Robin Pemantle
This article examines a recent body of work on stochastic processes indexed by a tree. Emphasis is on the application of this new framework to existing probability models. Proofs are largely omitted, with references provided.
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We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot's fractal percolation process.
We derive several explicit distributions of functionals of Brownian motion indexed by the Brownian tree. In particular, we give a direct proof of a result of Bousquet-Mélou and Janson identifying the distribution of the density at 0 of the integrated super-Brownian excursion.
We introduce a model of branching interlacements for general critical offspring distributions. It consists of a countable collection of infinite tree-indexed random walk trajectories on $Z^d,d\geq5$. We show that this model turns out to be the local limit of the tree-indexed random walk in a discrete torus, conditioned on the size proportional to the volume of the torus. This generalizes the previous results of Angel, Ráth and the author, for the critical geometric offspring distribution. Our model also includes the model of random interlacements introduced by Sznitman as a degenerate case. To…
We consider random walks indexed by arbitrary finite random or deterministic trees. We derive a simple sufficient criterion which ensures that the maximal displacement of the tree-indexed random walk is determined by a single large jump. This criterion is given in terms of four quantities : the tail and the expectation of the random walk steps, the height of the tree and the number of its vertices. The results are applied to critical Galton--Watson trees with offspring distributions in the domain of attraction of a stable law.
We construct the analogue of the local time -- at a fixed point $x$ -- for Markov processes indexed by Levy trees. We start by proving that Markov processes indexed by Levy trees satisfy a special Markov property which can be thought as a spatial version of the classical Markov property. Then, we construct the analogue of the local time by an approximation procedure and we characterize the support of its Lebesgue-Stieltjes measure. We also give an equivalent construction in terms of a special family of exit local times. Finally, combining these results, we show that the points at which the Mar…
Excursion theory for Markov processes indexed by Levy trees
本文为Lévy树索引的马尔可夫过程建立了远足理论，借助局部时刻画其离开正则瞬时点的演化，并证明远足家谱可由局部时编码的Lévy树表示，从而以新方法恢复了Abraham和Le Gall关于布朗树索引布朗运动的远足理论。
We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Holder exponent form a set with large intersection.
By introducing a new measure for the infinite Galton-Watson process and providing estimates for (discrete) Green's functions on trees, we establish the asymptotic behavior of the capacity of critical branching random walks: in high dimensions $d\ge 7$, the capacity grows linearly; and in the critical dimension $d=6$, it grows asymptotically proportional to $\frac{n}{\log n}$.
We study the behavior of Random Walk in Random Environment (RWRE) on trees in the critical case left open in previous work. Representing the random walk by an electrical network, we assume that the ratios of resistances of neighboring edges of a tree Gamma are i.i.d.random variables whose logarithms have mean zero and finite variance. Then the resulting RWRE is transient if simple random walk on Gamma is transient, but not vice versa. We obtain general transience criteria for such walks, which are sharp for symmetric trees of polynomial growth. In order to prove these criteria, we establish re…
We consider the model of Brownian motion indexed by the Brownian tree. For every $r\geq 0$ and every connected component of the set of points where Brownian motion is greater than $r$, we define the boundary size of this component, and we then show that the collection of these boundary sizes evolves when $r$ varies like a well-identified growth-fragmentation process. We then prove that the same growth-fragmentation process appears when slicing a Brownian disk at height $r$ and considering the perimeters of the resulting connected components.
The Markov property of local times of Brownian motion indexed by the Brownian tree
该研究针对布朗树索引布朗运动模型，证明密度及其导数构成的二元组为时间齐次马尔可夫过程，还得到一维超布朗运动局部时的类似结果，方法基于该模型的游程理论。
Poisson representations for tree-indexed Markov chains
本文研究树索引马尔可夫链的泊松可表示性，证明非路径有限树的可表示性依赖参数，并给出无限树族中非平凡参数可表示的实例，同时加强了已有结果。
We provide asymptotics for the range R(n) of a random walk on the d-dimensional lattice indexed by a random tree with n vertices. Using Kingman's subadditive ergodic theorem, we prove under general assumptions that R(n)/n converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension 4 and in the case of a symmetric random walk with exponential moments, we prove that R(n) grows like n/(log n). We apply our results to asymptotics for the range of branching random walk when the initial size of the population tends to infi…
We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical Itô theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connectedcomponent is itself a continuous tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with branching mechanism $ψ(u)=\sqrt{8/3}\,u^{3/2}$. Furthermo…
Ergodic theorem for branching Markov chains indexed by trees with arbitrary shape
本文证明了以任意形状树为索引的分支马尔可夫链的遍历定理，并展示了在特定条件下线图马尔可夫链具有最小方差。
A class of branching processes in varying environments is exhibited which become extinct almost surely even though the means M_n grow fast enough so that sum M_n^{-1} is finite. In fact, such a process is constructed for every offspring distribution of infinite variance, and this establishes the converse of a previously known fact: that if a distribution has finite variance then sum M_n^{-1}=infty is equivalent to almost sure extinction. This has as an immediate consequence the converse to a theorem on equipolarity of Galton-Watson trees.
A central limit theorem for the range of random walk indexed by an infinite tree
本文研究了临界分支随机游走范围大小的二阶波动，证明在不同维度下其范围满足中心极限定理，并结合多种数学工具进行证明。
In this paper is proved the limit theorem for randomly indexed sequence of random processes in the case where sequences of random index and random processes are independent, also the estimation of convergence rate is obtained.
We study the range $R_n$ of a random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ indexed by a random tree with $n$ vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension $d\leq3$ that $n^{-d/4}R_n$ converges in distribution to the Lebesgue measure of the support of the integrated super-Brownian excursion (ISE). An auxiliary result shows that the suitably rescaled local times of the tree-indexed random walk converge in distribution to the density process of ISE. We obtain similar results for the range of critical branching ran…
We consider a type of Markov property for set-indexed processes which is satisfied by all processes with independent increments and which allows us to introduce a transition system theory leading to the construction of the process. A set-indexed generator is defined such that it completely characterizes the distribution of the process.
We are interested in the increment stationarity property for $L^2$-indexed stochastic processes, which is a fairly general concern since many random fields can be interpreted as the restriction of a more generally defined $L^2$-indexed process. We first give a spectral representation theorem in the sense of \citet{Ito54}, and see potential applications on random fields, in particular on the $L^2$-indexed extension of the fractional Brownian motion. Then we prove that this latter process is characterized by its increment stationarity and self-similarity properties, as in the one-dimensional cas…
We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_σ)_{σ\in\mathcal T}$, indexed by the full binary tree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcal T_k$ is the $k$th generation of $\mathcal T$. In addition, let $(Σ_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $Σ_k \in \mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ with $\widetilde R_t^n := X_{Σ_{[tn]}}$, arising by observing the Markov chain $\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empir…
In this paper, we study the Hölder regularity of set-indexed stochastic processes defined in the framework of Ivanoff-Merzbach. The first key result is a Kolmogorov-like Hölder-continuity Theorem, whose novelty is illustrated on an example which could not have been treated with anterior tools. Increments for set-indexed processes are usually not simply written as $X_U-X_V$, hence we considered different notions of Hölder-continuity. Then, the localization of these properties leads to various definitions of Hölder exponents, which we compare to one another. In the case of Gaussian processes, al…
Some classes of increment martingales, and the corresponding localized classes, are studied. An increment martingale is indexed by the real line and its increment processes are martingales. We focus primarily on the behavior as time goes to minus infinity in relation to the quadratic variation or the predictable quadratic variation, and we relate the limiting behavior to the martingale property. Finally, integration with respect to an increment martingale is studied.
Asymptotic properties of the maximum likelihood estimator for Hidden Markov Models indexed by binary trees
本文研究以二叉树为索引、隐状态为一般度量空间的隐马尔可夫模型，在平稳与非平稳情形下证明最大似然估计量的强相合性与渐近正态性，并借助树索引马尔可夫链的遍历定理完成证明。
Intersections of branching random walks on $\mathbb{Z}^8$
本文研究$\u2124^8$上以无穷不变树为索引的分支随机游走的相交问题，得到两类不同索引的独立游走间不相交概率的精确阶数，并证明分支随机游走值域的分支容量满足弱大数定律。
A new family of stable processes indexed by metric spaces with stationary increments are introduced. They are special cases of a new family of set-indexed stable processes with Chentsov representation. At the heart of the representation, a result on the so-called measure definite kernels is of independent interest. A limit theorem for set-indexed processes is also established.
We study continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. Burdzy and Pal in their paper proposed a continuous version of graphical models -- processes indexed by time-like graphs. We construct a more general family of continuous processes on a wider family of graphs, find new properties and solve the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. We present new results showing connections to the stochastic heat equation, Markov random…
We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton--Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time t. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. The latter has the same generator as the Markov process along the branches plus additional jumps, ass…
Tree-indexed sums of Catalan numbers
本文研究由树索引的卡特兰数乘积的无限和，证明其为$1/\pi$的有理系数多项式，并给出有效算法，同时引入参数化提升，将其表示为完全椭圆积分的多项式，次数不超过树顶点数的一半，研究动机源于大型迷宫系统。
We derive in this preprint the moment and exponential tail estimates, sufficient conditions for the Non-Central Limit Theorem (NCLT) in the ordinary one-dimensional space as well as in the space of continuous functions for the properly (natural) normalized multi-indexed sums of function of random variables, processes or fields (r.f.), on the other words V-statistics, parametric, in general case. We construct also some examples in order to show the exactness of obtained estimates. We will use the theory of the so-called degenerate approximation of the functions of several variables as well as t…
We introduce block Markov chains (BMCs) indexed by an infinite rooted tree. It turns out that BMCs define a new class of tree-indexed Markovian processes. We clarify the structure of BMCs in connection with Markov chains (MCs) and Markov random fields (MRFs). Mainly, show that probability measures which are BMCs for every root are indeed Markov chains (MCs) and yet they form a strict subclass of Markov random fields (MRFs) on the considered tree. Conversely, a class of MCs which are BMCs is characterized. Furthermore, we establish that in the one-dimensional case the class of BMCs coincides wi…
We present a satisfactory definition of the important class of Lévy processes indexed by a general collection of sets. We use a new definition for increment stationarity of set-indexed processes to obtain different characterizations of this class. As an example, the set-indexed compound Poisson process is introduced. The set-indexed Lévy process is characterized by infinitely divisible laws and a Lévy-Khintchine representation. Moreover, the following concepts are discussed: projections on flows, Markov properties, and pointwise continuity. Finally the study of sample paths leads to a Lévy-Itô…
Let (S_n)_{n\in\N} be a Z-valued random walk with increments from the domain of attraction of some α-stable law and let (ξ(i))_{i\in\Z} be a sequence of iid random variables. We want to investigate U-statistics indexed by the random walk S_n, that is U_n:=\sum_{1\leq i<j\leq n}h(ξ(S_i),ξ(S_j)) for some symmetric bivariate function h. We will prove the weak convergence without assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the U-statistic U_n.
Given a finite typed rooted tree $T$ with $n$ vertices, the {\em empirical subtree measure} is the uniform measure on the $n$ typed subtrees of $T$ formed by taking all descendants of a single vertex. We prove a large deviation principle in $n$, with explicit rate function, for the empirical subtree measures of multitype Galton-Watson trees conditioned to have exactly $n$ vertices. In the process, we extend the notions of shift-invariance and specific relative entropy--as typically understood for Markov fields on deterministic graphs such as $\mathbb Z^d$--to Markov fields on random trees. We …
We study the accessibility percolation model on infinite trees. The model is defined by associating an absolute continuous random variable $X_v$ to each vertex $v$ of the tree. The main question to be considered is the existence or not of an infinite path of nearest neighbors $v_1,v_2,v_3\ldots$ such that $X_{v_1}<X_{v_2}<X_{v_3}<\cdots$ and which spans the entire graph. The event defined by the existence of such path is called {\it{percolation}}. We consider the case of the accessibility percolation model on a spherically symmetric tree with growth function given by $f(i)=\lceil (i+1)^ α\rcei…
Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] \times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Lévy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Hölder regularity prope…
We study the asymptotic behavior of empirical processes generated by measurable bounded functions of an infinite source Poisson transmission process when the session length have infinite variance. In spite of the boundedness of the function, the normalized fluctuations of such an empirical process converge to a non-Gaussian stable process. This phenomenon can be viewed as caused by the long-range dependence in the transmission process. Completing previous results on the empirical mean of similar types of processes, our results on non-linear bounded functions exhibit the influence of the limit …
We consider random arrays indexed by the leaves of an infinitary rooted tree of finite depth, with the distribution invariant under the rearrangements that preserve the tree structure. We call such arrays hierarchically exchangeable and prove that they satisfy an analogue of de Finetti's theorem. We also prove a more general result for arrays indexed by several trees, which includes a hierarchical version of the Aldous-Hoover representation.
In this article is introduced and studied a set-indexed Markov property named C-Markov. This new definition fulfils one important expectation for a Markov property: there exists a natural set-indexed generalization of the concept of transition operator which leads to characterization and construction theorems for C-Markov processes. Several other usual Markovian notions, including Feller and strong Markov properties, can also be developed in this framework. Actually, the C-Markov property turns out to be a natural extension of the two-parameter \ast-Markov property to the multiparameter and as…
We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class ${\mathcal{F}}$ is different from the class of functions $f$ for which we have good properties of the observables $(f(X_i))_{i\geq0}$. We introduce a new bracketing number to measure the size of the index…
We consider stochastic processes indexed by the vertices of an infinite binary tree having a simple recursive structure. The value at any vertex is some fixed function of the values at the two daughter vertices together with some independent innovation. Endogeny means the innovations are generating. When endogeny does not hold there exist dynamics in which the innovations are held fixed while some additional randomness on the boundary of the tree is perturbed.
Ergodicity and weak mixing for group-indexed infinitely divisible stationary processes
研究群索引无限可分平稳过程，通过对概率可分平稳过程进行随机连续扩展构造，将问题简化，证明了遍历无限可分平稳过程是弱混合的，揭示遍历性蕴含弱混合性是无限可分性固有属性，去除群结构假设。
We prove that a square-integrable set-indexed stochastic process is a set-indexed Brownian motion if and only if its projection on all the strictly increasing continuous sequences are one-parameter $G$-time-changed Brownian motions. In addition, we study the "sequence-independent variation" property for group stationary-increment stochastic processes in general and for a set-indexed Brownian motion in particular. We present some applications.
The purpose of this paper is to prove a weak convergence result for empirical processes indexed in general classes of functions and with an underlying $α$-mixing sequence of random variables. In particular the uniformly boundedness assumption on the function class, which is required in most of the existing literature, is spared. Furthermore under strict stationarity a weak convergence result for the sequential empirical process indexed in function classes is obtained, as a direct consequence. Two examples in mathematical statistics, that cannot be treated with existing results, are given as po…
Hausdorff dimensions of irreducible Markov hom tree-shifts
本文利用类型方法建立了以有根d-树为指标集的有限状态马尔可夫链的克拉梅尔定理，并由此得到样本均值的几乎必然类型收敛及不可约马尔可夫链符号空间的豪斯多夫维数公式。
Let G be a topological compact group acting on some space Y. We study a decomposition of Y-indexed stochastic processes, based on the orthogonality relations between the characters of the irreducible representations of G. In the particular case of a Gaussian process with a G-invariant law, such a decomposition gives a very general explanation of a classic identity in law - between quadratic functionals of a Brownian bridge - due to Watson (1961). Several relations with Karhunen-Loève expansions are discussed, and some applications and extensions are given - in particular related to Gaussian pr…
A family of log-correlated Gaussian processes
本文在条件负定度量空间上构造一类对数相关高斯过程，证明其为双分数布朗运动在 $K\downarrow0$ 时的缩放极限，并给出随机积分表示及聚合模型极限解释。
We study two empirical process of special structure: firstly, the centred multiplier process indexed by a class $F$, $f \to \left|\sum_{i=1}^N (ξ_i f(X_i) - \E ξf)\right|$, where the i.i.d. multipliers $(ξ_i)_{i=1}^N$ need not be independent of $(X_i)_{i=1}^N$, and secondly, $(f,h) \to \left|\sum_{i=1}^N (f(X_i)h(X_i)-\E f h) \right|$, the centred product process indexed by the classes $F$ and $H$. We use chaining methods to obtain high probability upper bounds on the suprema of the two processes using a natural variation of Talagrand's $γ$-functionals.
We are interested in the regularity of centered Gaussian processes (Z_x), x in M indexed by compact metric spaces M. It is shown that the almost everywhere Besov space regularity of such a process is (almost) equivalent to the Besov regularity of the covariance K(x,y) = E(Z_xZ_y) under the assumption that (i) there is an underlying Dirichlet structure on M which determines the Besov space regularity, and (ii) the operator K with kernel K(x,y) and the underlying operator A of the Dirichlet structure commute. As an application of this result we establish the Besov regularity of Gaussian processe…
Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes
本文基于8种离散变量多索引正交多项式，推导出精确可解的连续时间生灭过程及其离散时间版本，涵盖多种正交多项式类型。
Fully lifted interpolating comparisons of bilinearly indexed random processes
本文提出“完全提升”统计插值概念，建立双线性索引随机过程与完全解耦线性索引替代方案之间的联系，并证明该框架可广泛应用于多种随机结构与优化问题。
Let $A$ be a finite set with $|A|\geqslant 2$, let $n$ be a positive integer, and let $A^n$ denote the discrete $n$-dimensional hypercube (that is, $A^n$ is the Cartesian product of $n$ many copies of $A$). Given a family $\langle D_t:t\in A^n\rangle$ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $\langle D_t:t\in A^n\rangle$ are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to a mild "stationarity" condition. Our result has a numbe…
We investigate the simulation methods for a large family of stable random fields that appeared in the recent literature, known as the Karlin stable set-indexed processes. We exploit a new representation and implement the procedure introduced by Asmussen and Rosinski (2001) by first decomposing the random fields into large-jump and small-jump parts, and simulating each part separately. As special cases, simulations for several manifold-indexed processes are considered, and adjustments are introduced accordingly in order to improve the computational efficiency.
Explosion speed of continuous state branching processes indexed by the Esscher transform
本文构造由 Esscher 变换索引并几乎必然逼近非保守连续状态分支过程的保守过程族，刻画首达时收敛到爆炸时间的函数条件，并给出弱收敛、$L^1$ 收敛及几乎必然收敛条件。
Bilinearly indexed random processes -- \emph{stationarization} of fully lifted interpolation
本文提出完全提升统计插值机制的一种平稳化实现，揭示插值参数间的基本关系，并在特殊情形下给出可实用的简化形式，应用范围广泛。
A moment approach for the convergence of spatial branching processes to the Continuum Random Tree
本文研究离散时间空间分支过程，在临界与矩假设下证明其树结构收敛到布朗连续随机树，并提出基于计算矩推导系谱收敛的通用方法及新的多对少公式。
For every integer $k\geq 2$ let $[k]^{<\mathbb{N}}$ be the set of all words over $k$, that is, all finite sequences having values in $[k]:=\{1,...,k\}$. A Carlson-Simpson tree of $[k]^{<\mathbb{N}}$ of dimension $m\geq 1$ is a subset of $[k]^{<\mathbb{N}}$ of the form \[ \{w\}\cup \big\{w^{\smallfrown}w_0(a_0)^{\smallfrown}...^{\smallfrown}w_{n}(a_n): n\in \{0,...,m-1\} \text{ and } a_0,...,a_n\in [k]\big\} \] where $w$ is a word over $k$ and $(w_n)_{n=0}^{m-1}$ is a finite sequence of left variable words over $k$. We study the behavior of a family of measurable events in a probability space i…
Strong Approximations for Empirical Processes Indexed by Lipschitz Functions
本文提出了基于 $d$ 维随机向量的函数类索引的经验过程的一致 Gaussian 强近似，改进了 Lipschitz 函数类下的近似率，并扩展至乘法可分过程与 Haar 基序列，应用于非参数密度和回归估计。
We introduce tree representations for $ α$-determinantal point processes. The $ α$-determinantal point processes is introduced as a one parameter extension of the determinantal point process. In the previous paper with H.Osada, the tree representation was introduced for determinantal point processes. In this paper, we prove that the tree representation can be applied to $ α$-determinantal point processes.
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