Restriction of the cotangent bundle to elliptic curves and Hilbert-Kunz functions

We describe the possible restrictions of the cotangent bundle to an elliptic curve . We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R₊-primary ideal in the graded section ring .     

On location estimation for LARCH processes

We consider location estimation when the error process is a stationary LARCH process with long memory in the second moments. The asymptotic distribution of the sample mean and nonlinear M-estimators of the location parameter are derived. Essential assumptions for obtaining asymptotic normality with -rate of convergence are symmetry of the innovation distribution and skew-symmetry of the ψ-function.     

Simple braids for surface homeomorphisms

Let S be a compact, oriented surface with negative Euler characteristic and let 600071X-si1.gif"> be a homeomorphism isotopic to the identity. If there exists a periodic orbit with a non-zero rotation vector 600071X-si2.gif">, then there exists a simple braid with the same rotation vector.     

Total Variation Distance for Poisson Subset Numbers

Let n be an integer and A₀,..., A_k random subsets of {1,..., n} of fixed sizes a₀,..., a_k, respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between the distance from the random variable , the size of the intersection of the random sets, to a Poisson random variable Z with intensity λ = EW. In particular, the bound tends to zero when λ converges and for all j = 0,..., k, showing that W has an asymptotic Poisson distribution in this regime. Received February 24, 2005     

Strengthening classical results on convergence rates in strong limit theorems

Let X₁, X₂, . . . be i.i.d. random variables, and set S_n=X₁+ . . . +X_n. Several authors proved convergence of series of the type f(&#603;)=∑_nc_nP(|S_n|>&#603;a_n),&#603;>α, under necessary and sufficient conditions. We show that under the same conditions, in fact i.e. the finiteness of ∑_nc_nP(|S_n|>&#603;a_n),&#603;>α, is equivalent to the convergence of the double sum ∑_k∑_nc_nP(|S_n|>ka_n). Two exceptional series required deriving necessary and sufficient conditions for E[sup_n|S_n|(logn)^η/n]<∞,0≤η≤1.     

Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions

We give functional limit theorems for the fluctuations of the rescaled occupation time process of a critical branching particle system in R^d${\mathbb{R}}^d$ with symmetric α$\alpha$-stable motion in the cases of critical and large dimensions, d=2α$d=2\alpha$ and d>2α$d>2\alpha$. In a previous paper [T. Bojdecki, L.G. Gorostiza, A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems I: long-range dependence, Stochastic Process. Appl., this issue.] we treated the case of intermediate dimensions, α<d<2α$\alpha <d<2\alpha$, which leads to a long-range dependence limit process. In contrast, in the present cases the limits are generalized Wiener processes. We use the same space–time random field method of the previous paper, the main difference being that now the tightness requires a new approach and the proofs are more difficult. We also give analogous results for the system without branching in the cases d=α$d=\alpha$ and d>α$d>\alpha$.     

Weak existence and uniqueness for forward–backward SDEs

We aim to establish the existence and uniqueness of weak solutions to a suitable class of non-degenerate deterministic FBSDEs with a one-dimensional backward component. The classical Lipschitz framework is partially weakened: the diffusion matrix and the final condition are assumed to be space Hölder continuous whereas the drift and the backward driver may be discontinuous in x$x$. The growth of the backward driver is allowed to be at most quadratic with respect to the gradient term.     

Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle

Summary Given two pointsx, y60525m80/xxlarge8712.gif" alt="isin" align="MIDDLE" BORDER="0">S ¹ randomly chosen independently by a mixing absolutely continuous invariant measure 60525m80/xxlarge956.gif" alt="mgr" align="MIDDLE" BORDER="0"> of a piecewise expanding and smooth mapf of the circle, we consider for each 60525m80/xxlarge949.gif" alt="epsi" align="BASELINE" BORDER="0">>0 the point process obtained by recording the timesn>0 such that |f ⁿ (x)–f ⁿ (y)|60525m80/xxlarge8806.gif" alt="lE" align="MIDDLE" BORDER="0">60525m80/xxlarge949.gif" alt="epsi" align="BASELINE" BORDER="0">. With the further assumption that the density of 60525m80/xxlarge956.gif" alt="mgr" align="MIDDLE" BORDER="0"> is bounded away from zero, we show that when 60525m80/xxlarge949.gif" alt="epsi" align="BASELINE" BORDER="0"> tends to zero the above point process scaled by 60525m80/xxlarge949.gif" alt="epsi" align="BASELINE" BORDER="0">–1 converges in law to a marked Poisson point process with constant parameter measure. This parameter measure is given explicity by an average on the rate of expansion off.Partially supported by FAPESP grant number 90/3918-5     

Naturally reductive homogeneous Randers spaces

In this paper, we study naturally reductive Randers metrics on homogeneous manifolds. We first prove that naturally reductive Randers metrics are of Berwald type. We then give an explicit formula for the flag curvature of naturally reductive Randers metrics. Finally a necessary and sufficient condition for invariant Randers metrics on homogeneous manifolds being naturally reductive is given.