In this article, we derive the weak limiting distribution of the least squares estimator (LSE) of a convex probability mass function (p.m.f.) with a finite support. We show that it can be defined via a certain convex projection of a Gaussian vector. Furthermore, samples of any given size from this limit distribution can be generated using an efficient Dykstra-like algorithm.

The paper derives saddlepoint expansions for conditional expectations in the form of $\mathsf{E}[\overline{X}|\overline{\mathbf{Y}}=\mathbf{a}]$ and $\mathsf{E}[\overline{X}|\overline{\mathbf{Y}}\geq\mathbf{a}]$ for the sample mean of a continuous random vector $(X,\mathbf{Y}^{\top})$ whose joint moment generating function is available. Theses conditional expectations frequently appear in various applications, particularly in quantitative finance and risk management. Using the newly developed saddlepoint expansions, we propose fast and accurate methods to compute the sensitivities of risk measures such as value-at-risk and conditional value-at-risk, and the sensitivities of financial options with respect to a market parameter. Numerical studies are provided for the accuracy verification of the new approximations.

In this paper, we characterize (mixtures of) bridges of a continuous time random walk with values in a countable Abelian group. Our main tool is a conditional version of Mecke’s formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of such random processes. We observe in particular how their structure depends on the algebraic properties of the underlying group.

Predictive constructions are a powerful way of characterizing the probability laws of stochastic processes with certain forms of invariance, such as exchangeability or Markov exchangeability. When de Finetti-like representation theorems are available, the predictive characterization implicitly defines the prior distribution, starting from assumptions on the observables; moreover, it often helps in designing efficient computational strategies. In this paper we give necessary and sufficient conditions on the sequence of predictive distributions such that they characterize a Markov exchangeable probability law for a discrete valued process $\mathbf{X}$. Under recurrence, Markov exchangeable processes are mixtures of Markov chains. Our predictive conditions are in some sense minimal sufficient conditions for Markov exchangeability; we also provide predictive conditions for recurrence. We illustrate their application in relevant examples from the literature and in novel constructions.

Let $\{X(t)=(X_{1}(t),X_{2}(t))^{T},t\in\mathbb{R}^{N}\}$ be an $\mathbb{R}^{2}$-valued continuous locally stationary Gaussian random field with $\mathbb{E}[X(t)]=\mathbf{0}$. For any compact sets $A_{1},A_{2}\subset\mathbb{R}^{N}$, precise asymptotic behavior of the excursion probability \[\mathbb{P}(\max_{s\in A_{1}}X_{1}(s)>u,\max_{t\in A_{2}}X_{2}(t)>u)\qquad\mbox{as }u\rightarrow\infty\] is investigated by applying the double sum method. The explicit results depend not only on the smoothness parameters of the coordinate fields $X_{1}$ and $X_{2}$, but also on their maximum correlation $\rho$.

We consider the problem of testing whether two independent finite-dimensional random dot product graphs have generating latent positions that are drawn from the same distribution, or distributions that are related via scaling or projection. We propose a test statistic that is a kernel-based function of the estimated latent positions obtained from the adjacency spectral embedding for each graph. We show that our test statistic using the estimated latent positions converges to the test statistic obtained using the true but unknown latent positions and hence that our proposed test procedure is consistent across a broad range of alternatives. Our proof of consistency hinges upon a novel concentration inequality for the suprema of an empirical process in the estimated latent positions setting.

We establish general moment estimates for the discrete and continuous exit times of a general Itô process in terms of the distance to the boundary. These estimates serve as intermediate steps to obtain strong convergence results for the approximation of a continuous exit time by a discrete counterpart, computed on a grid. In particular, we prove that the discrete exit time of the Euler scheme of a diffusion converges in the ${\mathbf{L}}_{1}$ norm with an order $1/2$ with respect to the mesh size. This rate is optimal.

Motivated by recent work on ordinal embedding (In Proceedings of the 27th Conference on Learning Theory (2014) 40–67), we derive large sample consistency results and rates of convergence for the problem of embedding points based on triple or quadruple distance comparisons. We also consider a variant of this problem where only local comparisons are provided. Finally, inspired by (In Communication, Control, and Computing (Allerton), 2011 49th Annual Allerton Conference on (2011) 1077–1084 IEEE), we bound the number of such comparisons needed to achieve consistency.

The present paper considers the constrained optimal control problem with total undiscounted criteria for a continuous-time Markov decision process (CTMDP) in Borel state and action spaces. The cost rates are nonnegative. Under the standard compactness and continuity conditions, we show the existence of an optimal stationary policy out of the class of general nonstationary ones. In the process, we justify the reduction of the CTMDP model to a discrete-time Markov decision process (DTMDP) model based on the studies of the undiscounted occupancy and occupation measures. We allow that the controlled process is not necessarily absorbing, and the transition rates are not necessarily separated from zero, and can be arbitrarily unbounded; these features count for the main technical difficulties in studying undiscounted CTMDP models.

We introduce a class of two-sex branching processes in discrete time where, in each generation, mating between females and males is randomly governed by a set of Bernoulli distributions allowing polygamous behaviour with only perfect fidelity on the part of female. Moreover, mating as well as reproduction can be influenced by the number of females and males in the population. We study here, for any population whose dynamics is modeled by such processes, conditions leading to its extinction or to a possible persistence. Moreover, the behaviours of the female and male populations are analyzed more finely in case of persistence.

The problem of universal search and stop using an adaptive search policy is considered. When the unique target location is searched, the observation is distributed according to the target distribution, otherwise it is distributed according to the absence distribution. A universal scheme for search and stop is proposed using only the knowledge of the absence distribution, and its asymptotic performance is analyzed. The universal test is shown to yield a vanishing error probability, and to achieve the optimal reliability when the target is present, universally for every target distribution. Consequently, it is established that the knowledge of the target distribution is only useful for improving the reliability for detecting that the target is missing. It is also shown that a multiplicative gain for the search reliability equal to the number of searched locations is achieved by allowing adaptivity in the search.

We consider a branching-selection particle system on the real line. In this model, the total size of the population at time $n$ is limited by $\exp (an^{1/3})$. At each step $n$, every individual dies while reproducing independently, making children around their current position according to i.i.d. point processes. Only the $\exp (a(n+1)^{1/3})$ rightmost children survive to form the $(n+1)$th generation. This process can be seen as a generalisation of the branching random walk with selection of the $N$ rightmost individuals, introduced by Brunet and Derrida (Phys. Rev. E (3) 56 (1997) 2597–2604). We obtain the asymptotic behaviour of position of the extremal particles alive at time $n$ by coupling this process with a branching random walk with a killing boundary.

We propose a new empirical Bayes approach for inference in the $p\gg n$ normal linear model. The novelty is the use of data in the prior in two ways, for centering and regularization. Under suitable sparsity assumptions, we establish a variety of concentration rate results for the empirical Bayes posterior distribution, relevant for both estimation and model selection. Computation is straightforward and fast, and simulation results demonstrate the strong finite-sample performance of the empirical Bayes model selection procedure.

Copula modeling has become ubiquitous in modern statistics. Here, the problem of nonparametrically estimating a copula density is addressed. Arguably the most popular nonparametric density estimator, the kernel estimator is not suitable for the unit-square-supported copula densities, mainly because it is heavily affected by boundary bias issues. In addition, most common copulas admit unbounded densities, and kernel methods are not consistent in that case. In this paper, a kernel-type copula density estimator is proposed. It is based on the idea of transforming the uniform marginals of the copula density into normal distributions via the probit function, estimating the density in the transformed domain, which can be accomplished without boundary problems, and obtaining an estimate of the copula density through back-transformation. Although natural, a raw application of this procedure was, however, seen not to perform very well in the earlier literature. Here, it is shown that, if combined with local likelihood density estimation methods, the idea yields very good and easy to implement estimators, fixing boundary issues in a natural way and able to cope with unbounded copula densities. The asymptotic properties of the suggested estimators are derived, and a practical way of selecting the crucially important smoothing parameters is devised. Finally, extensive simulation studies and a real data analysis evidence their excellent performance compared to their main competitors.

Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimating functions under which estimators are consistent, rate optimal, and efficient under high frequency (in-fill) asymptotics. The asymptotic distributions of the estimators are shown to be normal variance-mixtures, where the mixing distribution generally depends on the full sample path of the diffusion process over the observation time interval. Utilising the concept of stable convergence, we also obtain the more easily applicable result that for a suitable data dependent normalisation, the estimators converge in distribution to a standard normal distribution. The theory is illustrated by a simulation study comparing an efficient and a non-efficient estimating function for an ergodic and a non-ergodic model.

We establish exponential bounds for the hypergeometric distribution which include a finite sampling correction factor, but are otherwise analogous to bounds for the binomial distribution due to León and Perron (Statist. Probab. Lett. 62 (2003) 345–354) and Talagrand (Ann. Probab. 22 (1994) 28–76). We also extend a convex ordering of Kemperman’s (Nederl. Akad. Wetensch. Proc. Ser. A 76 = Indag. Math. 35 (1973) 149–164) for sampling without replacement from populations of real numbers between zero and one: a population of all zeros or ones (and hence yielding a hypergeometric distribution in the upper bound) gives the extreme case.

This paper presents a novel algorithm, the particle-based, rapid incremental smoother (PaRIS), for efficient online approximation of smoothed expectations of additive state functionals in general hidden Markov models. The algorithm, which has a linear computational complexity under weak assumptions and very limited memory requirements, is furnished with a number of convergence results, including a central limit theorem. An interesting feature of PaRIS, which samples on-the-fly from the retrospective dynamics induced by the particle filter, is that it requires two or more backward draws per particle in order to cope with degeneracy of the sampled trajectories and to stay numerically stable in the long run with an asymptotic variance that grows only linearly with time.

We consider quantile regression incorporating polynomial spline approximation for single-index coefficient models. Compared to mean regression, quantile regression for this class of models is more technically challenging and has not been considered before. We use a check loss minimization approach and employed a projection/orthogonalization technique to deal with the theoretical challenges. Compared to previously used kernel estimation approach, which was developed for mean regression only, spline estimation is more computationally expedient and directly produces a smooth estimated curve. Simulations and a real data set is used to illustrate the finite sample properties of the proposed estimator.

In this article, we consider an unbiased simulation method for multidimensional diffusions based on the parametrix method for solving partial differential equations with Hölder continuous coefficients. This Monte Carlo method which is based on an Euler scheme with random time steps, can be considered as an infinite dimensional extension of the Multilevel Monte Carlo method for solutions of stochastic differential equations with Hölder continuous coefficients. In particular, we study the properties of the variance of the proposed method. In most cases, the method has infinite variance and therefore we propose an importance sampling method to resolve this issue.

In this paper, we give variational representations for decay parameters of Markov chains. In continuous-time cases, the representation involves Donsker–Varadhan’s famous $I$-functional, from which some dual representations are given, which are expected to he useful in estimating the lower and upper bounds of the decay parameter. As a consequence, dual representations for decay parameters of discrete time Markov chains are derived. For continuous-time chains with finite states, we also give another form of dual formulas, which can be regarded as a version of the one for the Perron–Frobenius eigenvalue, with nonnegative matrices replaced by $Q$-matrices of the chains. Connections with quasi-stationarity and quasi-ergodicity of absorbing Markov chains are discussed. An interpretation for the corresponding variational solutions is given.

We study the Wasserstein distance of order 1 between the empirical distribution and the marginal distribution of stationary $\alpha$-dependent sequences. We prove some moments inequalities of order $p$ for any $p\geq1$, and we give some conditions under which the central limit theorem holds. We apply our results to unbounded functions of expanding maps of the interval with a neutral fixed point at zero. The moment inequalities for the Wasserstein distance are similar to the well-known von Bahr–Esseen or Rosenthal bounds for partial sums, and seem to be new even in the case of independent and identically distributed random variables.