The problem of loss adaptation is investigated: given a fixed parameter, the goal is to construct an estimator that adapts to the loss function in the sense that the estimator is optimal both globally and locally at every point. Given the class of estimator sequences that achieve the minimax rate, over a fixed Besov space, for estimating the entire function a lower bound is given on the performance for estimating the function at each point. This bound is larger by a logarithmic factor than the usual minimax rate for estimation at a point when the global and local minimax rates of convergence differ. A lower bound for the maximum global risk is given for estimators that achieve optimal minimax rates of convergence at every point. An inequality concerning estimation in a two-parameter statistical problem plays a key role in the proof. It can be considered as a generalization of an inequality due to Brown and Low. This may be of independent interest. A particular wavelet estimator is constructed which is globally optimal and which attains the lower bound for the local risk provided by our inequality.

We prove limit theorems for rescaled occupation time fluctuations of a $(d,α,β)$-branching particle system (particles moving in $ℝ^d$ according to a spherically symmetric $α$-stable Lévy process, $(1+β)$-branching, $0<β<1$, uniform Poisson initial state), in the cases of critical dimension, $d=α(1+β)/β$, and large dimensions, $d>α(1+β)/β$. The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (tightness does not hold in the usual Skorohod topology). The results are in sharp contrast with those for intermediate dimensions, $α/β<d<d(1+β)/β$, where the limit process is continuous and has long-range dependence. The limit process is measure-valued for the critical dimension, and $\mathcal{S}^\prime(\mathbb{R}^{d})$-valued for large dimensions. We also raise some questions of interpretation of the different types of dimension-dependent results obtained in the present and previous papers in terms of properties of the particle system.

The connection between Hausdorff–Besicovitch dimension of graphs of trajectories and various Blumenthal–Getoor indices is well known for $α$-stable Lévy processes as well as for some stationary Gaussian processes possessing Orey index. We show that the same relationship holds for several classes of Lévy processes that are popular in financial mathematics models – in particular, the Carr–Geman–Madan–Yor, normal inverse Gaussian, generalized hyperbolic, generalized $z$ and Meixner processes.

Let $(M_n)$, $(N_n)$ be two Hilbert-space-valued martingales adapted to some filtration $(ℱ_n)$, with corresponding difference sequences $(d_n)$, $(e_n)$, respectively. We assume that $(N_n)$ weakly dominates $(M_n)$, that is, for any convex non-decreasing function $ϕ : ℝ_+→ℝ-_+$ and any $n=1,2,…$ we have, almost surely, $\mathrm{E}(ϕ(|d_n|)|ℱ_{n−1})\leqslant\mathrm{E}(ϕ(|e_n|)|ℱ_{n−1})$. We apply the Burkholder method to show that for a convex non-decreasing function $\mathbf{Φ} : ℝ_+→ℝ_+$ satisfying some extra conditions we have, for any $n=1,2,…$, $‖M_n‖_\mathbf{Φ}≤C_\mathbf{Φ}‖N_n‖_\mathbf{Φ}$, where $‖⋅‖_\mathbf{Φ}$ denotes an Orlicz norm with respect to $\mathbf{Φ}$ and $C_\mathbf{Φ}$ is a constant which depends only on $\mathbf{Φ}$. This approach unifies and extends the classical Burkholder inequalities for subordinated martingales and the inequalities for tangent martingales. The method leads to moment inequalities for Rosenthal-type dominated martingales and variance-dominated Gaussian martingales. All the constants obtained in the moment inequalities are of optimal order.

In this paper we study one-dimensional BSDE’s whose coefficient $f$ is monotonic in $y$ and non-Lipschitz in $z$. We obtain a general existence result when $f$ has at most quadratic growth in $z$ and $ξ$ is bounded. We study the special case $f(t,y,z)=|z|^p$ where $p∈(1,2]$. Finally, we study the case $f$ has a linear growth in $z$, general growth in $y$ and $ξ$ is not necessarily bounded.

This paper considers the estimation of the density of an absolutely continuous distribution with respect to an unknown baseline distribution $F$, and the estimation of $F$, from censored observations. For parametric and nonparametric densities, an $n^{1/2}$-consistent estimator of $F$ is defined from the two samples and the asymptotic distribution of the estimators is studied. The efficient score functions and the minimal variances of the estimators are established.

We consider the Skorokhod embedding problem in Brownian motion. In particular, we give a solution based on the local time at zero of a variably skewed Brownian motion related to the underlying Brownian motion. Special cases of the construction include the Azéma–Yor and Vallois embeddings. In turn, the construction has an interpretation in the Chacon–Walsh framework.

Generalizing to higher dimensions the classical gambler’s ruin estimates, we give pointwise estimates for the transition kernel corresponding to a spatially inhomogeneous random walk on the half-space. Our results hold under some strong but natural assumptions of symmetry, boundedness of the increments, and ellipticity. Among the most important steps in our proof are: discrete variants of the boundary Harnack estimate, as proven by Bauman, Bass and Burdzy, and Fabes et al., based on comparison arguments and potential-theoretical tools; the existence of a positive $\tilde{L}$-harmonic function globally defined in the half-space; and some Gaussian inequalities obtained by a treatment inspired by Varopoulos.

The class of type G distributions on $ℝ^d$ and its nested subclasses are studied. An analytic characterization in terms of Lévy measures for the class of type G distributions is known. In this paper, probabilistic characterizations by stochastic integral representations for all classes are shown, and analytic characterizations for the nested subclasses are given in terms of Lévy measures.

We investigate the first-order threshold moving-average model. We obtain a sufficient condition for a unique strictly stationary and ergodic solution of the model without the need to check irreducibility. We also establish necessary and sufficient conditions for its invertibility of first-order . Furthermore, we discuss the extension of the results to the first-order multiple threshold moving-average model and the higher-order threshold moving-average model.

Consider the following problem: if the maximum likelihood estimate of a location parameter of a population is given by the sample mean, is it true that the distribution is of normal type? The answer is positive and the proof was provided by Gauss, albeit without using the likelihood terminology. We revisit this result in a modern context and present a simple and rigorous proof. We also consider extensions to a $p$-dimensional population and to the case with a parameter additional to that of location.

We establish Edgeworth expansions for the distribution function of the standardized Hill estimator for the reciprocal of the index of regular variation of the tail of a distribution function. The expansions are used to derive expansions for coverage probabilities of confidence intervals for the tail index based on the Hill estimator.

The problem of simulation of multivariate Lévy processes is investigated. A method based on generalized shot noise series representations of Lévy processes combined with Gaussian approximation of the remainder is established in full generality. This method is applied to multivariate stable and tempered stable processes and formulae for their approximate simulation are obtained.

We consider the fragmentation at nodes of the Lévy continuous random tree introduced in a previous paper. In this framework we compute the asymptotic behaviour of the number of small fragments at time $θ$. This limit is increasing in $θ$ and discontinuous. In the $α$-stable case the fragmentation is self-similar with index $1/α$, with $α∈(1,2)$, and the results are close to those Bertoin obtained for general self-similar fragmentations but with an additional assumption which is not fulfilled here.

Recently there has been growing interest in applying elliptical distributions to risk management. Under certain conditions, Hult and Lindskog show that a random vector with an elliptical distribution is in the domain of attraction of a multivariate extreme value distribution. In this paper we study two estimators for the tail dependence function, which are based on extreme value theory and the structure of an elliptical distribution. After deriving second-order regular variation estimates and proving asymptotic normality for both estimators, we show that the estimator based on the structure of an elliptical distribution is better than that based on extreme value theory in terms of both asymptotic variance and optimal asymptotic mean squared error. Our simulation study further confirms this.

Layered stable (multivariate) distributions and processes are defined and studied. A layered stable process combines stable trends of two different indices, one of them possibly Gaussian. More precisely, over short intervals it is close to a stable process, while over long intervals it approximates another stable (possibly Gaussian) process. The absolute continuity of a layered stable process with respect to its short-range limiting stable process is also investigated. A series representation of layered stable processes is derived, giving insights into the structure both of the sample paths and of the short- and long-range behaviours of the process. This series representation is further used for simulation of sample paths.

We investigate the problem of estimating the best binary decision tree approximation to the baseline hazard function in the Cox proportional hazards model. Our motivation is to find an effective way of condensing key functional information in the baseline hazard into a small number of estimable parameters. The parameters consist of a threshold and two hazard levels, one to the left of the threshold and one to the right, defined in terms of the best $L^2$ approximation to the nonparametric baseline hazard function. Estimators of these parameters are introduced and shown to converge at cube-root rate to a non-normal limit distribution. Two alternate ways of constructing confidence intervals for the threshold are compared. Results from a simulation study and an example concerning a threshold for the age of onset of schizophrenia in a large cohort study are discussed.