Related papers: Cram\'{e}r asymptotics for finite time first passa…

Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n \leq x + H} \lambda(n) e(-P(n)) \Big |\ dx = o ( X H)$$ for…

We give a new proof of Heath-Brown's full asymptotic expansion for the second moment of Dirichlet L-functions and we obtain a corresponding asymptotic expansion for a twisted first moment of Hecke-Maass L-functions.

In this paper we study the large time asymptotics of the flow of a dynamical system $X'=b(X)$ posed in the $d$-dimensional torus. Rather than using the classical unique ergodicity condition which is not fulfilled if $b$ vanishes at…

In this paper we present some new limit theorems for power variation of $k$th order increments of stationary increments L\'evy driven moving averages. In this infill sampling setting, the asymptotic theory gives very surprising results,…

We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy tailed steps, the limiting…

In a recent work on the bipartite Erd\H{o}s-R\'{e}nyi graph, Do et al. (2023) established upper bounds on the number of connected labeled bipartite graphs with a fixed surplus. We use some recent encodings of bipartite random graphs in…

In this paper, long time and high order moment asymptotics for super-Brownian motions (sBm's) are studied. By using a moment formula for sBm's (e.g. Theorem 3.1, Hu et al. Ann. Appl. Probab. 2023+), precise upper and lower bounds for all…

Let $\xi_1,\xi_2,\ldots$ be independent, identically distributed random variables with infinite mean $\mathbf E[|\xi_1|]=\infty.$ Consider a random walk $S_n=\xi_1+\cdots+\xi_n$, a stopping time $\tau=\min\{n\ge 1: S_n\le 0\}$ and let…

Consider a finite renewal process in the sense that interrenewal times are positive i.i.d. variables and the total number of renewals is a random variable, independent of interrenewal times. A finite point process can be obtained by…

Distributional identities for a L\'evy process $X_t$, its quadratic variation process $V_t$ and its maximal jump processes, are derived, and used to make "small time" (as $t\downarrow0$) asymptotic comparisons between them. The…

For non uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds…

This paper addresses heavy-tailed large deviation estimates for the distribution tail of functionals of a class of spectrally one-sided L\'evy process. Our contribution is to show that these estimates remain valid in a near-critical regime.…

For sequences of non-lattice weakly dependent random variables, we obtain asymptotic expansions for Large Deviation Principles. These expansions, commonly referred to as strong large deviation results, are in the spirit of Edgeworth…

Let $X$ be a L\'evy process with regularly varying L\'evy measure $\nu$. We obtain sample-path large deviations for scaled processes $\bar X_n(t) \triangleq X(nt)/n$ and obtain a similar result for random walks. Our results yield detailed…

Associated to each complex-valued random variable satisfying appropriate integrability conditions, we introduce a different generalization of the Stirling numbers of the second kind. Various equivalent definitions are provided. Attention,…

We consider a finite collection of reinforced stochastic processes with a general network-based interaction among them. We provide sufficient and necessary conditions in order to have some form of almost sure asymptotic synchronization,…

We introduce the concept of asymptotic period for an irreducible and aperiodic, discrete-time Markov chain X on a countable state space, and develop the theory leading to its formal definition. The asymptotic period of X equals one - its…

Let $X_t^\sharp$ be a multivariate process of the form $X_t =Y_t - Z_t$, $X_0=x$, killed at some terminal time $T$, where $Y_t$ is a Markov process having only jumps of the length smaller than $\delta$, and $Z_t$ is a compound Poisson…

Assume that we observe a stochastic process $(X(t))_{t\in[-r,T]}$, which satisfies the linear stochastic delay differential equation \[ \mathrm{d} X(t) = \vartheta \int_{[-r,0]} X(t + u) \, a(\mathrm{d} u) \, \mathrm{d} t + \mathrm{d} W(t)…

We study the asymptotic tail behaviour of the first-passage time over a moving boundary for asymptotically $\alpha$-stable L\'evy processes with $\alpha<1$. Our main result states that if the left tail of the L\'evy measure is regularly…