Marginal densities of the “true” self-repelling motion

Let X(t)$X\left(t\right)$ be the true self-repelling motion (TSRM) constructed by Tóth and Werner (1998) [22], L(t,x)$L(t,x)$ its occupation time density (local time) and H(t):=L(t,X(t))$H\left(t\right):=L(t,X\left(t\right))$ the height of the local time profile at the actual position of the motion. The joint distribution of (X(t),H(t))$(X\left(t\right),H\left(t\right))$ was identified by Tóth (1995) [20] in somewhat implicit terms. Now we give explicit formulas for the densities of the marginal distributions of X(t)$X\left(t\right)$ and H(t)$H\left(t\right)$. The distribution of X(t)$X\left(t\right)$ has a particularly surprising shape: its density has a sharp local minimum   with discontinuous derivative at 0$0$. As a consequence we also obtain a precise version of the large deviation estimate of Dumaz (2011) [5].     

A double generalized Pareto distribution

Papastathopoulos and Tawn [Papastathopoulos, I., Tawn, J.A., 2013. A generalized Student’s t$t$-distribution. Statistics & Probability Letters 83, 70–77] proposed a generalization of Student’s t$t$ distribution to account for negative degrees of freedom. Here, an alternative distribution that has simpler mathematical properties is discussed. Several advantages are established for using the alternative distribution over Papastathopoulos and Tawn’s generalization.     

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 ∞$\infty$.     

On finite capacity queues with time dependent arrival rates

We consider the finite capacity M/M/1−K$M/M/1-K$ queue with a time dependent arrival rate λ(t)$\lambda \left(t\right)$. Assuming that the capacity K$K$ is large and that the arrival rate varies slowly with time (as t/K$t/K$), we construct asymptotic approximations to the probability of finding n$n$ customers in the system at time t$t$, as well as the mean number. We consider various time ranges, where the system is nearly empty, nearly full, or is filled to a fraction of its capacity. Extensive numerical studies are used to back up the asymptotic analysis.     

A limit theorem for the time of ruin in a Gaussian ruin problem

For certain Gaussian processes X(t)$X\left(t\right)$ with trend −ct^β$-c{t}^{\beta }$ and variance V²(t)$V^2\left(t\right)$, the ruin time is analyzed where the ruin time is defined as the first time point t$t$ such that X(t)−ct^β≥u$X\left(t\right)-c{t}^{\beta }\ge u$. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications, e.g. in telecommunications where it indicates the first time of an overflow. We derive the asymptotic distribution of the ruin time as u→∞$u\to \infty$ showing that the limiting distribution depends on the parameters β$\beta$, V(t)$V\left(t\right)$ and the correlation function of X(t)$X\left(t\right)$.     

Quasi-stationary distributions and Yaglom limits of self-similar Markov processes

We discuss the existence and characterization of quasi-stationary distributions and Yaglom limits of self-similar Markov processes that reach 0 in finite time. By Yaglom limit, we mean the existence of a deterministic function g$g$ and a non-trivial probability measure ν$\nu$ such that the process rescaled by g$g$ and conditioned on non-extinction converges in distribution towards ν$\nu$. We will see that a Yaglom limit exists if and only if the extinction time at 0$0$ of the process is in the domain of attraction of an extreme law and we will then treat separately three cases, according to whether the extinction time is in the domain of attraction of a Gumbel, Weibull or Fréchet law. In each of these cases, necessary and sufficient conditions on the parameters of the underlying Lévy process are given for the extinction time to be in the required domain of attraction. The limit of the process conditioned to be positive is then characterized by a multiplicative equation which is connected to a factorization of the exponential distribution in the Gumbel case, a factorization of a Beta distribution in the Weibull case and a factorization of a Pareto distribution in the Fréchet case.     

Asymptotic results for the two-parameter Poisson–Dirichlet distribution

The two-parameter Poisson–Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and gamma subordinators with the two parameters, α$\alpha$ and θ$\theta$, corresponding to the stable component and the gamma component respectively. The moderate deviation principle is established for the distribution when θ$\theta$ approaches infinity, and the large deviation principle is established when both α$\alpha$ and θ$\theta$ approach zero.     

The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let _ηt${\eta }_t$ be a Poisson point process of intensity t≥1$t\ge 1$ on some state space Y$\mathbb{Y}$ and let f$f$ be a non-negative symmetric function on Y^k${\mathbb{Y}}^k$ for some k≥1$k\ge 1$. Applying f$f$ to all k$k$-tuples of distinct points of _ηt${\eta }_t$ generates a point process _ξt${\xi }_t$ on the positive real half-axis. The scaling limit of _ξt${\xi }_t$ as t$t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m$m$-th smallest point of _ξt${\xi }_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k$k$-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.     

Median,concentration and fluctuations for Lévy processes

We estimate a median of f(_Xt)$f\left(X_t\right)$ where f$f$ is a Lipschitz function, X$X$ is a Lévy process and t$t$ is an arbitrary time. This leads to concentration inequalities for f(_Xt)$f\left(X_t\right)$. In turn, corresponding fluctuation estimates are obtained under assumptions typically satisfied if the process has a regular behavior in small time and a, possibly different, regular behavior in large time.     

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.     

Interval versions of statistical techniques with applications to environmental analysis,bioinformatics, and privacy in statistical databases

In many areas of science and engineering, it is desirable to estimate statistical characteristics (mean, variance, covariance, etc.) under interval uncertainty. For example, we may want to use the measured values x(t)$x(t)$ of a pollution level in a lake at different moments of time to estimate the average pollution level; however, we do not know the exact values x(t)$x(t)$—e.g., if one of the measurement results is 0, this simply means that the actual (unknown) value of x(t)$x(t)$ can be anywhere between 0 and the detection limit (DL). We must, therefore, modify the existing statistical algorithms to process such interval data.     

Time homogeneous diffusions with a given marginal at a deterministic time

In this article, it is proved that for any probability law μ$\mu$ over R$\mathbb{R}$ with finite first moment and a given deterministic time t>0$t>0$, there exists a gap diffusion with law μ$\mu$ at the prescribed time t$t$.     

Characteristic vector fields of generic distributions of corank 2

We study generic distributions D⊂TM$D\subset TM$ of corank 2 on manifolds M   of dimension n?5$n?5$. We describe singular curves of such distributions, also called abnormal curves. For n   even the singular directions (tangent to singular curves) are discrete lines in D(x)$D(x)$, while for n   odd they form a Veronese curve in a projectivized subspace of D(x)$D(x)$, at generic x∈M$x\in M$. We show that singular curves of a generic distribution determine the distribution on the subset of M where they generate at least two different directions. In particular, this happens on the whole of M if n is odd. The distribution is determined by characteristic vector fields and their Lie brackets of appropriate order. We characterize pairs of vector fields which can appear as characteristic vector fields of a generic corank 2 distribution, when n is even.     

The Deodhar decomposition of the Grassmannian and the regularity of KP solitons

Given a point A$A$ in the real Grassmannian, it is well-known that one can construct a soliton solution _uA(x,y,t)$u_A(x,y,t)$ to the KP equation. The contour plot   of such a solution provides a tropical approximation to the solution when the variables x$x$, y$y$, and t$t$ are considered on a large scale and the time t$t$ is fixed. In this paper we use several decompositions of the Grassmannian in order to gain an understanding of the contour plots of the corresponding soliton solutions. First we use the positroid stratification   of the real Grassmannian in order to characterize the unbounded line-solitons in the contour plots at y?0$y?0$ and y?0$y?0$. Next we use the Deodhar decomposition   of the Grassmannian–a refinement of the positroid stratification–to study contour plots at t?0$t?0$. More specifically, we index the components of the Deodhar decomposition of the Grassmannian by certain tableaux which we call Go-diagrams  , and then use these Go-diagrams to characterize the contour plots of solitons solutions when t?0$t?0$. Finally we use these results to show that a soliton solution _uA(x,y,t)$u_A(x,y,t)$ is regular for all times t$t$ if and only if A$A$ comes from the totally non-negative part of the Grassmannian.     

Global attraction to solitary waves for Klein–Gordon equation with mean field interaction

We consider a U(1)$\mathbf{U}(1)$-invariant nonlinear Klein–Gordon equation in dimension n?1$n?1$, self-interacting via the mean field mechanism. We analyze the long-time asymptotics of finite energy solutions and prove that, under certain generic assumptions, each solution converges as t→±∞$t\to \pm \infty$ to the two-dimensional set of all “nonlinear eigenfunctions” of the form ?(x)e−iωt$?(x)e^{-i\omega t}$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.