If the inter-arrival time distribution of a renewal process is regularly varying with index $α\in\left( 0,1\right) $ (i.e. the inter-arrival times have infinite mean) and if $A\left( t\right) $ is the associated age process at time $t$. Then we show that if $C\left( t\right) $ is the length of the current cycle at time $t$, \[ A\left( t\right) /C\left( t\right) \Rightarrow U^{1/α}, \] where $U$ is $U\left( 0,1\right) $. This extends a classical result in renewal theory in the finite mean case which indicates that the limit is $U\left( 0,1\right) $.
We consider the {following} coverage model on $\mathbb{N}$. For each site $i\in \mathbb{N}$ we associate a pair $(ξ_i, R_i)$ where $\{ξ_0, ξ_1, \ldots \}$ is a 1-dimensional {undelayed} discrete renewal point process and $\{R_0,R_1,\ldots\}$ is an i.i.d. sequence of $\mathbb{N}$-valued random variables. At each site where $ξ_i=1$ we start an interval of length $R_i$. Coverage occurs if every site of $\mathbb{N}$ is covered by some interval. We obtain sharp conditions for both, positive and null probability of coverage. As corollaries, we extend results of the literature of rumor processes and …
Consider a coin tossing experiment which consists of tossing one of two coins at a time, according to a renewal process. The first coin is fair and the second has probability $1/2 + θ$, $θ\in [-1/2,1/2]$, $θ$ unknown but fixed, of head. The biased coin is tossed at the renewal times of the process, and the fair one at all the other times. The main question about this experiment is whether or not it is possible to determine $θ$ almost surely as the number of tosses increases, given only the probabilities of the renewal process and the observed sequence of heads and tails. We will construct a co…
The Doeblin Graph of a countable state space Markov chain describes the joint pathwise evolutions of the Markov dynamics starting from all possible initial conditions, with two paths coalescing when they reach the same point of the state space at the same time. Its Bridge Doeblin subgraph only contains the paths starting from a tagged point of the state space at all possible times. In the irreducible, aperiodic, and positive recurrent case, the following results are known: the Bridge Doeblin Graph is an infinite tree that is unimodularizable. Moreover, it contains a single bi-infinite path, wh…
Perron-Frobenius theory developed for irreducible non-negative kernels deals with so-called $R$-positive recurrent kernels. If kernel $M$ is $R$-positive recurrent, then the main result determines the limit of the scaled kernel iterations $R^nM^n$ as $n\to\infty$. In the Nummelin's monograph this important result is proven using a regeneration method whose major focus is on $M$ having an atom. In the special case when $M=P$ is a stochastic kernel with an atom, the regeneration method has an elegant explaination in terms of an associated split chain. In this paper we give a new probabilistic in…
We introduce a new class of asymmetric random walks on the one-dimensional infinite lattice. In this walk the direction of the jumps (positive or negative) is determined by a discrete-time renewal process which is independent of the jumps. We call this discrete-time counting process the `it generator process' of the walk. We refer the so defined walk to as `Asymmetric Discrete-Time Random Walk' (ADTRW). We highlight connections of the waiting-time density generating functions with Bell polynomials. We derive the discrete-time renewal equations governing the time-evolution of the ADTRW and anal…
On discrete-time arrival processes and related random motions
Politecnico di Torino（都灵理工大学）；Sorbonne Université, CNRS, Institut Jean Le Rond d’Alembert（索邦大学，法国国家科学研究中心，让·勒朗·达朗贝尔研究所）；University of Torino（都灵大学）；Universidad Nacional de Colombia（哥伦比亚国立大学）
Linear fractional Galton-Watson branching processes in i.i.d.~random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s.~convergent counterpart in the positive recurrent case given by the corresponding backward iteration…
We consider autoregressive sequences $X_n=aX_{n-1}+ξ_n$ and $M_n=\max\{aM_{n-1},ξ_n\}$ with a constant $a\in(0,1)$ and with positive, independent and identically distributed innovations $\{ξ_k\}$. It is known that if $\mathbf P(ξ_1>x)\sim\frac{d}{\log x}$ with some $d\in(0,-\log a)$ then the chains $\{X_n\}$ and $\{M_n\}$ are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index $-1-d/\log a$. We also prove limit theorems for $\{X_n\}$ and $\…
We consider the problem of the first passage time to the origin of a spatially non-homogeneous random walk with a position-dependent drift, known as the Gillis random walk, in the presence of resetting. The walk starts from an initial site $ x_0 $ and, with fixed probability $ r $, at each step may be relocated to a given site $ x_r $. From a general perspective, we first derive a series of results regarding the first and the second moment of the first hitting time distribution, valid for a wide class of processes, including random walks lacking the property of translational invariance; we the…
In this paper we study a particular class of Piecewise deterministic Markov processes (PDMP's) which are semi-stochastic catastrophe versions of deterministic population growth models. In between successive jumps the process follows a flow describing deterministic population growth. Moreover, at random jump times, governed by state-dependent rates, the size of the population shrinks by a random amount of its current size, an event possibly leading to instantaneous local (or total) extinction. A special separable shrinkage transition kernel is investigated in more detail, including the case of …
Catastrophe Markov chain population models have received a lot of attention in the recent past. We herewith consider two special cases of such models involving total disasters, both in discrete and in continuous-time. Depending on the parameters range, the two models can show up a recurrence/transience transition and, in the critical case, a positive/null recurrence transition. The collapse transition probabilities are chosen in such a way that the models are exactly solvable and, in case of positive recurrence, intimately related to the extended Sibuya and Pareto-Zipf distributions whose divi…
We study the norm of the two-dimensional Brownian motion conditioned to stay outside the unit disk at all times. By conditioning the process is changed from barely recurrent to slightly transient. We obtain sharp results on the rate of escape to infinity of the process of future minima: (i) we find an integral test on the function $g$ so that the future minima process drops beyond the barrier $\exp \{ \ln t \times g(\ln \ln t)\}$ at arbitrary large times; (ii) we show that the future minima process exceeds $K \sqrt{ t \times \ln \ln \ln t}$ at arbitrary large times with probability 0 [resp., 1…
Consider a class of null-recurrent randomly biased walks on a super-critical Gaton-Watson tree. We obtain the rates of convergence of the local times and the quenched local probability for the biased walk in the sub-diffusive case. These results are a consequence of a sharp estimate on the return time, whose analysis is driven by a family of concave recursive equations on trees.
This paper is concerned with an extended Galton-Watson process so as to allow individuals to live and reproduce for more than one unit time. We assume that each individual can live $k$ seasons (time-units) with probability $h_k$, and produce $m$ offspring with probability $p_m$ during each season. These can be seen as Galton-Watson processes with countably infinitely many types in which particles of type $i$ may only have offspring of type $i+1$ and type $1$. Let $\textbf{M}$ be its mean progeny matrix and $γ$ be the convergence radius of the power series $\sum_{k\geq 0}r^k(\textbf{M}^k)_{ij}$…
