Related papers: A CLT for empirical processes involving time-depen…
When the limiting compensator of a sequence of martingales is continuous, we obtain a weak convergence theorem for the martingales; the limiting process can be written as a Brownian motion evaluated at the compensator and we find sufficient…
Let $\{(X_t)_{t\geq 0}, \mathbb{P}_{\delta_x}, x\in E\}$ be a supercritical branching Markov process (which is not necessary symmetric) on a locally compact metric measure space $(E,\mu)$ with spatially dependent local branching mechanism.…
We consider the problem of tracking an unstable stochastic process $X_t$ by using causal knowledge of another stochastic process $Y_t$. We obtain necessary conditions and sufficient conditions for maintaining a finite tracking error. We…
A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching L\'evy process, and its law is characterized by a triplet $(\sigma^2,a,\Lambda)$. We…
Models of particle dynamics based on Brownian motion and its variants are a rich source of insights into the stochastic behaviour of complex condensed phase systems. In this paper we use one such variant - a breathing parabola with an…
We consider sequences of symmetric $U$-statistics, not necessarily Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove quantitative central limit theorems (CLTs) based on the use of {\it contraction operators}. Our…
For real L\'{e}vy processes $(X\_t)\_{t \geq 0}$ having no Brownian component with Blumenthal-Getoor index $\beta$, the estimate $\E \sup\_{s \leq t} | X\_s - a\_p s |^p \leq C\_p t$ for every $t \in [0,1]$ and suitable $a\_p \in \R$ has…
We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under…
In this paper we investigate the class of grey Brownian motions $B_{\alpha,\beta}$ ($0<\alpha<2$, $0<\beta\leq1$). We show that grey Brownian motion admits different representations in terms of certain known processes, such as fractional…
In this paper, we develop a theory of common decomposition for two correlated Brownian motions, in which, by using change of time method, the correlated Brownian motions are represented by a triplet of processes, $(X,Y,T)$, where $X$ and…
We consider a strictly stationary and ergodic sequence of random elements taking values in some Hilbert space. Our target is to study the weak convergence of the discrete Fourier transforms under sharp conditions. As a side-result we obtain…
We consider $n$ independent, identically distributed one-dimensional Brownian motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth density function $f$. The empirical quantiles, or pointwise order statistics, are denoted by…
Empirical processes for stationary, causal sequences are considered. We establish empirical central limit theorems for classes of indicators of left half lines, absolutely continuous functions and piecewise differentiable functions. Sample…
In this paper we study the functional central limit theorem for stationary Markov chains with self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n; and…
In this work, we provide a refinement of the selective CLT result of Tian and Taylor (2015), which allows for selective inference in non-parametric settings by adjusting for the asymptotic Gaussian limit for selection. Under some regularity…
A time-varying empirical spectral process indexed by classes of functions is defined for locally stationary time series. We derive weak convergence in a function space, and prove a maximal exponential inequality and a…
When the underlying random variables are Gaussian, the classical Central Limit Theorem (CLT) is trivial, but the functional CLT is not. The objective of the paper is to investigate the functional CLT for stationary Gaussian processes in the…
In the case of monotone independence, the transparent understanding of the mechanism to validate the central limit theorem (CLT) has been lacking, in sharp contrast to commutative, free and Boolean cases. We have succeeded in clarifying it…
Let $Z_t^{(0,\infty)}$ be the point process formed by the positions of all particles alive at time $t$ in a branching Brownian motion with drift and killed upon reaching 0. We study the asymptotic expansions of $Z_t^{(0,\infty)}(A)$ for $A=…
This work improves the existing central limit theorems (CLTs) for geometric functionals of Gibbs processes in three aspects. First, we derive a CLT for weakly stabilizing functionals, thereby improving on the previously used assumption of…