The previous, longer version 1 has been split in two parts: the present paper, and a more group-theoretic one with the title "Kesten's theorem for Invariant Random Subgroups"
We give explicit estimates between the spectral radius and the densities of short cycles for finite d-regular graphs. This allows us to show that the essential girth of a finite d-regular Ramanujan graph G is at least c log log |G|.
We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Using Benjamini-Schramm convergence this leads to a rigidity result saying that if most eigenvalues of a d-regular finite graph G fall in the Alon-Boppana region, then the eigenvalue distribution of G is close to the spectral measure of the d-regular tree.
Kesten showed that if a Cayley graph has the same spectral radius as its universal cover, then it must be a tree. We generalize this to unimodular random graphs.
We consider a class of perpetuities which admit direct characterization of asymptotics of the key truncated moment. The class contains perpetuities without polynomial decay of tail probabilities and thus not satisfying Kesten's theorem. We show how to apply this result in deriving a new weak law of large numbers for solutions to stochastic recurrence equations and a new central limit theorem for GARCH(1,1) processes in the critical case.
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 $(Z_n,n\geq 0)$ be a supercritical Galton-Watson process whose offspring distribution $μ$ has mean $λ>1$ and is such that $\int x(\log(x))_+ dμ(x)<+\infty$. According to the famous Kesten \& Stigum theorem, $(Z_n/λ^n)$ converges almost surely, as $n\to+\infty$. The limiting random variable has mean~1, and its distribution is characterised as the solution of a fixed point equation. \par In this paper, we consider a family of Galton-Watson processes $(Z_n(λ), n\geq 0)$ defined for~$λ$ ranging in an interval $I\subset (1, \infty)$, and where we interpret $λ$ as the time (when $n$ is the gener…
We study supercritical branching processes in which all particles evolve according to some general Markovian motion (which may possess absorbing states) and branch independently at a fixed constant rate. Under fairly natural assumptions on the asymptotic distribution of the underlying motion, we first show using only probablistic tools that there is convergence in $L^2$ of the empirical measure (normalized by the mean number of particles) if and only if an associated additive martingale is bounded in $L^2$. This is a significant improvement over previous results, which were mainly restricted t…
We generalize Brooks's theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d \geq 3$ which contains no $(d+1)$-cliques, then $G$ admits a $μ$-measurable $d$-coloring with respect to any Borel probability measure $μ$ on $X$, and a Baire measurable $d$-coloring with respect to any compatible Polish topology on $X$. The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of …
Cramér's theorem in measurable locally convex spaces
我们给出了独立同分布序列的经验平均值的Cramér大偏差定理的一个一般性条件。随机向量，其中包含了巴拿赫空间中的Cramér定理和Sanov定理。- Nous établissons un cadre général pour le théorème de Cramér sur les grandes deviations des moyennes empiriques d'une suite de vectoraléatoires i.i.d.，在Banach空间中包含Cramer的理论和Sanov的理论。
We generalize theorems of Kesten and Deuschel-Pisztora about the connectedness of the exterior boundary of a connected subset of $\mathbb{Z}^d$, where "connectedness" and "boundary" are understood with respect to various graphs on the vertices of $\mathbb{Z}^d$. We provide simple and elementary proofs of their results. It turns out that the proper way of viewing these questions is graph theory, instead of topology.
In this paper we study a representation problem first considered in a simpler version by Bank and El Karoui [2004]. A key ingredient to this problem is a random measure $μ$ on the time axis which in the present paper is allowed to have atoms. Such atoms turn out to not only pose serious technical challenges in the proof of the representation theorem, but actually have significant meaning in its applications, for instance, in irreversible investment problems. These applications also suggest to study the problem for processes which are measurable with respect to a Meyer-$σ$-field that lies betwe…
We prove a semi-invertible Oseledets theorem for cocycles acting on measurable fields of Banach spaces, i.e. we only assume invertibility of the base, not of the operator. As an application, we prove an invariant manifold theorem for nonlinear cocycles acting on measurable fields of Banach spaces.
In the spirit of the famous KOMLÓS (1967) theorem, every sequence of nonnegative, measurable functions $\{ f_n \}_{n \in \N}$ on a probability space, contains a subsequence which - along with all its subsequences - converges a.e. in CESÀRO mean to some measurable $f_* : Ω\to [0, \infty]$. This result of VON WEIZSÄCKER (2004) is proved here using a new methodology and elementary tools; these sharpen also a theorem of DELBAEN & SCHACHERMAYER (1994), replacing general convex combinations by CESÀRO means.
