Let ($X_t, t \geq 0)$ be a Lévy process with compound Poisson process and $τ_x$ be the first passage time of a fixed level $x > 0$ by ($X_t, t \geq 0$). We prove that the law of $τ_x$ has a density (defective when $\mathbb{E}(X_{1})\ < 0)$ with respect to the Lebesgue measure.

Consider the linear stochastic differential equation (SDE) on $ℝ^n$: $$\mathrm{d}X_t = AX_t \mathrm{d}t + B \mathrm{d}L_t,$$ where $A$ is a real $n × n$ matrix, $B$ is a real $n × d$ real matrix and $L_t$ is a Lévy process with Lévy measure $ν$ on $ℝ^d$. Assume that $ν(\mathrm{d}z) ≥ ρ_0(z)\mathrm{d}z$ for some ${ρ_0} \geq 0$. If $A \leq 0$, Rank$(B) = n$ and $∫_{\{|z−z_0| \leq ε\}}ρ_0(z)^{−1} \mathrm{d}z< \infty$ holds for some $z_0 ∈ ℝ^d$ and some $ε > 0$, then the associated Markov transition probability $P_t(x, \mathrm{d}y)$ satisfies $$‖P_t(x, ⋅) − P_t(y, ⋅)‖_{\mathrm{var}} \le \frac{C(1 + |x − y|)}{√t}, x, y ∈ ℝ^d, t > 0,$$ for some constant $C > 0$, which is sharp for large $t$ and implies that the process has successful couplings. The Harnack inequality, ultracontractivity and the strong Feller property are also investigated for the (conditional) transition semigroup.

In this paper we study the asymptotic behaviour of power and multipower variations of processes $Y$: $$Y_t = ∫_{−∞}^tg(t − s)σ_sW(\mathrm{d}s) + Z_t,$$ where $g : (0, ∞) → ℝ$ is deterministic, $σ > 0$ is a random process, $W$ is the stochastic Wiener measure and $Z$ is a stochastic process in the nature of a drift term. Processes of this type serve, in particular, to model data of velocity increments of a fluid in a turbulence regime with spot intermittency $σ$. The purpose of this paper is to determine the probabilistic limit behaviour of the (multi)power variations of $Y$ as a basis for studying properties of the intermittency process $σ$. Notably the processes $Y$ are in general not of the semimartingale kind and the established theory of multipower variation for semimartingales does not suffice for deriving the limit properties. As a key tool for the results, a general central limit theorem for triangular Gaussian schemes is formulated and proved. Examples and an application to the realised variance ratio are given.

Approximations of fractional Brownian motion using Poisson processes whose parameter sets have the same dimensions as the approximated processes have been studied in the literature. In this paper, a special approximation to the one-parameter fractional Brownian motion is constructed using a two-parameter Poisson process. The proof involves the tightness and identification of finite-dimensional distributions.

We study scaling limits of non-increasing Markov chains with values in the set of non-negative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from $n$ and appropriately rescaled, converges in distribution, as $n → ∞$, to a non-increasing self-similar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the self-similar Markov process reaches first 0.

We discuss various applications to the study of random walks with a barrier, of the number of collisions in $Λ$-coalescents that do not descend from infinity and of non-consistent regenerative compositions. Further applications to the scaling limits of Markov branching trees are developed in our paper, Scaling limits of Markov branching trees, with applications to Galton–Watson and random unordered trees (2010).

This paper deals with some self-interacting diffusions $(X_t, t \geq 0)$ living on $ℝ^d$. These diffusions are solutions to stochastic differential equations: $$\mathrm{d}X_t = \mathrm{d}B_t − g(t)∇V(X_t − \overline μ_t) \mathrm{d}t,$$ where $\overline μ_t$ is the empirical mean of the process $X, V$ is an asymptotically strictly convex potential and $g$ is a given function. We study the ergodic behaviour of $X$ and prove that it is strongly related to $g$. Actually, we show that $X$ is ergodic (in the limit quotient sense) if and only if $\overline μ_t$ converges a.s. We also give some conditions (on $g$ and $V$) for the almost sure convergence of $X$.

We consider a class of observation-driven Poisson count processes where the current value of the accompanying intensity process depends on previous values of both processes. We show under a contractive condition that the bivariate process has a unique stationary distribution and that a stationary version of the count process is absolutely regular. Moreover, since the intensities can be written as measurable functionals of the count variables, we conclude that the bivariate process is ergodic. As an important application of these results, we show how a test method previously used in the case of independent Poisson data can be used in the case of Poisson count processes.

Consider a finite renewal process in the sense that interrenewal times are positive i.i.d. variables and the total number of renewals is a random variable, independent of interrenewal times. A finite point process can be obtained by probabilistic sampling of the finite renewal process, where each renewal is sampled with a fixed probability and independently of other renewals. The problem addressed in this work concerns statistical inference of the original distributions of the total number of renewals and interrenewal times from a sample of i.i.d. finite point processes obtained by sampling finite renewal processes. This problem is motivated by traffic measurements in the Internet in order to characterize flows of packets (which can be seen as finite renewal processes) and where the use of packet sampling is becoming prevalent due to increasing link speeds and limited storage and processing capacities.

In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if $f$, $g$ are martingales satisfying \[ |\mathrm{d}g_n|\leq|\mathrm{d}f_n|,\qquad n=0, 1, 2, \ldots, \] almost surely, then \[ \Bigl\|\sup_{n\geq0} |g_n|\Bigr\|_p \leq p \|f\|_p,\qquad p\geq2, \] and the inequality is sharp. Furthermore, if $\alpha\in[0,1]$, $f$ is a non-negative submartingale and $g$ satisfies \[ |\mathrm{d}g_n|\leq|\mathrm{d}f_n|\quad \mbox{and}\quad |\mathbb{E}(\mathrm{d}g_{n+1}|\mathcal {F}_n)|\leq\alpha\mathbb{E} (\mathrm{d}f_{n+1}|\mathcal{F}_n),\qquad n=0, 1, 2, \ldots, \] almost surely, then \[ \Bigl\|\sup_{n\geq0} |g_n|\Bigr\|_p \leq(\alpha+1)p \|f\|_p,\qquad p\geq2, \] and the inequality is sharp. As an application, we establish related estimates for stochastic integrals and Itô processes. The inequalities strengthen the earlier classical results of Burkholder and Choi.

There is a wide literature on change point tests, but the case of variables with infinite variances is essentially unexplored. In this paper we address this problem by studying the asymptotic behavior of trimmed CUSUM statistics. We show that in a location model with i.i.d. errors in the domain of attraction of a stable law of parameter $0 < α < 2$, the appropriately trimmed CUSUM process converges weakly to a Brownian bridge. Thus, after moderate trimming, the classical method for detecting change points remains valid also for populations with infinite variance. We note that according to the classical theory, the partial sums of trimmed variables are generally not asymptotically normal and using random centering in the test statistics is crucial in the infinite variance case. We also show that the partial sums of truncated and trimmed random variables have different asymptotic behavior. Finally, we discuss resampling procedures which enable one to determine critical values in the case of small and moderate sample sizes.

This paper studies $ℓ_1$ regularization with high-dimensional features for support vector machines with a built-in reject option (meaning that the decision of classifying an observation can be withheld at a cost lower than that of misclassification). The procedure can be conveniently implemented as a linear program and computed using standard software. We prove that the minimizer of the penalized population risk favors sparse solutions and show that the behavior of the empirical risk minimizer mimics that of the population risk minimizer. We also introduce a notion of classification complexity and prove that our minimizers adapt to the unknown complexity. Using a novel oracle inequality for the excess risk, we identify situations where fast rates of convergence occur.

The conventional definition of a depth function is vector-based. In this paper, a novel projection depth (PD) technique directly based on tensors, such as matrices, is instead proposed. Tensor projection depth (TPD) is still an ideal depth function and its computation can be achieved through the iteration of PD. Furthermore, we also discuss the cases for sparse samples and higher order tensors. Experimental results in data classification with the two projection depths show that TPD performs much better than PD for data with a natural tensor form, and even when the data have a natural vector form, TPD appears to perform no worse than PD.

An additive growth curve model with orthogonal design matrices is proposed in which observations may have different profile forms. The proposed model allows us to fit data and then estimate parameters in a more parsimonious way than the traditional growth curve model. Two-stage generalized least-squares estimators for the regression coefficients are derived where a quadratic estimator for the covariance of observations is taken as the first-stage estimator. Consistency, asymptotic normality and asymptotic independence of these estimators are investigated. Simulation studies and a numerical example are given to illustrate the efficiency and parsimony of the proposed model for model specifications in the sense of minimizing Akaike’s information criterion (AIC).

For multivariate data, Tukey’s half-space depth is one of the most popular depth functions available in the literature. It is conceptually simple and satisfies several desirable properties of depth functions. The Tukey median, the multivariate median associated with the half-space depth, is also a well-known measure of center for multivariate data with several interesting properties. In this article, we derive and investigate some interesting properties of half-space depth and its associated multivariate median. These properties, some of which are counterintuitive, have important statistical consequences in multivariate analysis. We also investigate a natural extension of Tukey’s half-space depth and the related median for probability distributions on any Banach space (which may be finite- or infinite-dimensional) and prove some results that demonstrate anomalous behavior of half-space depth in infinite-dimensional spaces.