Weak convergence of censored and reflected stable processes

It is shown that if a sequence of open n$n$-sets _Dk$D_k$ increases to an open n$n$-set D$D$ then reflected stable processes in _Dk$D_k$ converge weakly to the reflected stable process in D$D$ for every starting point x$x$ in D$D$. The same result holds for censored α$\alpha$-stable processes for every x$x$ in D$D$ if D$D$ and _Dk$D_k$ satisfy the uniform Hardy inequality. Using the method in the proof of the above results, we also prove the weak convergence of reflected Brownian motions in unbounded domains.     

Positional graphs and conditional structure of weakly null sequences

We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer n$n$ there are normalized weakly-null sequences of length _ωn${\omega }_n$ without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that _ωω${\omega }_{\omega }$ is the minimal cardinal κ$\kappa$ that could possibly have the property that every weakly null κ$\kappa$-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κ$\kappa$ which is smaller than the first ω$\omega$-Erd?s cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either _c0$c_0$ or _?p${?}_p$, with p≥1$p\ge 1$.     

Tournaments associated with multigraphs and a theorem of Hakimi

A tournament of order n$n$ is usually considered as an orientation of the complete graph _Kn$K_n$. In this note, we consider a more general definition of a tournament that we call aC$C$-tournament, where C$C$ is the adjacency matrix of a multigraph G$G$, and a C$C$-tournament is an orientation of G$G$. The score vector of a C$C$-tournament is the vector of outdegrees of its vertices. In 1965 Hakimi obtained necessary and sufficient conditions for the existence of a C$C$-tournament with a prescribed score vector R$R$ and gave an algorithm to construct such a C$C$-tournament which required, however, some backtracking. We give a simpler and more transparent proof of Hakimi’s theorem, and then provide a direct construction of such a C$C$-tournament which works even for weighted graphs.     

On Ocneanu's theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams

We give an exposition of Ocneanu's theory of double triangle algebras for subfactors and its application to the classification of irreducible bi-unitary connections on the Dynkin diagrams _An$A_n$, _Dn$D_n$, _E6$E_6$, _E7$E_7$ and _E8$E_8$. More precisely, we give a detailed proof of the complete classification of irreducible K–L$K–L$ bi-unitary connections up to gauge choice, where K and L   represent the two horizontal graphs which are among the A–D–E$A–D–E$ Dynkin diagrams. The result also provides a simple proof of the flatness of _D2n$D_{2n}$, _E6$E_6$ and _E8$E_8$ connections as well as an easy computation of the flat part of _E7$E_7$ as an application.     

A unified proof of Brooks’ theorem and Catlin’s theorem

Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let d$d$ be an integer at least 3, and let G$G$ be a graph with maximum degree d$d$. If G$G$ does not contain _Kd+1$K_{d+1}$ as a subgraph, then G$G$ has a d$d$-coloring in which one color class has size α(G)$\alpha \left(G\right)$. Here α(G)$\alpha \left(G\right)$ denotes the independence number of G$G$. We give a unified proof of Brooks’ theorem and Catlin’s theorem.     

Randomly weighted self-normalized Lévy processes

Let (_Ut,_Vt)$(U_t,V_t)$ be a bivariate Lévy process, where _Vt$V_t$ is a subordinator and _Ut$U_t$ is a Lévy process formed by randomly weighting each jump of _Vt$V_t$ by an independent random variable _Xt$X_t$ having cdf F$F$. We investigate the asymptotic distribution of the self-normalized Lévy process _Ut/_Vt$U_t/V_t$ at 0 and at ∞$\infty$. We show that all subsequential limits of this ratio at 0 (∞$\infty$) are continuous for any nondegenerate F$F$ with finite expectation if and only if _Vt$V_t$ belongs to the centered Feller class at 0 (∞$\infty$). We also characterize when _Ut/_Vt$U_t/V_t$ has a non-degenerate limit distribution at 0 and ∞$\infty$.     

Mosco convergence of the sets of fixed points for one-parameter nonexpansive semigroups

We consider the Mosco convergence of the sets of fixed points for one-parameter strongly continuous semigroups of nonexpansive mappings. One of our main results is the following: Let C$C$ be a closed convex subset of a Hilbert space E$E$. Let {T(t):t≥0}$\{T\left(t\right):t\ge 0\}$ be a strongly continuous semigroup of nonexpansive mappings on C$C$. The set of all fixed points of T(t)$T\left(t\right)$ is denoted by F(T(t))$F\left(T\left(t\right)\right)$ for each t≥0$t\ge 0$. Let τ$\tau$ be a nonnegative real number and let {_tn}$\left\{t_n\right\}$ be a sequence in R$\mathbb{R}$ satisfying τ+_tn≥0$\tau +t_n\ge 0$ and _tn≠0$t_n\ne 0$ for n∈N$n\in \mathbb{N}$, and _limntn=0${lim}_n{t}_n=0$. Then {F(T(τ+_tn))}$\left\{F\left(T(\tau +t_n)\right)\right\}$ converges to _?t≥0F(T(t))${?}_{t\ge 0}F\left(T\left(t\right)\right)$ in the sense of Mosco.     

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.     

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.     

Meixner polynomials in several variables satisfying bispectral difference equations

We construct a set _Md${\mathfrak{M}}_d$ whose points parametrize families of Meixner polynomials in d   variables. There is a natural bispectral involution b$\mathfrak{b}$ on _Md${\mathfrak{M}}_d$ which corresponds to a symmetry between the variables and the degree indices of the polynomials. We define two sets of d   commuting partial difference operators diagonalized by the polynomials. One of the sets consists of difference operators acting on the variables of the polynomials and the other one on their degree indices, thus proving their bispectrality. The two sets of partial difference operators are naturally connected via the involution b$\mathfrak{b}$.     

The Helmholtz equation in a non-smooth inclusion

In this paper, we study the Helmholtz equation in a non-smooth inclusion, i.e., in a doubly connected bounded domain B$B$ in R²$R^2$ with boundary ∂B$\partial B$ that consists of two disjoint closed curves Γ$\Gamma$ and _Γ0${\Gamma }_0$. The existence and uniqueness of a solution to the Helmholtz equation for mixed boundary conditions on Γ$\Gamma$ are obtained by using Riesz–Fredholm theory.     

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.     

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.     

Polynomial identities for the Jordan algebra of a degenerate symmetric bilinear form

Let _Jn$J_n$ be the Jordan algebra of a degenerate symmetric bilinear form. In the first section we classify all possible G  -gradings on _Jn$J_n$ where G   is any group, while in the second part we restrict our attention to a degenerate symmetric bilinear form of rank n−1$n-1$, where n is the dimension of the vector space V   defining _Jn$J_n$. We prove that in this case the algebra _Jn$J_n$ is PI-equivalent to the Jordan algebra of a nondegenerate bilinear form.     

A degree by degree recursive construction of Hermite spline interpolants

Based on the classical Hermite spline interpolant _H2n−1$H_{2n-1}$, which is the piecewise interpolation polynomial of class Cⁿ⁻¹$C^{n-1}$ and degree 2n−1$2n-1$, a piecewise interpolation polynomial _H2n$H_{2n}$ of degree 2n$2n$ is given. The formulas for computing _H2n$H_{2n}$ by _H2n−1$H_{2n-1}$ and computing _H2n+1$H_{2n+1}$ by _H2n$H_{2n}$ are shown. Thus a simple recursive method for the construction of the piecewise interpolation polynomial set {_Hj}$\left\{H_j\right\}$ is presented. The piecewise interpolation polynomial _H2n$H_{2n}$ satisfies the same interpolation conditions as the interpolant _H2n−1$H_{2n-1}$, and is an optimal approximation of the interpolant _H2n+1$H_{2n+1}$. Some interesting properties are also proved.