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In recent years there has been some focus in work by Bertoin, Chaumont and Doney on the behaviour of one-dimensional Lévy processes and random walks conditioned to stay positive. The resulting conditioned process is transient. In earlier literature, however, one encounters for special classes of random walks and Lévy processes a similar, but nonetheless different, type of asymptotic conditioning to stay positive which results in a limiting quasi-stationary distribution. We extend this theme into the general setting of a Lévy process fulfilling certain types of conditions which are analogues of known classes in the random walk literature. Our results generalize those of E.K. Kyprianou for special types of one-sided compound Poisson processes with drift and of Martínez and San Martín for Brownian motion with drift, and complement the results due to Iglehart, Doney, and Bertoin and Doney for random walks.
We study so-called augmented GARCH sequences, which include many submodels of considerable interest, such as polynomial and exponential GARCH. To model the returns of speculative assets, it is particularly important to understand the behaviour of the squares of the observations. The main aim of this paper is to present a strong approximation for the sum of the squares. This will be achieved by an approximation of the volatility sequence with a sequence of blockwise independent random variables. Furthermore, we derive a necessary and sufficient condition for the existence of a unique (strictly) stationary solution of the general augmented GARCH equations. Also, necessary and sufficient conditions for the finiteness of moments are provided.
Weak Besov spaces play an important role in statistics as maxisets of classical procedures or for measuring the sparsity of signals. The goal of this paper is to study weak Besov balls from the statistical point of view by using the minimax Bayes method. In particular, we compare weak and strong Besov balls statistically. By building an optimal Bayes wavelet thresholding rule, we first establish that, under suitable conditions, the rate of convergence of the minimax risk for is the same as for the strong Besov ball that is contained in . However, we show that the asymptotically least favourable priors of that are based on Pareto distributions cannot be asymptotically least favourable priors for . Finally, we present sample paths of such priors that provide representations of the worst functions to be estimated for classical procedures and we give an interpretation of the roles of the parameters , and of .
We study the problem of nonparametric, completely data-driven estimation of the intensity of counting processes satisfying the Aalen multiplicative intensity model. To do so, we use model selection techniques and, specifically, penalized projection estimators for a random inner product. For histogram estimators, under some assumptions on the process, we obtain adaptive results for the minimax risk. In general, for more intricate (predictable) models, we only obtain oracle inequalities. The study is complemented by some simulations in the right-censoring model.
We study a family of importance sampling estimators of the probability of level crossing when the crossing level is large or the intensity of the noise is small. We develop a method which gives explicitly the asymptotics of the second-order moment. Some of the results apply to fractional Brownian motion, some are more general. The main tools are refined versions of classical large-deviations results.
A specific bootstrap method is introduced for positive recurrent Markov chains, based on the regenerative method and the Nummelin splitting technique. This construction involves generating a sequence of approximate pseudo-renewal times for a Harris chain X from data X1,...,Xn and the parameters of a minorization condition satisfied by its transition probability kernel and then applying a variant of the methodology proposed by Datta and McCormick for bootstrapping additive functionals of type n-1∑i=1nf(Xi) when the chain possesses an atom. This novel methodology mainly consists in dividing the sample path of the chain into data blocks corresponding to the successive visits to the atom and resampling the blocks until the (random) length of the reconstructed trajectory is at least n, so as to mimic the renewal structure of the chain. In the atomic case we prove that our method inherits the accuracy of the bootstrap in the independent and identically distributed case up to OP(n-1) under weak conditions. In the general (not necessarily stationary) case asymptotic validity for this resampling procedure is established, provided that a consistent estimator of the transition kernel may be computed. The second-order validity is obtained in the stationary case (up to a rate close to OP(n-1) for regular stationary chains). A data-driven method for choosing the parameters of the minorization condition is proposed and applications to specific Markovian models are discussed.
We consider the asymptotic behaviour of the realized power variation of processes of the form , where is a fractional Brownian motion with Hurst parameter , and is a process with finite -variation, . We establish the stable convergence of the corresponding fluctuations. These results provide new statistical tools to study and detect the long-memory effect and the Hurst parameter.
Consider an M/G/1 queue with unknown service-time distribution and unknown traffic intensity ρ. Given systematically sampled observations of the workload, we construct estimators of ρ and of the service-time distribution function, and we study asymptotic properties of these estimators.