We consider microstructure as an arbitrary contamination of the underlying latent securities price, through a Markov kernel Q. Special cases include additive error, rounding and combinations thereof. Our main result is that, subject to smoothness conditions, the two scales realized volatility is robust to the form of contamination Q. To push the limits of our result, we show what happens for some models that involve rounding (which is not, of course, smooth) and see in this situation how the robustness deteriorates with decreasing smoothness. Our conclusion is that under reasonable smoothness, one does not need to consider too closely how the microstructure is formed, while if severe non-smoothness is suspected, one needs to pay attention to the precise structure and also the use to which the estimator of volatility will be put.

Exponential families are the workhorses of parametric modelling theory. One reason for their popularity is their associated inference theory, which is very clean, both from a theoretical and a computational point of view. One way in which this set of tools can be enriched in a natural and interpretable way is through mixing. This paper develops and applies the idea of local mixture modelling to exponential families. It shows that the highly interpretable and flexible models which result have enough structure to retain the attractive inferential properties of exponential families. In particular, results on identification, parameter orthogonality and log-concavity of the likelihood are proved.

Consider a parametric statistical model P(dx|θ) and an improper prior distribution ν(dθ) that together yield a (proper) formal posterior distribution Q(dθ|x). The prior is called strongly admissible if the generalized Bayes estimator of every bounded function of θ is admissible under squared error loss. Eaton [Ann. Statist. 20 (1992) 1147–1179] has shown that a sufficient condition for strong admissibility of ν is the local recurrence of the Markov chain whose transition function is R(θ, dη)=∫ Q(dη|x)P(dx|θ). Applications of this result and its extensions are often greatly simplified when the Markov chain associated with R is irreducible. However, establishing irreducibility can be difficult. In this paper, we provide a characterization of irreducibility for general state space Markov chains and use this characterization to develop an easily checked, necessary and sufficient condition for irreducibility of Eaton’s Markov chain. All that is required to check this condition is a simple examination of P and ν. Application of the main result is illustrated using two examples.

We derive high-resolution upper bounds for optimal product quantization of pathwise continuous Gaussian processes with respect to the supremum norm on [0, T]^d. Moreover, we describe a product quantization design which attains this bound. This is achieved under very general assumptions on random series expansions of the process. It turns out that product quantization is asymptotically only slightly worse than optimal functional quantization. The results are applied to fractional Brownian sheets and the Ornstein–Uhlenbeck process.

Given a sample from a discretely observed compound Poisson process, we consider estimation of the density of the jump sizes. We propose a kernel type nonparametric density estimator and study its asymptotic properties. An order bound for the bias and an asymptotic expansion of the variance of the estimator are given. Pointwise weak consistency and asymptotic normality are established. The results show that, asymptotically, the estimator behaves very much like an ordinary kernel estimator.

In this note we introduce the notion of Newton–Côtes functionals corrected by Lévy areas, which enables us to consider integrals of the type ∫ f(y) dx, where f is a C^2m function and x, y are real Hölderian functions with index α>1/(2m+1) for all m∈ℕ^*. We show that this concept extends the Newton–Côtes functional introduced in Gradinaru et al., to a larger class of integrands. Then we give a theorem of existence and uniqueness for differential equations driven by x, interpreted using the symmetric Russo–Vallois integral.

Cohen, Guyon, Perrin and Pontier have given assumptions under which the second-order quadratic variations of a Gaussian process converge almost surely to a deterministic limit. In this paper we present two new convergence results about these variations: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we apply these results to identify two-parameter fractional Brownian motion and anisotropic fractional Brownian motion.

Let {X_n, n≥1} be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call X_n a δ-record if X_n>max {X₁, …, X_n−1}+δ, where δ is an integer constant. We use martingale arguments to show that the counting process of δ-records among the first n observations, suitably centered and scaled, is asymptotically normally distributed for δ≠0. In particular, taking δ=−1 we obtain a central limit theorem for the number of weak records.

In this paper, we present new Poisson-type deviation inequalities for continuous-time Markov chains whose Wasserstein curvature or Γ-curvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and yield different deviation bounds. In the case of birth–death processes, we provide some conditions on the transition rates of the associated generator for such curvatures to be bounded below and we extend the deviation inequalities established [Ané, C. and Ledoux, M. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 (2000) 573–602] for continuous-time random walks, seen as models in null curvature. Some applications of these tail estimates are given for Brownian-driven Ornstein–Uhlenbeck processes and M/M/1 queues.

We investigate statistical properties for a broad class of modern kernel-based regression (KBR) methods. These kernel methods were developed during the last decade and are inspired by convex risk minimization in infinite-dimensional Hilbert spaces. One leading example is support vector regression. We first describe the relationship between the loss function L of the KBR method and the tail of the response variable. We then establish the L-risk consistency for KBR which gives the mathematical justification for the statement that these methods are able to “learn”. Then we consider robustness properties of such kernel methods. In particular, our results allow us to choose the loss function and the kernel to obtain computationally tractable and consistent KBR methods that have bounded influence functions. Furthermore, bounds for the bias and for the sensitivity curve, which is a finite sample version of the influence function, are developed, and the relationship between KBR and classical M estimators is discussed.

Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83–109] prove an extension of Itô’s formula for F(X_t, t), where F(x, t) has a locally square-integrable derivative in x that satisfies a mild continuity condition in t and X is a one-dimensional diffusion process such that the law of X_t has a density satisfying certain properties. This formula was expressed using quadratic covariation. Following the ideas of Eisenbaum [Potential Anal. 13 (2000) 303–328] concerning Brownian motion, we show that one can re-express this formula using integration over space and time with respect to local times in place of quadratic covariation. We also show that when the function F has a locally integrable derivative in t, we can avoid the mild continuity condition in t for the derivative of F in x.

In this paper, we give quantitative bounds on the f-total variation distance from convergence of a Harris recurrent Markov chain on a given state space under drift and minorization conditions implying ergodicity at a subgeometric rate. These bounds are then specialized to the stochastically monotone case, covering the case where there is no minimal reachable element. The results are illustrated with two examples, from queueing theory and Markov Chain Monte Carlo theory.

We establish estimates for the local and uniform moduli of continuity of the local time of multifractional Brownian motion, B^H=(B^H(t)(t), t∈ℝ⁺). An analogue of Chung’s law of the iterated logarithm is studied for B^H and used to obtain the pointwise Hölder exponent of the local time. A kind of local asymptotic self-similarity is proved to be satisfied by the local time of B^H.

We observe stationary random tessellations X={Ξ_n}_n≥1 in ℝ^d through a convex sampling window W that expands unboundedly and we determine the total (k−1)-volume of those (k−1)-dimensional manifold processes which are induced on the k-facets of X (1≤k≤d−1) by their intersections with the (d−1)-facets of independent and identically distributed motion-invariant tessellations X_n generated within each cell Ξ_n of X. The cases of X being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In both cases, however, we obtain that all of the total volumes measured in W are approximately normally distributed when W is sufficiently large. Structural formulae for mean values and asymptotic variances are derived and explicit numerical values are given for planar Poisson–Voronoi tessellations (PVTs) and Poisson line tessellations (PLTs).