For the Ornstein-Uhlenbeck process, the asymptotic behavior of the maximum likelihood estimator of the drift parameter is totally different in the stable, unstable, and explosive cases. Notwithstanding of this trichotomy, we investigate sharp large deviation principles for this estimator in the three situations. In the explosive case, we exhibit a very unusual rate function with a shaped flat v...

The genetic distance between biological sequences is a fundamental quantity in molecular evolution. It pertains to questions of rates of evolution, existence of a molecular clock, and phylogenetic inference. Under the class of continuous-time substitution models, the distance is commonly defined as the expected number of substitutions at any site in the sequence. We eschew the almost ubiquitous...

Observing a load process above high thresholds, modeling it as a pulse process with random occurrence times and magnitudes, and extrapolating life-time maximum or design loads from the data is a common task in structural reliability analyses. In this paper, we consider a stationary live load sequence that arrive according to a dependent point process and allow for a weakened mixing-type depende...

The aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known about the context in which such processes can arise. To our knowledge, discretization and convergence theorems are available only in the case of stable Lévy motions and fra...

We present a novel approach for non-stationary Gaussian process regression (GPR), where the three key parameters – noise variance, signal variance and lengthscale – can be simultaneously input-dependent. We develop gradient-based inference methods to learn the unknown function and the non-stationary model parameters, without requiring any model approximations. For inferring the full posterior d...

We give an example of the moderate deviations for the family (X n t) t≥0 , n ≥ 1 with ˙ X n t = a(X n t) + b(X n t) n √ n k+1 ηtn, where ηt is a stationary process obeying the Wold – decomposition: ηt = R t −∞ h(t − s)dN s with respect to a process Nt with homogeneous independent square integrable increments and 1/2 < k < 1.

Let {X(t) : t ∈ [0,∞)} be a centered Gaussian process with stationary increments and variance function σ X(t). We study the exact asymptotics of P(supt∈[0,T ] X(t) > u), as u → ∞, where T is an independent of {X(t)} nonnegative Weibullian random variable. As an illustration we work out the asymptotics of supremum distribution of fractional Laplace motion.

Let {X1, . . . , Xm} and {Y1, . . . , Yn} be two samples independent of each other, but the random variables within each sample are stationary associated with one dimensional marginal distribution functions F and G, respectively. We study the properties of the classical WilcoxonMann-Whitney statistic for testing for stochastic dominance in the above set up.