The aims of this paper are twofold. Firstly, we derive some probabilistic representation for the constant which appears in the one-dimensional case of Kesten's renewal theorem. Secondly, we estimate the tail of some related random variable which plays an essential role in the description of the stable limit law of one-dimensional transient sub-ballistic random walks in random environment.
In this article we study a diophantine property of probability measures on R. We will always assume that the considered measures have an exponential moment and a drift. We link this property to the points in C close to the imaginary axis where the Fourier-Laplace transform of those measures take the value 1 and finally, we apply this to the study of the speed in Kesten's renewal theorem on R.
Consider the perpetuity equation $X \stackrel{\mathcal{D}}{=} A X + B$, where $(A,B)$ and $X$ on the right-hand side are independent. The Kesten--Grincevičius--Goldie theorem states that $P \{ X > x \} \sim c x^{-κ}$ if $E A^κ= 1$, $E A^κ\log_+ A < \infty$, and $E |B|^κ< \infty$. We assume that $E |B|^ν< \infty$ for some $ν> κ$, and consider two cases (i) $E A^κ= 1$, $E A^κ\log_+ A = \infty$; (ii) $E A^κ< 1$, $E A^t = \infty$ for all $t > κ$. We show that under appropriate additional assumptions on $A$ the asymptotic $P \{ X > x \} \sim c x^{-κ} \ell(x) $ holds, where $\ell$ is a nonconstant s…
Let $d >1$ and $(A_n)_{n \ge 1}$ be a sequence of independent identically distributed random matrices with nonnegative entries and no zero column. This induces a Markov chain $M_n = A_n M_{n-1}$ on the cone of d-vectors with nonnegative entries. We study harmonic functions of this Markov chain. Considering a polar decomposition $M_n = X_n \exp(S_n)$, where $X_n$ is a vector of unit length, and $S_n$ a real valued random variable, it is in particular shown that all "compound" harmonic functions $L(x,s)=f(x)g(s)$ are constant. The idea of the proof is originally due to Kesten [Renewal theory for…
We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cramér condition the stationary distribution has a power law tail. We determine the tail process of the stationary Markov chain, prove point process convergence, and convergence of the partial sums. The original motivation comes from Kesten, Kozlov and Spitzer seminal 1975 paper, which connects a random walk in a random environment model to a special Galton-Watson process with immigration in a random environment. We obtain new …
This paper takes the so-called probabilistic approach to the Strong Renewal Theorem (SRT) for multivariate distributions in the domain of attraction of a stable law. A version of the SRT is obtained that allows any kind of lattice-nonlattice composition of a distribution. A general bound is derived to control the so-called "small-$n$ contribution", which arises from random walk paths that have a relatively small number of steps but make large cumulative moves. The asymptotic negligibility of the small-$n$ contribution is essential to the SRT. Applications of the SRT are given, including some t…
Given a sequence of i.i.d. random functions $Ψ_{n}:\mathbb{R}\to\mathbb{R}$, $n\in\mathbb{N}$, we consider the iterated function system and Markov chain which is recursively defined by $X_{0}^{x}:=x$ and $X_{n}^{x}:=Ψ_{n-1}(X_{n-1}^{x})$ for $x\in\mathbb{R}$ and $n\in\mathbb{N}$. Under the two basic assumptions that the $Ψ_{n}$ are a.s. continuous at any point in $\mathbb{R}$ and asymptotically linear at the "endpoints" $\pm\infty$, we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie's implicit …
This paper studies the first hitting times of generalized Poisson processes $N^f(t)$, related to Bernstein functions $f$. For the space-fractional Poisson processes, $N^α(t)$, $t>0$ (corresponding to $f= x^α$), the hitting probabilities $P\{T_k^α<\infty\}$ are explicitly obtained and analyzed. The processes $N^f(t)$ are time-changed Poisson processes $N(H^f(t))$ with subordinators $H^f(t)$ and here we study $N\left(\sum_{j=1}^n H^{f_j}(t)\right)$ and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form $N(|\mathcal{G}_{H…
Let $F$ be a probability measure on $\mathbb{R}$ in the domain of attraction of a stable law with exponent $α\in (0, 1)$. We establish integral criteria on $F$ that significantly expand the probabilistic approach to Strong Renewal Theorems (SRTs). The criterion for $α\in (0,1/2]$ is much weaker than currently available ones and in some cases provides sufficient and necessary conditions for the SRT. The criterion for $α\in (1/2, 1)$ establishes the SRT in full generality and in a unified way, barring the Limit Local Theorems employed. As an application, for infinitely divisible $F$, an integral…
Kesten's incipient infinite cluster for the three-dimensional, metric-graph Gaussian free field, from critical level-set percolation, and for the Villain model, from random cluster geometries and a Swendsen-Wang type algorithm
Let $G$ be a multiplicative subsemigroup of the general linear group $\Gl(\mathbb{R}^d)$ which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a $G$--valued random matrix $A$, we consider the following generalized multidimensional affine equation R\stackrel{\mathcal{D}}{=}\sum_{i=1}^N A_iR_i+B, where $N\ge2$ is a fixed natural number, $A_1,...,A_N$ are independent copies of $A$, $B\in\mathbb{R}^d$ is a random vector with positive entries, and $R_1,...,R_N$ are independent copies of $R\in\mathbb{R}^d$, which have also p…
Given a sequence $(M_{n},Q_{n})_{n\ge 1}$ of i.i.d. random variables with generic copy $(M,Q)$ such that $M$ is a regular $d\times d$ matrix and $Q$ takes values in $\mathbb{R}^{d}$, we consider the random difference equation (RDE) $R_{n}=M_{n}R_{n-1}+Q_{n}$, $n\ge 1$. Under suitable assumptions, this equation has a unique stationary solution $R$ such that, for some $κ>0$ and some finite positive and continuous function $K$ on $S^{d-1}:=\{x \in \mathbb{R}^{d}:|x|=1\}$, $ \lim_{t \to \infty} t^κ P(xR>t)=K(x)$ for all $x \in S^{d-1} $ holds true. This result is originally due to Kesten and Le Pa…
This short survey is aimed at sketching the ergodic-theoretic aspects of the author's recent studies on Weyl's eigenvalue asymptotics for a \emph{"geometrically canonical" Laplacian} defined by the author on some self-conformal circle packing fractals including the classical \emph{Apollonian gasket}. The main result being surveyed is obtained by applying Kesten's renewal theorem [\emph{Ann.\ Probab.}\ \textbf{2} (1974), 355--386, Theorem 2] for functionals of Markov chains on general state spaces and provides an alternative probabilistic proof of the result by Oh and Shah [\emph{Invent.\ Math.…
In recent years, there has been an interest in deriving certain important probabilistic results as consequences of deterministic ones; see for instance \cite{beig} and \cite{acc}. In this work, we continue on this path by deducing a well known equivalence between the speed of random walks on the integers and the growth of the size of their ranges. This result is an immediate consequence of the Kesten-Spitzer-Whitman theorem, and by appearances is probabilistic in nature, but we will show that it follows easily from an elementary deterministic result. We also investigate the common property of …
We study the following microscopic model of infection or epidemic reaction: red and blue particles perform independent nearest-neighbor continuous-time symmetric random walks on the integer lattice $\mathbb{Z}$ with jump rates $D_R$ for red particles and $D_B$ for blue particles, the interaction rule being that blue particles turn red upon contact with a red particle. The initial condition consists of i.i.d. Poisson particle numbers at each site, with particles at the left of the origin being red, while particles at the right of the origin are blue. We are interested in the dynamics of the fro…
We study properties of compactly supported, 4 parameter \newline $(ρ_{12},ρ_{23},ρ_{13},q)\in (-1,1)^{\times 4}$ family of continuous type 3 dimensional distributions, that have the property that for $q\rightarrow 1^{-}$ this family tends to some 3 dimensional Normal distribution. For $q=0$ we deal with 3 dimensional generalization of Kesten--McKay distribution. In a very special case when $ρ_{12}ρ_{13}ρ_{23}=q$ all one dimensional marginals are identical, semicircle distributions. We find both all marginal as well as all conditional distributions. Moreover, we find also families of polynomial…
For the random interval partition of $[0,1]$ generated by the uniform stick-breaking scheme known as GEM$(1)$, let $u_k$ be the probability that the first $k$ intervals created by the stick-breaking scheme are also the first $k$ intervals to be discovered in a process of uniform random sampling of points from $[0,1]$. Then $u_k$ is a renewal sequence. We prove that $u_k$ is a rational linear combination of the real numbers $1, ζ(2), \ldots, ζ(k)$ where $ζ$ is the Riemann zeta function, and show that $u_k$ has limit $1/3$ as $k \to \infty$. Related results provide probabilistic interpretations …
Modeling of polymer chains has received a lot of attention in mathematics. In fact, probabilistic models that naturally arise in statistical mechanics have been widely studied by mathematicians for the very challenging and novel problems that they pose. The physical situation that we consider in this thesis is that of a polymer in the proximity of an interface between two selective solvents, in the case when the interaction of the monomers with the solvents and the interface may vary from monomer to monomer (inhomogeneous polymer). In interesting cases thee is a phase transition between a stat…
