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We define a time dependent empirical process based on $n$ i.i.d.~fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for…
We establish asymptotic normality of weighted sums of periodograms of a stationary linear process where weights depend on the sample size. Such sums appear in numerous statistical applications and can be regarded as a discretized versions…
We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider e.g., maximum relative to…
We define generalized innovations associated with generalized error models having arbitrary distributions, that is, distributions that can be mixtures of continuous and discrete distributions. These models include stochastic volatility…
Exchangeable arrays are natural tools to model common forms of dependence between units of a sample. Jointly exchangeable arrays are well suited to dyadic data, where observed random variables are indexed by two units from the same…
It is well known that the distribution of extreme values of strictly stationary sequences differ from those of independent and identically distributed sequences in that extremal clustering may occur. Here we consider non-stationary but…
Continuously-observed event occurrences, often exhibit self- and mutually-exciting effects, which can be well modeled using temporal point processes. Beyond that, these event dynamics may also change over time, with certain periodic trends.…
In this paper, we prove maximal inequalities and study the functional central limit theorem for the partial sums of linear processes generated by dependent innovations. Due to the general weights, these processes can exhibit long-range…
This article considers multivariate linear processes whose components are either short- or long-range dependent. The functional central limit theorems for the sample mean and the sample autocovariances for these processes are investigated,…
We consider the adaptive test for the parameter change in discretely observed ergodic diffusion processes based on the cusum test. Using two test statistics based on the two quasi-log likelihood functions of the diffusion parameter and the…
We study a continuous-time simple random walk on a regular rooted tree of depth $n$ in two settings: either the walk is started from a leaf vertex and run until the tree root is first hit or it is started from the root and run until it has…
The limitations resulting from the dichtomisation of continuous outcomes have been extensively described. But the need to present results based on binary outcomes in particular in health science remains. Alternatives based on the…
We consider a borderline case: the central limit theorem for a strictly stationary time series with infinite variance but a Gaussian limit. In the iid case a well-known sufficient condition for this central limit theorem is regular…
We introduce a process where a connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When a vertex splits, its incoming edges are randomly assigned between its offspring and a…
Drees and Rootz\'en (2010) have established limit theorems for a general class of empirical processes of statistics that are useful for the extreme value analysis of time series, but do not apply to statistics of sliding blocks, including…
In many complex systems studied in statistical physics, inter-arrival times between events such as solar flares, trades and neuron voltages follow a heavy-tailed distribution. The set of event times is fractal-like, being dense in some time…
A classical limit theorem of stochastic process theory concerns the sample cumulative distribution function (CDF) from independent random variables. If the variables are uniformly distributed then these centered CDFs converge in a suitable…
Extreme events are by nature rare and difficult to predict, yet are often much more important than frequent, typical events. An interesting counterpoint to the prediction of such events is their retrodiction -- given a process in an outlier…
We present a new proof of an Ergodic theorem for Wide-Sense Stationary Random Processes added with a new canonical sampling theorem of mine for finite time duration signals in the frequency domain (periodograms) which is free from the…
We present a new method to sample conditioned trajectories of a system evolving under Langevin dynamics, based on Brownian bridges. The trajectories are conditioned to end at a certain point (or in a certain region) in space. The bridge…