A necessary and sufficient condition for a Lévy process $X$ to stay positive, in probability, near 0, which arises in studies of Chung-type laws for $X$ near 0, is given in terms of the characteristics of $X$.

A functional limit theorem for the partial maxima of a long memory stable sequence produces a limiting process that can be described as a $\beta$-power time change in the classical Fréchet extremal process, for $\beta$ in a subinterval of the unit interval. Any such power time change in the extremal process for $0<\beta<1$ produces a process with stationary max-increments. This deceptively simple time change hides the much more delicate structure of the resulting process as a self-affine random sup measure. We uncover this structure and show that in a certain range of the parameters this random measure arises as a limit of the partial maxima of the same long memory stable sequence, but in a different space. These results open a way to construct a whole new class of self-similar Fréchet processes with stationary max-increments.

Determinantal point processes (DPPs) have recently proved to be a useful class of models in several areas of statistics, including spatial statistics, statistical learning and telecommunications networks. They are models for repulsive (or regular, or inhibitive) point processes, in the sense that nearby points of the process tend to repel each other. We consider two ways to quantify the repulsiveness of a point process, both based on its second-order properties, and we address the question of how repulsive a stationary DPP can be. We determine the most repulsive stationary DPP, when the intensity is fixed, and for a given $R>0$ we investigate repulsiveness in the subclass of $R$-dependent stationary DPPs, that is, stationary DPPs with $R$-compactly supported kernels. Finally, in both the general case and the $R$-dependent case, we present some new parametric families of stationary DPPs that can cover a large range of DPPs, from the stationary Poisson process (the case of no interaction) to the most repulsive DPP.

Cramér type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cramér type moderate deviation theorem for general self-normalized processes which include many well-known Studentized nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized $U$-statistics.

Zellner’s $g$-prior is a popular prior choice for the model selection problems in the context of normal regression models. Wang and Sun [J. Statist. Plann. Inference 147 (2014) 95–105] recently adopt this prior and put a special hyper-prior for $g$, which results in a closed-form expression of Bayes factor for nested linear model comparisons. They have shown that under very general conditions, the Bayes factor is consistent when two competing models are of order $O(n^{\tau})$ for $\tau <1$ and for $\tau=1$ is almost consistent except a small inconsistency region around the null hypothesis. In this paper, we study Bayes factor consistency for nonnested linear models with a growing number of parameters. Some of the proposed results generalize the ones of the Bayes factor for the case of nested linear models. Specifically, we compare the asymptotic behaviors between the proposed Bayes factor and the intrinsic Bayes factor in the literature.

We prove a local limit theorem, that is, a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional Gaussian framework are played by the Ornstein–Uhlenbeck semigroup and Wick product, respectively. We proceed by establishing a necessary condition on the density of the random variables for the local limit theorem to hold true. We then reverse the implication and prove under an additional assumption the desired $\mathcal{L}^{1}$-convergence of the density of $\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}$. We close the paper comparing our result with certain Berry–Esseen bounds for multidimensional central limit theorems.

We generalize the SiZer of Chaudhuri and Marron (J. Amer. Statist. Assoc. 94 (1999) 807–823; Ann. Statist. 28 (2000) 408–428) for the detection of shape parameters of densities on the real line to the case of circular data. It turns out that only the wrapped Gaussian kernel gives a symmetric, strongly Lipschitz semi-group satisfying “circular” causality, that is, not introducing possibly artificial modes with increasing levels of smoothing. Some notable differences between Euclidean and circular scale space theory are highlighted. Based on this, we provide an asymptotic theory to make inference about the persistence of shape features. The resulting circular mode persistence diagram is applied to the analysis of early mechanically-induced differentiation in adult human stem cells from their actin-myosin filament structure. As a consequence, the circular SiZer based on the wrapped Gaussian kernel (WiZer) allows the verification at a controlled error level of the observation reported by Zemel et al. (Nat. Phys. 6 (2010) 468–473): Within early stem cell differentiation, polarizations of stem cells exhibit preferred directions in three different micro-environments.

We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, that is, they are iteratively forwarded from participant to participant over distances smaller than the communication radius until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model.

We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random and a global, deterministic mechanism, and we give a formula for the limiting behaviour.

A prime example of the movement schemes that we consider is the well-known random waypoint model. Here, we give a negative upper bound for the decay rate, in the limit of large time horizons, of the probability of the event that the portion of the connection time is less than the expectation.

Let $(X_{1},\ldots,X_{n})$ be an i.i.d. sequence of random variables in $\mathbb{R}^{d}$, $d\geq 1$. We show that, for any function $\varphi:\mathbb{R}^{d}\rightarrow\mathbb{R}$, under regularity conditions,

\[n^{1/2}(n^{-1}\sum_{i=1}^{n}\frac{\varphi(X_{i})}{\widehat{f}(X_{i})}-\int \varphi(x)\,dx){\longrightarrow}^{\mathbb{P}}0,\] where $\widehat{f}$ is the classical kernel estimator of the density of $X_{1}$. This result is striking because it speeds up traditional rates, in root $n$, derived from the central limit theorem when $\widehat{f}=f$. Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions $\varphi$ and $f$, the dimension $d$ and the bandwidth of the kernel estimator $\widehat{f}$. Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of $\varphi$. As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations.

We consider a stochastic model describing a constant size $N$ population that may be seen as a directed polymer in random medium with $N$ sites in the transverse direction. The population dynamics is governed by a noisy traveling wave equation describing the evolution of the individual fitnesses. We show that under suitable conditions the generations are independent and the model is characterized by an extended Wright–Fisher model, in which the individual $i$ has a random fitness $\eta_{i}$ and the joint distribution of offspring $(\nu_{1},\ldots,\nu_{N})$ is given by a multinomial law with $N$ trials and probability outcomes $\eta_{i}$’s. We then show that the average coalescence times scales like $\log N$ and that the limit genealogical trees are governed by the Bolthausen–Sznitman coalescent, which validates the predictions by Brunet, Derrida, Mueller and Munier for this class of models. We also study the extended Wright–Fisher model, and show that, under certain conditions on $\eta_{i}$, the limit may be Kingman’s coalescent, a coalescent with multiple collisions, or a coalescent with simultaneous multiple collisions.

A stochastic algorithm is proposed, finding some elements from the set of intrinsic $p$-mean(s) associated to a probability measure $\nu$ on a compact Riemannian manifold and to $p\in[1,\infty)$. It is fed sequentially with independent random variables $(Y_{n})_{n\in\mathbb{N}}$ distributed according to $\nu$, which is often the only available knowledge of $\nu$. Furthermore, the algorithm is easy to implement, because it evolves like a Brownian motion between the random times when it jumps in direction of one of the $Y_{n}$, $n\in\mathbb{N}$. Its principle is based on simulated annealing and homogenization, so that temperature and approximations schemes must be tuned up (plus a regularizing scheme if $\nu$ does not admit a Hölderian density). The analysis of the convergence is restricted to the case where the state space is a circle. In its principle, the proof relies on the investigation of the evolution of a time-inhomogeneous $\mathbb{L}^{2}$ functional and on the corresponding spectral gap estimates due to Holley, Kusuoka and Stroock. But it requires new estimates on the discrepancies between the unknown instantaneous invariant measures and some convenient Gibbs measures.

We characterize the combinatorial structure of conditionally-i.i.d. sequences of negative binomial processes with a common beta process base measure. In Bayesian nonparametric applications, such processes have served as models for latent multisets of features underlying data. Analogously, random subsets arise from conditionally-i.i.d. sequences of Bernoulli processes with a common beta process base measure, in which case the combinatorial structure is described by the Indian buffet process. Our results give a count analogue of the Indian buffet process, which we call a negative binomial Indian buffet process. As an intermediate step toward this goal, we provide a construction for the beta negative binomial process that avoids a representation of the underlying beta process base measure. We describe the key Markov kernels needed to use a NB-IBP representation in a Markov Chain Monte Carlo algorithm targeting a posterior distribution.

Distributional identities for a Lévy process $X_{t}$, its quadratic variation process $V_{t}$ and its maximal jump processes, are derived, and used to make “small time” (as $t\downarrow0$) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of $X$. Apart from providing insight into the connections between $X$, $V$, and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study “self-normalised” versions of $X_{t}$, that is, $X_{t}$ after division by $\sup_{0<s\le t}\Delta X_{s}$, or by $\sup_{0<s\le t}\vert \Delta X_{s}\vert $. Thus, we obtain necessary and sufficient conditions for $X_{t}/\sup_{0<s\le t}\Delta X_{s}$ and $X_{t}/\sup_{0<s\le t}\vert \Delta X_{s}\vert $ to converge in probability to 1, or to $\infty$, as $t\downarrow0$, so that $X$ is either comparable to, or dominates, its largest jump. The former situation tends to occur when the singularity at 0 of the Lévy measure of $X$ is fairly mild (its tail is slowly varying at 0), while the latter situation is related to the relative stability or attraction to normality of $X$ at 0 (a steeper singularity at 0). An important component in the analyses is the way the largest positive and negative jumps interact with each other. Analogous “large time” (as $t\to\infty$) versions of the results can also be obtained.

This paper establishes expectation and variance asymptotics for statistics of the Poisson–Voronoi approximation of general sets, as the underlying intensity of the Poisson point process tends to infinity. Statistics of interest include volume, surface area, Hausdorff measure, and the number of faces of lower-dimensional skeletons. We also consider the complexity of the so-called Voronoi zone and the iterated Voronoi approximation. Our results are consequences of general limit theorems proved with an abstract Steiner-type formula applicable in the setting of sums of stabilizing functionals.

We introduce the notions of scaling transition and distributional long-range dependence for stationary random fields $Y$ on $\mathbb{Z}^{2}$ whose normalized partial sums on rectangles with sides growing at rates $O(n)$ and $O(n^{\gamma})$ tend to an operator scaling random field $V_{\gamma}$ on $\mathbb{R}^{2}$, for any $\gamma>0$. The scaling transition is characterized by the fact that there exists a unique $\gamma_{0}>0$ such that the scaling limits $V_{\gamma}$ are different and do not depend on $\gamma$ for $\gamma>\gamma_{0}$ and $\gamma<\gamma_{0}$. The existence of scaling transition together with anisotropic and isotropic distributional long-range dependence properties is demonstrated for a class of $\alpha$-stable $(1<\alpha\le2)$ aggregated nearest-neighbor autoregressive random fields on $\mathbb{Z}^{2}$ with a scalar random coefficient $A$ having a regularly varying probability density near the “unit root” $A=1$.

The aim of this paper is to study the asymptotic expansion in total variation in the central limit theorem when the law of the basic random variable is locally lower-bounded by the Lebesgue measure (or equivalently, has an absolutely continuous component): we develop the error in powers of $n^{-1/2}$ and give an explicit formula for the approximating measure.

We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of Itô’s integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlled rough path integral. We prove that every “typical price path” has a naturally associated Itô rough path, and justify the application of the controlled rough path integral in finance by showing that it is the limit of non-anticipating Riemann sums, a new result in itself. Compared to the first approach, rough paths have the disadvantage of severely restricting the space of integrands, but the advantage of being a Banach space theory.

Both approaches are based entirely on financial arguments and do not require any probabilistic structure.

In decision theoretic estimation of parameters in Euclidean space $\mathbb{R}^{p}$, the action space is chosen to be the convex closure of the estimand space. In this paper, the concept has been extended to the estimation of circular parameters of distributions having support as a circle, torus or cylinder. As directional distributions are of curved nature, existing methods for distributions with parameters taking values in $\mathbb{R}^{p}$ are not immediately applicable here. A circle is the simplest one-dimensional Riemannian manifold. We employ concepts of convexity, projection, etc., on manifolds to develop sufficient conditions for inadmissibility of estimators for circular parameters. Further invariance under a compact group of transformations is introduced in the estimation problem and a complete class theorem for equivariant estimators is derived. This extends the results of Moors [J. Amer. Statist. Assoc. 76 (1981) 910–915] on $\mathbb{R}^{p}$ to circles. The findings are of special interest to the case when a circular parameter is truncated. The results are implemented to a wide range of directional distributions to obtain improved estimators of circular parameters.

For $\alpha\in (1,2)$, we present a generalized central limit theorem for $\alpha$-stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for partial integro-differential equations (PIDEs). A classical generalized central limit theorem is recovered as a special case, provided a mild but natural additional condition holds. Our approach contrasts with previous arguments for the result in the linear setting which have typically relied upon tools that are non-existent in the sublinear framework, for example, characteristic functions.

We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein–Uhlenbeck processes driven by Lévy motion and their finite and infinite superpositions. We present the general conditions for the $\mathcal{L}_{q}$ convergence of cumulative processes to the limiting processes and investigate their $q$th order moments and Rényi functions, which are non-linear, hence displaying the multifractality of the processes as constructed. We also establish the corresponding scenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normal tempered stable scenarios.