An extension of a logarithmic form of Cramér’s ruin theorem to some FARIMA and related processes

Cramér’s theorem provides an estimate for the tail probability of the maximum of a random walk with negative drift and increments having a moment generating function finite in a neighborhood of the origin. The class of (g,F)$(g,F)$-processes generalizes in a natural way random walks and fractional ARIMA models used in time series analysis. For those (g,F)$(g,F)$-processes with negative drift, we obtain a logarithmic estimate of the tail probability of their maximum, under conditions comparable to Cramér’s. Furthermore, we exhibit the most likely paths as well as the most likely behavior of the innovations leading to a large maximum.     

Occupation time distributions for the telegraph process

For the one-dimensional telegraph process, we obtain explicitly the distribution of the occupation time of the positive half-line. The long-term limiting distribution is then derived when the initial location of the process is in the range of subnormal or normal deviations from the origin; in the former case, the limit is given by the arcsine law. These limit theorems are also extended to the case of more general occupation-type functionals.     

Weakly dependent chains with infinite memory

We prove the existence of a weakly dependent strictly stationary solution of the equation _Xt=F(_Xt−1,_Xt−2,_Xt−3,…;_ξt)$X_t=F(X_{t-1},X_{t-2},X_{t-3},\dots ;{\xi }_t)$ called a chain with infinite memory. Here the innovations  _ξt${\xi }_t$ constitute an independent and identically distributed sequence of random variables. The function F$F$ takes values in some Banach space and satisfies a Lipschitz-type condition. We also study the interplay between the existence of moments, the rate of decay of the Lipschitz coefficients of the function F$F$ and the weak dependence properties. From these weak dependence properties, we derive strong laws of large number, a central limit theorem and a strong invariance principle.     

Weighted branching and a pathwise renewal equation

This paper is devoted to the study of a pathwise renewal equation for stochastic processes which are functions of a weighted tree defined in a general weighted branching model. Motivated by applications in the analysis of certain stochastic fixed-point equations and in the theory of general (Crump–Mode–Jagers) branching processes, we analyze the solutions to the equation under several conditions, the main result being a characterization of the set of solutions satisfying appropriate integrability conditions.     

The point process approach for fractionally differentiated random walks under heavy traffic

We prove some heavy-traffic limit theorems for some nonstationary linear processes which encompass the fractionally differentiated random walk as well as some FARIMA processes, when the innovations are in the domain of attraction of a non-Gaussian stable distribution. The results are based on an extension of the point process methodology to linear processes with nonsummable coefficients and make use of a new maximal type inequality.     

Geodesics in CAT(0) cubical complexes

We describe an algorithm to compute the geodesics in an arbitrary CAT(0) cubical complex. A key tool is a correspondence between cubical complexes of global non-positive curvature and posets with inconsistent pairs. This correspondence also gives an explicit realization of such a complex as the state complex of a reconfigurable system, and a way to embed any interval in the integer lattice cubing of its dimension.     

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.     

Large deviations for statistics of the Jacobi process

This paper aims to derive large deviations for statistics of the Jacobi process already conjectured by M. Zani in her thesis. To proceed, we write in a simpler way the Jacobi semi-group density. Being given by a bilinear sum involving Jacobi polynomials, it differs from Hermite and Laguerre cases by the quadratic form of its eigenvalues. Our attempt relies on subordinating the process using a suitable random time change. This gives a Mehler-type formula whence we recover the desired semi-group density. Once we do, an adaptation of Zani’s result [M. Zani, Large deviations for squared radial Ornstein–Uhlenbeck processes, Stochastic. Process. Appl. 102 (1) (2002) 25–42] to the non-steep case will provide the required large deviations principle.     

On diffusivity of a tagged particle in asymmetric zero-range dynamics

Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑jp(j)≠0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d?1, and at most order t in d=1 and d?3 for a wide class of rates g. Also, in d=1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g(k) increases, and g(k)/k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.     

Poisson type approximations for the Markov binomial distribution

The Markov binomial distribution is approximated by the Poisson distribution with the same mean, by a translated Poisson distribution and by two-parametric Poisson type signed measures. Using an adaptation of Le Cam’s operator technique, estimates of accuracy are proved for the total variation, local and Wasserstein norms. In a special case, asymptotically sharp constants are obtained. For some auxiliary results, we used Stein’s method.     

White noise driven SPDEs with reflection: Existence,uniqueness and large deviation principles

In the first part of this paper, we prove the uniqueness of the solutions of SPDEs with reflection, which was left open in the paper [C. Donati-Martin, E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields 95 (1993) 1–24]. We also obtain the existence of the solution for more general coefficients depending on the past with a much shorter proof. In the second part of the paper, we establish a large deviation principle for SPDEs with reflection. The weak convergence approach is proven to be very efficient on this occasion.     

A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter

By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we illustrate our results by simulations.     

Intrinsic volumes of the maximal polytope process in higher dimensional STIT tessellations

Stationary and isotropic iteration stable random tessellations are considered, which are constructed by a random process of iterative cell division. The collection of maximal polytopes at a fixed time t within a convex window W⊂R^d is regarded and formulas for mean values, variances and a characterization of certain covariance measures are proved. The focus is on the case d≥3, which is different from the planar one, treated separately in Schreiber and Thäle (2010) [12]. Moreover, a limit theorem for suitably rescaled intrinsic volumes is established, leading — in sharp contrast to the situation in the plane — to a non-Gaussian limit.     

On the ruin probability in a dependent discrete time risk model with insurance and financial risks

This paper considers the discrete-time risk model with insurance risk and financial risk in some dependence structures. Under assumptions that the insurance risks are heavy tailed (belong to the intersection of the long-tailed class and the dominatedly varying-tailed class) and the financial risks satisfy some moment conditions, the asymptotic and uniformly asymptotic relations for the finite-time and ultimate ruin probabilities are derived.     

An asymptotic theory for sample covariances of Bernoulli shifts

Covariances play a fundamental role in the theory of stationary processes and they can naturally be estimated by sample covariances. There is a well-developed asymptotic theory for sample covariances of linear processes. For nonlinear processes, however, many important problems on their asymptotic behaviors are still unanswered. The paper presents a systematic asymptotic theory for sample covariances of nonlinear time series. Our results are applied to the test of correlations.     

Localization and completion of nilpotent groups of automorphisms

We study nilpotent subgroups of automorphism groups in the category of groups and the homotopy category of spaces. We establish localization and completion theorems for nilpotent groups of automorphisms of nilpotent groups. We then apply these algebraic theorems to prove analogous results for certain groups of self-homotopy equivalences of spaces.