In two earlier papers, two of the present authors (A.G. and U.S.) extended Lai’s [Ann. Probab. 2 (1974) 432–440] law of the single logarithm for delayed sums to a multiindex setting in which the edges of the $\mathbf{n}$th window grow like $|\mathbf{n}|^α$, or with different $α$’s, where the $α$’s belong to $(0, 1)$. In this paper, the edge of the $n$th window typically grows like $n / \log n$, thus at a higher rate than any power less than one, but not quite at the LIL-rate.

We derive new explicit bounds for the total variation distance between two convolution products of $n∈ℕ$ probability distributions, one of which having identical convolution factors. Approximations by finite signed measures of arbitrary order are considered as well. We are interested in bounds with magic factors, that is, roughly speaking n also appears in the denominator. Special emphasis is given to the approximation by the $n$-fold convolution of the arithmetic mean of the distributions under consideration. As an application, we consider the multinomial approximation of the generalized multinomial distribution. It turns out that here the order of some bounds given in Roos (Theory Probab. Appl. 46 (2001) 103–117) and Loh (Ann. Appl. Probab. 2 (1992) 536–554) can significantly be improved. In particular, it follows that a dimension factor can be dropped. Moreover, better accuracy is achieved in the context of symmetric distributions with finite support. In the course of proof, we use a basic Banach algebra technique for measures on a measurable Abelian group. Though this method was already used by Le Cam (Pacific J. Math. 10 (1960) 1181–1197), our central arguments seem to be new. We also derive new smoothness bounds for convolutions of probability distributions, which might be of independent interest.

We consider a positive stationary generalized Ornstein–Uhlenbeck process $$V_t=e^{−ξ_t} \left( ∫_0^te^{ ξ_{s−}}dη_s+V_0 \right) \mathrm{for}\quad t≥0,$$ and the increments of the integrated generalized Ornstein–Uhlenbeck process $I_{k}=\int_{k-1}^{k}\sqrt{V_{t-}}\,\mathrm{d}L_{t}$, $k∈ℕ$, where $(ξ_t, η_t, L_t)_{t≥0}$ is a three-dimensional Lévy process independent of the starting random variable $V_0$. The genOU model is a continuous-time version of a stochastic recurrence equation. Hence, our models include, in particular, continuous-time versions of ARCH(1) and GARCH(1, 1) processes. In this paper we investigate the asymptotic behavior of extremes and the sample autocovariance function of $(V_t)_{t≥0}$ and $(I_k)_{k∈ℕ}$. Furthermore, we present a central limit result for $(I_k)_{k∈ℕ}$. Regular variation and point process convergence play a crucial role in establishing the statistics of $(V_t)_{t≥0}$ and $(I_k)_{k∈ℕ}$. The theory can be applied to the COGARCH$(1, 1)$ and the Nelson diffusion model.

Multivariate COGARCH(1, 1) processes are introduced as a continuous-time models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate Lévy process and the latent time-varying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices.

After defining the COGARCH(1, 1) process, we analyze its probabilistic properties. We show a sufficient condition for the existence of a stationary distribution for the stochastic covariance matrix process and present criteria ensuring the finiteness of moments. Under certain natural assumptions on the moments of the driving Lévy process, explicit expressions for the first and second-order moments and (asymptotic) second-order stationarity of the covariance matrix process are obtained. Furthermore, we study the stationarity and second-order structure of the increments of the multivariate COGARCH(1, 1) process and their “squares”.

This paper deals with the ergodicity (convergence of the marginals) and the law of large numbers for adaptive MCMC algorithms built from transition kernels that are not necessarily geometrically ergodic. We develop a number of results that significantly broaden the class of adaptive MCMC algorithms for which rigorous analysis is now possible. As an example, we give a detailed analysis of the adaptive Metropolis algorithm of Haario et al. [Bernoulli 7 (2001) 223–242] when the target distribution is subexponential in the tails.

The study of properties of mean functionals of random probability measures is an important area of research in the theory of Bayesian nonparametric statistics. Many results are now known for random Dirichlet means, but little is known, especially in terms of posterior distributions, for classes of priors beyond the Dirichlet process. In this paper, we consider normalized random measures with independent increments (NRMI’s) and mixtures of NRMI. In both cases, we are able to provide exact expressions for the posterior distribution of their means. These general results are then specialized, leading to distributional results for means of two important particular cases of NRMI’s and also of the two-parameter Poisson–Dirichlet process.

A common belief in high-dimensional data analysis is that data are concentrated on a low-dimensional manifold. This motivates simultaneous dimension reduction and regression on manifolds. We provide an algorithm for learning gradients on manifolds for dimension reduction for high-dimensional data with few observations. We obtain generalization error bounds for the gradient estimates and show that the convergence rate depends on the intrinsic dimension of the manifold and not on the dimension of the ambient space. We illustrate the efficacy of this approach empirically on simulated and real data and compare the method to other dimension reduction procedures.

We study Markov bases of decomposable graphical models consisting of primitive moves (i.e., square-free moves of degree two) by determining the structure of fibers of sample size two. We show that the number of elements of fibers of sample size two are powers of two and we characterize primitive moves in Markov bases in terms of connected components of induced subgraphs of the independence graph of a hierarchical model. This allows us to derive a complete description of minimal Markov bases and minimal invariant Markov bases for decomposable models.

A large deviation principle is established for a general class of stochastic flows in the small noise limit. This result is then applied to a Bayesian formulation of an image matching problem, and an approximate maximum likelihood property is shown for the solution of an optimization problem involving the large deviations rate function.

In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator: \[ \cases{ \dfrac{\partial u}{\partial t}(t,x )+\mathcal{L}_{t}u(t,x )+f (t,x,u (t,x )) \in\partial\varphi (u (t,x)), & $\quad t\in[ 0,T) ,x\in\mathbb{R}^{d},$\cr u( T,x ) =h(x), & $\quad x\in\mathbb{R}^{d},$ } \] where $\partial\varphi$ is the subdifferential operator of the proper convex lower semicontinuous function $\varphi:\mathbb{R}^{k}\rightarrow (-\infty,+\infty]$ and $\mathcal{L}_{t}$ is a second differential operator given by $\mathcal{L}_{t}v_{i}(x)=\frac{1}{2}\operatorname{Tr}% [\sigma(t,x)\sigma^{\ast}(t,x)\mathrm{D}^{2}v_{i}(x) ]+ \langle b(t,x),\nabla v_{i}(x) \rangle$, $i\in\overline{1,k}$.

We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution $u: [ 0,T ] \times\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$ of the above parabolic variational inequality.

Measurement error data or errors-in-variable data have been collected in many studies. Natural criterion functions are often unavailable for general functional measurement error models due to the lack of information on the distribution of the unobservable covariates. Typically, the parameter estimation is via solving estimating equations. In addition, the construction of such estimating equations routinely requires solving integral equations, hence the computation is often much more intensive compared with ordinary regression models. Because of these difficulties, traditional best subset variable selection procedures are not applicable, and in the measurement error model context, variable selection remains an unsolved issue. In this paper, we develop a framework for variable selection in measurement error models via penalized estimating equations. We first propose a class of selection procedures for general parametric measurement error models and for general semi-parametric measurement error models, and study the asymptotic properties of the proposed procedures. Then, under certain regularity conditions and with a properly chosen regularization parameter, we demonstrate that the proposed procedure performs as well as an oracle procedure. We assess the finite sample performance via Monte Carlo simulation studies and illustrate the proposed methodology through the empirical analysis of a familiar data set.