A quenched CLT for super-Brownian motion with random immigration

Random walks with unbounded jumps among random conductances II: Conditional quenched CLT

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched conditional invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$.

Random walks with unbounded jumps among random conductances I: Uniform quenched CLT

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{uniform} invariance principle for the random walk. This means that the rescaled trajectory of length $n$ is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval of length $O(\sqrt{n})$ around the origin.

Heavy tailed branching process with immigration

In this paper we analyze a branching process with immigration defined recursively by $X_t=θ_t\circ X_{t-1}+B_t$ for a sequence $(B_t)$ of i.i.d. random variables and random mappings $ θ_t\circ x:=θ_t(x)=\sum_{i=1}^xA_i^{(t)}, $ with $(A_i^{(t)})_{i\in \mathbb{N}_0}$ being a sequence of $\mathbb{N}_0$-valued i.i.d. random variables independent of $B_t$. We assume that one of generic variables $A$ and $B$ has a regularly varying tail distribution. We identify the tail behaviour of the distribution of the stationary solution $X_t$. We also prove CLT for the partial sums that could be further gene…

Quenched CLT for random toral automorphism

We establish a quenched Central Limit Theorem (CLT) for a smooth observable of random sequences of iterated linear hyperbolic maps on the torus. To this end we also obtain an annealed CLT for the same system. We show that, almost surely, the variance of the quenched system is the same as for the annealed system. Our technique is the study of the transfer operator on an anisotropic Banach space specifically tailored to use the cone condition satisfied by the maps.

Conditional and uniform quenched CLTs for one-dimensional random walks among random conductances

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{conditional} invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$. One of the main tools for proving these conditional limit la…

Quenched central limit theorems for a stationary linear process

We find a sufficient condition under which a central limit theorem for a stationary linear process is quenched. We find a stationary linear process szatisfying the Maxwell-Woodroofe condition for which the variances of partial sums are o(n), there is a CLT with a convergence towards N(0,1) when dividing by standard deviation of the partial sums, and the CLT is not quenched. The weak invariance principle does not hold.

Logistic分支随机游走祖先谱系的淬火中心极限定理

Quenched CLT for ancestral lineages of logistic branching random walks

该论文证明在种群密度足够高的动态随机环境中，由Logistic分支随机游走时间反转生成的随机游走满足淬火中心极限定理，方法包括粗粒化、再生时间构造及耦合技术。

Quenched limit theorems for Fourier transforms and periodogram

In this paper, we study the quenched central limit theorem for the discrete Fourier transform. We show that the Fourier transform of a stationary ergodic process, suitable centered and normalized, satisfies the quenched CLT conditioned by the past sigma algebra. For functions of Markov chains with stationary transitions, this means that the CLT holds with respect to the law of the chain started at a point for almost all starting points. It is necessary to emphasize that no assumption of irreducibility with respect to a measure or other regularity conditions are imposed for this result. We also…

随机环境中单周期阿兹特克钻石的淬火与退火中心极限定理

Quenched and Annealed CLTs for the one-periodic Aztec diamond in random environment

研究随机环境中单周期阿兹特克钻石随机二聚体覆盖的渐近行为，通过分析带随机参数的Schur过程，证明了淬火涨落收敛到高斯自由场，并推广了退火中心极限定理。

Functional CLT for martingale-like nonstationary dependent structures

In this paper we develop non-stationary martingale techniques for dependent data. We shall stress the non-stationary version of the projective Maxwell-Woodroofe condition, which will be essential for obtaining maximal inequalities and functional central limit theorem for the following examples: nonstationary \r{ho}-mixing sequences, functions of linear processes with non-stationary innovations, quenched version of the functional central limit theorem for a stationary sequence, evolutions in random media such as a process sampled by a shifted Markov chain.

平稳随机场在射影准则下的淬火泛函中心极限定理

On the Quenched Functional Central Limit Theorem for Stationary Random Fields under Projective Criteria

本文在Orlicz空间弱假设下，将Zhang等人和Peligrad与Volný的淬火泛函中心极限定理推广至满足Hannan型射影条件的平稳随机场，并推导Rosenthal型不等式，应用于多种随机过程。

Optimal rates of convergence for quenched central limit theorem rates for hitting times of one-dimensional random walks in random environments

We consider the rates of convergence of the quenched central limit theorem for hitting times of one-dimensional random walks in a random environment. Previous results had identified polynomial upper bounds for the rates of decay which are sometimes slower than $n^{-1/2}$ (the optimal rate in the classical Berry-Esseen estimates). Here we prove that the previous upper bounds are in fact the best possible polynomial rates for the quenched CLT.

Quenched central limit theorem rates of convergence for one-dimensional random walks in random environments

Unlike classical simple random walks, one-dimensional random walks in random environments (RWRE) are known to have a wide array of potential limiting distributions. Under certain assumptions, however, it is known that CLT-like limiting distributions hold for the walk under both the quenched and averaged measures. We give upper bounds on the rates of convergence for the quenched central limit theorems for both the hitting time and position of the RWRE with polynomial rates of convergence that depend on the distribution on environments.

Fluctuations of the quenched mean of a planar random walk in an i.i.d. random environment with forbidden direction

We consider an i.i.d. random environment with a strong form of transience on the two dimensional integer lattice. Namely, the walk always moves forward in the y-direction. We prove a functional CLT for the quenched expected position of the random walk indexed by its level crossing times. We begin with a variation of the Martingale Central Limit Theorem. The main part of the paper checks the conditions of the theorem for our problem.

On the functional CLT for stationary Markov Chains started at a point

We present a general functional central limit theorem started at a point also known under the name of quenched. As a consequence, we point out several new classes of stationary processes, defined via projection conditions, which satisfy this type of asymptotic result. One of the theorems shows that if a Markov chain is stationary ergodic and reversible, this result holds for bounded additive functionals of the chain which have a martingale coboundary in L_1 representation. Our results are also well adapted for strongly mixing sequences providing for this case an alternative, shorter approach t…

Quenched invariance principles for orthomartingale-like sequences

In this paper we study the central limit theorem and its functional form for random fields which are not started from their equilibrium, but rather under the measure conditioned by the past sigma field. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random fields by approximating them, in some sense, with an orthomartingale. We construct an example which shows that there are orthomartingales which satisfy the CLT but not its quenched form. This example also clarifies the optimality of the moment conditions used for the validi…

钉扎模型局域相中的集中与涨落现象

Concentration and fluctuation phenomena in the localized phase of the pinning model

本工作针对满足矩生成函数在原点邻域有界的独立同分布位点无序钉扎模型局域相，建立自由能密度的正则性估计，导出接触数的淬火集中界与CLT，还为无序依赖的中心化序列建立CLT，推广了前人的分析。

Central limit theorem started at a point for additive functionals of reversible Markov chains

In this paper we study the almost sure central limit theorem started from a point for additive functionals of a stationary and ergodic Markov chain via a martingale approximation in the almost sure sense. As a consequence we derive the quenched CLT for additive functionals or reversible Markov chains.

A spectral approach for quenched limit theorems for random expanding dynamical systems

We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem (LCLT) for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of \emph{twisted transfer operator cocycles} with respect to a twist parameter. While some versions of the LDP and CLT have previo…

Functional CLT for random walk among bounded random conductances

We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of i.i.d. random nearest-neighbor conductances $ω_{xy}\in[0,1]$. Apart from the requirement that the bonds with positive conductances percolate, we pose no restriction on the law of the $ω$'s. We prove that, for a.e. realization of the environment, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. The quenched functional CLT holds despite the fact that the local CLT may fail in $d\ge5$ due to anomalously slow decay of the probability that the walk ret…

关于保守IID随机动力系统淬灭与退火统计性质的等价性

On equivalence of quenched and annealed statistical properties for conservative IID random dynamical systems

该研究针对保守IID随机动力系统，证明两点运动的退火指数混合性可推出单点运动的淬灭指数混合性，且带多项式收敛速率的退火CLT结合该混合性可推出单点运动的淬灭CLT，结果适用于所有正指数Hölder与Sobolev空间。

Central limit theorems with a rate of convergence for time-dependent intermittent maps

We study dynamical systems arising as time-dependent compositions of Pomeau-Manneville-type intermittent maps. We establish central limit theorems for appropriately scaled and centered Birkhoff-like partial sums, with estimates on the rate of convergence. For maps chosen from a certain parameter range, but without additional assumptions on how the maps vary with time, we obtain a self-normalized CLT provided that the variances of the partial sums grow sufficiently fast. When the maps are chosen randomly according to a shift-invariant probability measure, we identify conditions under which the …

Simple Transient Random Walks in One-dimensional Random Environment: the Central Limit Theorem

We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the Central Limit Theorem (CLT) holds for the position of the walk. Verifying these conditions leads to a complete solution of the problem in the case of independent identically distributed environments as well as in the case of uniformly ergodic (and thus also weakly mixing) environments.

Central limit theorems for biased randomly trapped random walks on Z

We prove CLTs for biased randomly trapped random walks in one dimension. In particular, we will establish an annealed invariance principal by considering a sequence of regeneration times under the assumption that the trapping times have finite second moment. In a quenched environment, an environment dependent centring is determined which is necessary to achieve a central limit theorem. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive and prove a tight bound on the bias required to obtain such limiting behaviour.

A Central Limit Theorem for biased random walks on Galton-Watson trees

Let ${\cal T}$ be a rooted Galton-Watson tree with offspring distribution $\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails. Consider the $λ$-biased random walk $\{X_n\}_{n\geq 0}$ on ${\cal T}$; this is the nearest neighbor random walk which, when at a vertex $v$ with $d_v$ offspring, moves closer to the root with probability $λ/(λ+d_v)$, and moves to each of the offspring with probability $1/(λ+d_v)$. It is known that this walk has an a.s. constant speed $\v=\lim_n |X_n|/n$ (where $|X_n|$ is the distance of $X_n$ from the root), with $\v>0$ for $ 0<λ<m$ and $\v=0$ for $λ…

Random walks in a one-dimensional Lévy random environment

We consider a generalization of a one-dimensional stochastic process known in the physical literature as Lévy-Lorentz gas. The process describes the motion of a particle on the real line in the presence of a random array of marked points, whose nearest-neighbor distances are i.i.d. and long-tailed (with finite mean but possibly infinite variance). The motion is a continuous-time, constant-speed interpolation of a symmetric random walk on the marked points. We first study the quenched random walk on the point process, proving the CLT and the convergence of all the accordingly rescaled moments. …

Quenched and averaged large deviation rate functions for random walks in random environments: the impact of disorder

In 2003, Varadhan [V03] developed a robust method for proving quenched and averaged large deviations for random walks in a uniformly elliptic and i.i.d. environment (RWRE) on $\mathbb Z^d$. One fundamental question which remained open was to determine when the quenched and averaged large deviation rate functions agree, and when they do not. In this article we show that for RWRE in uniformly elliptic and i.i.d. environment in $d\geq 4$, the two rate functions agree on any compact set contained in the interior of their domain which does not contain the origin, provided that the disorder of the e…

The random conductance model with Cauchy tails

We consider a random walk in an i.i.d. Cauchy-tailed conductances environment. We obtain a quenched functional CLT for the suitably rescaled random walk, and, as a key step in the arguments, we improve the local limit theorem for $p^ω_{n^2t}(0,y)$ in [Ann. Probab. (2009). To appear], Theorem 5.14, to a result which gives uniform convergence for $p^ω_{n^2t}(x,y)$ for all $x,y$ in a ball.

非均匀随机图上伊辛模型的标度极限及其应用

Scaling Limits for Ising Models on Inhomogeneous Random Graphs and Applications

本文推导了非均匀随机图上伊辛模型的淬火标度极限，将其应用于贝叶斯神经网络与因果推断，得到了相关极限结果与渐近性质。

随机动力系统中有效的淬火线性响应

Effective quenched linear response for random dynamical systems

本文研究了非均匀扩展随机动力系统中有效的线性响应，探讨了其在CLT方差可微性和退化线性响应中的应用，提供了多种一维和高维映射示例。

Central limit theorem for products of toral automorphisms

Let $(τ_n)$ be a sequence of toral automorphisms $τ_n : x \rightarrow A_n x \hbox{mod}\ZZ^d$ with $A_n \in {\cal A}$, where ${\cal A}$ is a finite set of matrices in $SL(d, \mathbb{Z})$. Under some conditions the method of "multiplicative systems" of Komlòs can be used to prove a Central Limit Theorem for the sums $\sum_{k=1}^n f(τ_k \circ τ_{k-1} \cdots \circ τ_1 x)$ if $f$ is a Hölder function on $\mathbb{T}^d$. These conditions hold for $2\times 2$ matrices with positive coefficients. In dimension $d$ they can be applied when $A_n= A_n(ω)$, with independent choices of $A_n(ω)$ in a finite s…

随机磁场伊辛模型中的涨落

Fluctuations in random field Ising models

本文在高温区域建立了线性统计量的中心极限定理，并给出了贝叶斯-埃辛界限。研究应用至具有离散和连续自旋的随机磁场伊辛模型，并推导了淬火和退火的CLT，包括Erdős-Rényi等图结构的伊辛模型和Hopfield自旋玻璃模型。

Central limit theorem for biased random walk on multi-type Galton-Watson trees

Let T be a rooted supercritical multi-type Galton-Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. The lambda-biased random walk (X_t, t>=0) on T is the nearest-neighbor random walk which, when at a vertex v with d(v) offspring, moves closer to the root with probability lambda/[lambda+d(v)], and to each of the offspring with probability 1/[lambda+d(v)]. This walk is recurrent for lambda>=rho and transient for 0<lambda<rho, with rho the Perron-Frobenius eigenvalue for the (assumed) irreducible matrix of expected offspring numbers. Subject to finite mome…