n except the lowest one (Weq,D,0) vanish identically. Off-equilibrium, one expects that, for long times (for suitable initial conditions): (i) all non-equilibrium moments Wn (except the lowest moment W0), will approach zero, while (ii) the lowest non-equilibrium moment W0 will tend to Weq,D,0(≠0). To complete the justification, we outline how the approximate long-time non-equilibrium theories determined by W0 for the different models (D) yield Smoluchowski equations and irreversible evolutions of the qBp towards thermal equilibrium.  相似文献   

17.
We consider a diffusion (ξ t ) t≥0 whose drift contains some deterministic periodic signal. Its shape being fixed and known, up to scaling in time, the periodicity of the signal is the unknown parameter ? of interest. We consider sequences of local models at ? corresponding to continuous observation of the process ξ on the time interval [0, n] as n → ∞, with suitable choice of local scale at ?. Our tools - under an ergodicity condition — are path segments of ξ corresponding to the period ?, and limit theorems for certain functionals of the process ξ, which are not additive functionals. When the signal is smooth, with local scale n ?3/2 at ?, we have local asymptotic normality (LAN) in the sense of Le Cam [21]. When the signal has a finite number of discontinuities, with local scale n ?2 at ?, we obtain a limit experiment of different type, studied by Ibragimov and Khasminskii [14], where smoothness of the parametrization (in the sense of Hellinger distance) is Hölder 1/2.  相似文献   

18.
Let (ξ n ) nN be a sequence of arbitrarily dependent random variables. In this paper, a generalized strong limit theorem of the delayed average of (ξ n ) nN is investigated, then some limit theorems for arbitrary information sources follow. As corollaries, some known results are generalized.  相似文献   

19.
Let (Zn)nN be a d-dimensional random walk in random scenery, i.e., with (Sk)kN0 a random walk in Zd and (Y(z))zZd an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of for various choices of sequences n(bn) in [1,∞). Depending on n(bn) and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497-527] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004].  相似文献   

20.
We classify the reverse process {Xn} of a multitype Galton-Watson process {Zn}. In the positive recurrent cases we give the stationary measure for {Xn} explicitly, and in the critical case, supposing that all the second moments of Z1 are finite, we establish the convergence in law to a gamma distribution. Limit distributions of {Zcn}, 0 < c < 1, conditioned on Zn, are also given in the subcritical, supercritical and critical cases, respectively. These extend the previous one-type work of W. W. Esty.  相似文献   

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