Related papers: Zooming in on a L\'evy process at its supremum

In this paper, we investigate the asymptotic behavior of supercritical branching Markov processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are L\'evy processes with regularly varying tails. Recently, Ren et al. [Appl. Probab. 61…

For domains in $\mathbb{R}^d$, $d\geq 2$, we prove universal upper and lower bounds on the product of the bottom of the spectrum for the Laplacian to the power $p>0$ and the supremum over all starting points of the $p$-moments of the exit…

We consider a L\'evy process $Y(t)$ that is not permanently observed, but rather inspected at Poisson($\omega$) moments only, over an exponentially distributed time $T_\beta$ with parameter $\beta$. The focus lies on the analysis of the…

We give some approximations of the local time process $(L_t^x)_{t\geqslant 0}$ at level $x$ of the real Brownian motion $(X_t)$. We prove that $ \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon)\wedge t}^+ \indi_{\{X_u \leqslant 0\}} du +…

For given two standard processes with no positive jumps, we construct, using the excursion theory, a Markov process whose positive and negative motions have the same law as the two processes. The resulting process is a generalization of…

Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of L\'evy processes, which includes subordinated stable and symmetric L\'evy processes. We apply this characterisaiton to…

We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to -infinity and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we…

We study the asymptotic behaviour of the tail of the distribution of the first passage time of a L\'evy process over a one-sided moving boundary. Our main result states that if the boundary behaves as $t^{\gamma}$ for large $t$ for some…

In this paper we consider convergence of moments in the small-time limit theorems for L\'evy processes. We provide precise asymptotics for all the absolute moments of positive order. The convergence of moments in limit theorems holds…

Consider on a manifold the solution $X$ of a stochastic differential equation driven by a L\'evy process without Brownian part. Sufficient conditions for the smoothness of the law of $X_t$ are given, with particular emphasis on noncompact…

In contrast to their seemingly simple and shared structure of independence and stationarity, L\'evy processes exhibit a wide variety of behaviors, from the self-similar Wiener process to piecewise-constant compound Poisson processes.…

We consider a spectrally positive L\'evy process $X$ that does not drift to $+\infty$, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process…

This paper is concerned with the behaviour of a L\'{e}vy process when it crosses over a positive level, $u$, starting from 0, both as $u$ becomes large and as $u$ becomes small. Our main focus is on the time, $\tau_u$, it takes the process…

The maximal (or Hilbertian) correlation coefficient between two random variables X and Y, denoted by \{X:Y\}, is the supremum of the |Corr(f(X),g(Y))| for real measurable functions f, g, where "Corr" denotes Pearson's correlation…

Let (B^{(1)}_t ;B^{(2)}_t ;B^{(3)}_t + \mu t) be a three-dimensional Brownian motion with drift \mu, starting at the origin. Then X_t = ||(B^{(1)}_t ;B^{(2)}_t ;B^{(3)}_t +\mu t)||, its distance from the starting point, is a diffusion with…

We study a random walk on a point process given by an ordered array of points $(\omega_k, \, k \in \mathbb{Z})$ on the real line. The distances $\omega_{k+1} - \omega_k$ are i.i.d. random variables in the domain of attraction of a…

Let $(X_1, \xi_1), (X_2,\xi_2),\ldots$ be i.i.d.~copies of a pair $(X,\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $\xi$ is a positive random variable. Define $S_k := \xi_1+\ldots+\xi_k$, $k \in…

We study positive random variables whose moments can be expressed by products and quotients of Gamma functions; this includes many standard distributions. General results are given on existence, series expansion and asymptotics of density…

We establish the global asymptotic equivalence between a pure jumps L\'evy process $\{X_t\}$ on the time interval $[0,T]$ with unknown L\'evy measure $\nu$ belonging to a non-parametric class and the observation of $2m^2$ Poisson…

In this work we derive limit theorems for trawl processes. First,we study the asymptotic behaviour of the partial sums of the discretized trawl process $(X_{i\Delta_{n}})_{i=0}^{\lfloor nt\rfloor-1}$, under the assumption that as…