Related papers: Functional limit laws for the increments of the qu…

Let $s:[1,\infty) \to \C $ be a locally integrable function in Lebesgue's sense. The logarithmic (also called harmonic) mean of the function $s$ is defined by [\tau(t) := \frac 1{\log t} \int_1^t \frac {s(x)}{x} dx, \qquad t>1,] where the…

A uniform law of large numbers and a central limit theorem are established via a martingale approach for a univariate Hawkes process with immigration given by a renewal process. The results are obtained for renewal processes with absolutely…

In this paper, by using the exact tail asymptotics derived by Debicki, Hashorva and Ji (Ann. Probab. 2014), we proved the Gumbel limit theorem for the maximum of a class of non-homogeneous Gaussian random fields. By using the obtained…

We obtain the empirical strong law of large numbers, empirical Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various nonparametric Bayesian priors which include the Dirichlet process with general…

We prove a law of large numbers and functional central limit theorem for a class of multivariate Hawkes processes with time-dependent reproduction rate. We address the difficulties induced by the use of non-convolutive Volterra processes by…

In homogenization theory and multiscale modeling, typical functions satisfy the scaling law $f^{\epsilon}(x) = f(x,x/\epsilon)$, where $f$ is periodic in the second variable and $\epsilon$ is the smallest relevant wavelength,…

Edgeworth-type expansions for convolutions of probability densities and powers of the characteristic functions with non-uniform error terms are established for i.i.d. random variables with finite (fractional) moments of order $s \geq 2$,…

We present two limit theorems, a mean ergodic and a central limit theorem, for a specific class of one-dimensional diffusion processes that depend on a small-scale parameter $\varepsilon$ and converge weakly to a homogenized diffusion…

Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Under purification and irreducibility assumptions, these Markov processes admit a unique invariant measure…

We study long time behavior of integrated trawl processes introduced by Barndorff-Nielsen. The trawl processes form a class of stationary infinitely divisible processes, described by an infinitely divisible random measure (L\'evy base) and…

We give functional laws of large numbers for a class of marked Hawkes processes and marked compound Hawkes processes with a general mark space. Our results provide some complement to those presented previously in the literature. As an…

We consider sequences of $U$-processes based on symmetric kernels of a fixed order, that possibly depend on the sample size. Our main contribution is the derivation of a set of analytic sufficient conditions, under which the aforementioned…

Let $(X_{k})_{k \in \mathbb Z }$ be a linear process with values in a separable Hilbert space $\mathbb{H}$ given by $X_{k} =\sum_{j=0}^{\infty} (j+1)^{-N}\varepsilon_{k-j}$ for each $k \in \mathbb Z$, where $N:\mathbb{H} \to \mathbb{H}$ is…

We prove a Central Limit Theorem for probability measures defined via the variation of the sum-of-digits function, in base $b\ge 2$. For $r\ge 0$ and $d \in \mathbb{Z}$, we consider $\mu^{(r)}(d)$ as the density of integers $n\in…

A short proof is given of a necessary and sufficient condition for the normalized occupation measure of a L\'evy process in a metrizable compact group to be asymptotically uniform with probability one.

The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the…

Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, finance and many other fields. In this paper, we obtain a functional…

We survey some geometrical properties of trajectories of $d$-dimensional random walks via the application of functional limit theorems. We focus on the functional law of large numbers and functional central limit theorem (Donsker's…

We prove functional limit theorems for dynamical systems in the presence of clusters of large values which, when summed and suitably normalised, get collapsed in a jump of the limiting process observed at the same time point. To keep track…

It is shown that two conjectures put forward in the recent article Iksanov and Kostohryz (2025) are true. Namely, we prove a functional central limit theorem (FCLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series…