Pull-back morphisms for reflexive differential forms

Let f:X→Y$f:X\to Y$ be a morphism between normal complex varieties, where Y$Y$ is Kawamata log terminal. Given any differential form σ$\sigma$, defined on the smooth locus of Y$Y$, we construct a “pull-back form” on X$X$. The pull-back map obtained by this construction is _?Y${?}_Y$-linear, uniquely determined by natural universal properties and exists even in cases where the image of f$f$ is entirely contained in the singular locus of Y$Y$.     

Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains (_Xn,_ηn)$(X_n,{\eta }_n)$ on _Z+×S${\mathbb{Z}}_{+}\times S$, where _Z+${\mathbb{Z}}_{+}$ is the non-negative integers and S$S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of _Xn$X_n$, and that, roughly speaking, _ηn${\eta }_n$ is close to being Markov when _Xn$X_n$ is large. This departure from much of the literature, which assumes that _ηn${\eta }_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for _Xn$X_n$ given _ηn${\eta }_n$. We give a recurrence classification in terms of increment moment parameters for _Xn$X_n$ and the stationary distribution for the large- X$X$ limit of _ηn${\eta }_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between _Xn$X_n$ (rescaled) and _ηn${\eta }_n$. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on _Z+${\mathbb{Z}}_{+}$ (the case where S$S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where _ηn${\eta }_n$ tracks an internal state of the system.     

Characterizations of hypersurfaces of a Danielewski type

Let k$k$ be a field of characteristic zero and R$R$ a factorial affine k$k$-domain. Let B$B$ be an affineR$R$-domain. In terms of locally nilpotent derivations, we give criteria for B$B$ to be R$R$-isomorphic to the residue ring of a polynomial ring R[_X1,_X2,Y]$R[X_1,X_2,Y]$ over R$R$ by the ideal (_X1X2−φ(Y))$(X_1{X}_2-\phi \left(Y\right))$ for φ(Y)∈R[Y]?R$\phi \left(Y\right)\in R\left[Y\right]?R$.     

The homotopy type of the polyhedral product for shifted complexes

We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K$K$ is a shifted simplicial complex on n$n$ vertices, _X1,…,_Xn$X_1,\dots ,X_n$ are pointed connected CW$CW$-complexes and C_Xi$C{X}_i$ is the cone on  _Xi$X_i$, then the polyhedral product determined by K$K$ and the pairs (C_Xi,_Xi)$(C{X}_i,X_i)$ is homotopy equivalent to a wedge of suspensions of smashes of the _Xi$X_i$’s. Earlier work of the authors dealt with the special case where each _Xi$X_i$ is a loop space. New techniques are introduced to prove the general case. These have the advantage of simplifying the earlier results and of being sufficiently general to show that the conjecture holds for a substantially larger class of simplicial complexes. We discuss connections between polyhedral products and toric topology, combinatorics, and classical homotopy theory.     

Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(_Xt,α)$b(X_t,\alpha )$ and diffusion coefficient εa(_Xt,β)$\epsilon a(X_t,\beta )$ where α$\alpha$ and β$\beta$ are two unknown parameters, while ε$\epsilon$ is known. For a high frequency sample of observations of the diffusion at the time points k/n$k/n$, k=1,…,n$k=1,\dots ,n$, we propose a class of contrast functions and thus obtain estimators of (α,β)$(\alpha ,\beta )$. The estimators are shown to be consistent and asymptotically normal when n→∞$n\to \infty$ and ε→0$\epsilon \to 0$ in such a way that ε⁻¹n^−ρ${\epsilon }^{-1}{n}^{-\rho }$ remains bounded for some ρ>0$\rho >0$. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.     

Parametric inference for discretely observed multidimensional diffusions with small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(α,_Xt)$b(\alpha ,X_t)$ and diffusion coefficient ?σ(β,_Xt)$?\sigma (\beta ,X_t)$. The diffusion sample path is discretely observed at times _tk=kΔ$t_k=k\Delta$ for k=1…n$k=1\dots n$ on a fixed interval [0,T]$[0,T]$. We study minimum contrast estimators derived from the Gaussian process approximating X$X$ for small ?$?$. We obtain consistent and asymptotically normal estimators of α$\alpha$ for fixed Δ$\Delta$ and ?→0$?\to 0$ and of (α,β)$(\alpha ,\beta )$ for Δ→0$\Delta \to 0$ and ?→0$?\to 0$ without any condition linking ?$?$ and Δ$\Delta$. We compare the estimators obtained with various methods and for various magnitudes of Δ$\Delta$ and ?$?$ based on simulation studies. Finally, we investigate the interest of using such methods in an epidemiological framework.     

On the removal lemma for linear systems over Abelian groups

In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the k$k$-determinantal of an integer (k×m)$(k\times m)$ matrix A$A$ is coprime with the order n$n$ of a group G$G$ and the number of solutions of the system Ax=b$Ax=b$ with _x1∈_X1,…,_xm∈_Xm$x_1\in X_1,\dots ,x_m\in X_m$ is o(n^m−k)$o\left(n^{m-k}\right)$, then we can eliminate o(n)$o\left(n\right)$ elements in each set to remove all these solutions.     

Abelian theorems for stochastic volatility models with application to the estimation of jump activity

In this paper, we prove a kind of Abelian theorem for a class of stochastic volatility models (X,V)$(X,V)$ where both the state process X$X$ and the volatility process V$V$ may have jumps. Our results relate the asymptotic behavior of the characteristic function of _XΔ$X_{\Delta }$ for some Δ>0$\Delta >0$ in a stationary regime to the Blumenthal–Getoor indexes of the Lévy processes driving the jumps in X$X$ and V$V$. The results obtained are used to construct consistent estimators for the above Blumenthal–Getoor indexes based on low-frequency observations of the state process X$X$. We derive convergence rates for the corresponding estimator and show that these rates cannot be improved in general.     

Viscosity approximation method for accretive operator in Banach space

Let X$X$ be a uniformly smooth Banach space, C$C$ be a closed convex subset of X$X$, and A$A$ an m-accretive operator with a zero. Consider the iterative method that generates the sequence {_xn}$\left\{x_n\right\}$ by the algorithm

_xn+1=_αnf(_xn)+(1−_αn)_Jrnxn,$x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)J_{r_n}{x}_n\text{,}$

Nonparametric estimation of the stationary density and the transition density of a Markov chain

In this paper, we study first the problem of nonparametric estimation of the stationary density f$f$ of a discrete-time Markov chain (_Xi)$\left(X_i\right)$. We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized contrast. The same technique enables us to estimate the density g$g$ of (_Xi,_Xi+1)$(X_i,X_{i+1})$ and so to provide an adaptive estimator of the transition density π=g/f$\pi =g/f$. We give bounds in L²$L^2$ norm for these estimators and we show that they are adaptive in the minimax sense over a large class of Besov spaces. Some examples and simulations are also provided.     

Coalescence in the recent past in rapidly growing populations

In a rapidly growing population one expects that two individuals chosen at random from the n$n$th generation are unlikely to be closely related if n$n$ is large. In this paper it is shown that for a broad class of rapidly growing populations this is not the case. For a Galton–Watson branching process with an offspring distribution {_pj}$\left\{p_j\right\}$ such that _p0=0$p_0=0$ and ψ(x)=_{∑jpjI{j≥x}}$\psi \left(x\right)={\sum }_j{p}_j{I}_{\{j\ge x\}}$ is asymptotic to x^−αL(x)$x^{-\alpha }L\left(x\right)$ as x→∞$x\to \infty$ where L(⋅)$L(\cdot )$ is slowly varying at ∞$\infty$ and 0<α<1$0<\alpha <1$ (and hence the mean m=∑j_pj=∞$m=\sum j{p}_j=\infty$) it is shown that if _Xn$X_n$ is the generation number of the coalescence of the lines of descent backwards in time of two randomly chosen individuals from the n$n$th generation then n−_Xn$n-X_n$ converges in distribution to a proper distribution supported by N={1,2,3,…}$\mathbb{N}=\{1,2,3,\dots \}$. That is, in such a rapidly growing population coalescence occurs in the recent past rather than the remote past. We do show that if the offspring mean m$m$ satisfies 1<m≡∑j_pj<∞$1<m\equiv \sum j{p}_j<\infty$ and _p0=0$p_0=0$ then coalescence time _Xn$X_n$ does converge to a proper distribution as n→∞$n\to \infty$, i.e., coalescence does take place in the remote past.     

Non-parametric adaptive estimation of the drift for a jump diffusion process

In this article, we consider a jump diffusion process _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ observed at discrete times t=0,Δ,…,nΔ$t=0,\Delta ,\dots ,n\Delta$. The sampling interval Δ$\Delta$ tends to 0 and nΔ$n\Delta$ tends to infinity. We assume that _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ is ergodic, strictly stationary and exponentially β$\beta$-mixing. We use a penalised least-square approach to compute two adaptive estimators of the drift function b$b$. We provide bounds for the risks of the two estimators.     

New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems

Let T:D⊂X→X$T:D\subset X\to X$ be an iteration function in a complete metric space X$X$. In this paper we present some new general complete convergence theorems for the Picard iteration _xn+1=T_xn$x_{n+1}=T{x}_n$ with order of convergence at least r≥1$r\ge 1$. Each of these theorems contains a priori and a posteriori error estimates as well as some other estimates. A central role in the new theory is played by the notions of a function of initial conditions   of T$T$ and a convergence function   of T$T$. We study the convergence of the Picard iteration associated to T$T$ with respect to a function of initial conditions E:D→X$E:D\to X$. The initial conditions in our convergence results utilize only information at the starting point _x0$x_0$. More precisely, the initial conditions are given in the form E(_x0)∈J$E\left(x_0\right)\in J$, where J$J$ is an interval on _R+${\mathbb{R}}_{+}$ containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete ω$\omega$-versions of the famous semilocal Newton–Kantorovich theorem as well as a complete version of the famous semilocal α$\alpha$-theorem of Smale for analytic functions.     

Nonautonomous stochastic search for global minimum in continuous optimization

Various iterative stochastic optimization schemes can be represented as discrete-time Markov processes defined by the nonautonomous equation _Xt+1=_Tt(_Xt,_Yt)$X_{t+1}=T_t(X_t,Y_t)$, where _Yt$Y_t$ is an independent sequence and _Tt$T_t$ is a sequence of mappings. This paper presents a general framework for the study of the stability and convergence of such optimization processes. Some applications are given: the mathematical convergence analysis of two optimization methods, the elitist evolution strategy (μ+λ)$(\mu +\lambda )$ and the grenade explosion method, is presented.