Conditional Random Walk Research Articles

Scaling limit of the local time of random walks conditioned to stay positive

AbstractWe prove that the local time of random walks conditioned to stay positive converges to the corresponding local time of three-dimensional Bessel processes by proper scaling. Our proof is based on Tanaka’s pathwise construction for conditioned random walks and the derivation of asymptotics for mixed moments of the local time.

Refined behaviour of a conditioned random walk in the large deviations regime

Refined behaviour of a conditioned random walk in the large deviations regime

Analysis of the injury severity of motor vehicle–pedestrian crashes at urban intersections using spatiotemporal logistic regression models

This paper conducted a comprehensive study on the injury severity of motor vehicle–pedestrian crashes at 489 urban intersections across a dense road network based on high-resolution accident data recorded by the police from 2010 to 2019 in Hong Kong. Given that accounting for the spatial and temporal correlations simultaneously among crash data can contribute to unbiased parameter estimations for exogenous variables and improved model performance, we developed spatiotemporal logistic regression models with various spatial formulations and temporal configurations. The results indicated that the model with the Leroux conditional autoregressive prior and random walk structure outperformed other alternatives in terms of goodness-of-fit and classification accuracy. According to the parameter estimates, pedestrian age, head injury, pedestrian location, pedestrian actions, driver maneuvers, vehicle type, first point of collision, and traffic congestion status significantly affected the severity of pedestrian injuries. On the basis of our analysis, a range of targeted countermeasures integrating safety education, traffic enforcement, road design, and intelligent traffic technologies were proposed to improve the safe mobility of pedestrians at urban intersections. The present study provides a rich and sound toolkit for safety analysts to deal with spatiotemporal correlations when modeling crashes aggregated at contiguous spatial units within multiple years.

Asymptotic behaviour of the first positions of uniform parking functions

AbstractIn this paper we study the asymptotic behaviour of a random uniform parking function $\pi_n$ of size n. We show that the first $k_n$ places $\pi_n(1),\ldots,\pi_n(k_n)$ of $\pi_n$ are asymptotically independent and identically distributed (i.i.d.) and uniform on $\{1,2,\ldots,n\}$ , for the total variation distance when $k_n = {\rm{o}}(\sqrt{n})$ , and for the Kolmogorov distance when $k_n={\rm{o}}(n)$ , improving results of Diaconis and Hicks. Moreover, we give bounds for the rate of convergence, as well as limit theorems for certain statistics such as the sum or the maximum of the first $k_n$ parking places. The main tool is a reformulation using conditioned random walks.

Brownian bridges for stochastic chemical processes-An approximation method based on the asymptotic behavior of the backward Fokker-Planck equation.

A Brownian bridge is a continuous random walk conditioned to end in a given region by adding an effective drift to guide paths toward the desired region of phase space. This idea has many applications in chemical science where one wants to control the endpoint of a stochastic process-e.g., polymer physics, chemical reaction pathways, heat/mass transfer, and Brownian dynamics simulations. Despite its broad applicability, the biggest limitation of the Brownian bridge technique is that it is often difficult to determine the effective drift as it comes from a solution of a Backward Fokker-Planck (BFP) equation that is infeasible to compute for complex or high-dimensional systems. This paper introduces a fast approximation method to generate a Brownian bridge process without solving the BFP equation explicitly. Specifically, this paper uses the asymptotic properties of the BFP equation to generate an approximate drift and determine ways to correct (i.e., re-weight) any errors incurred from this approximation. Because such a procedure avoids the solution of the BFP equation, we show that it drastically accelerates the generation of conditioned random walks. We also show that this approach offers reasonable improvement compared to other sampling approaches using simple bias potentials.

Integrals of incomplete beta functions, with applications to order statistics, random walks and string enumeration

We study the probability that one beta-distributed random variable exceeds the maximum of two others, allowing all three to have general parameters. This amounts to studying Euler transforms of products of two incomplete beta functions. We provide a closed form for the general problem in terms of Kampé de Fériet functions and a variety of simpler closed forms in special cases. The results are applied to derive the moments of the maximum of two independent beta-distributed random variables and to find inner products of incomplete beta functions. Restricted to positive integer parameters, our results are applied to determine an expected exit time for a conditioned random walk and also to a combinatorial problem of enumerating strings comprised of three different letters, subject to constraints.

On representations and simulation of conditioned random walks on integer lattices

On representations and simulation of conditioned random walks on integer lattices

Localization at the boundary for conditioned random walks in random environment in dimensions two and higher

We introduce the notion of localization at the boundary for conditioned random walks in i.i.d. and uniformly elliptic random environment on Zd, in dimensions two and higher. If d=2 or 3, we prove localization for (almost) all walks. In contrast, for d≥4, there is a phase transition for environments of the form ωe(x,e)=α(e)(1+eξ(x,e)), where {ξ(x)}x∈Zd is an i.i.d. sequence of random variables, and e represents the amount of disorder with respect to a simple random walk.

Identifying Spatiotemporal Clusters by Means of Agglomerative Hierarchical Clustering and Bayesian Regression Analysis with Spatiotemporally Varying Coefficients: Methodology and Application to Dengue Disease in Bandung, Indonesia

Dengue disease has serious health and socioeconomic consequences. Preventive actions are needed to avoid outbreaks. Bayesian spatiotemporal models with conditional autoregressive (CAR) and random walk (RW) priors are two common smoothing approaches used in disease mapping to develop early warning systems and action plans. However, this approach can lead to over‐smoothing, such that discontinuities in the risk surface are “averaged” out. Moreover, local variation in the relationship between disease risk and spatiotemporal risk factors can be concealed by the assumption of homogeneous regression effects. To overcome these problems, we propose a two‐stage spatiotemporal clustering approach. In the first stage, an agglomerative hierarchical clustering algorithm is applied to map out the complete set of cluster configurations. In the second stage, the optimal cluster configuration is selected using a spatiotemporal model with spatiotemporally varying coefficients. The methodology is applied to data for 30 districts in the city of Bandung, Indonesia, for the period January 2012 to December 2016. Weather factors are used as the risk factors. High‐risk and low‐risk clusters are identified that strongly vary across space and time, with sunshine duration, precipitation, evaporation, and humidity as the most important risk factors.

Mixing via controllability for randomly forced nonlinear dissipative PDEs

We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that the unperturbed equation has exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.

Conditioned Two-Dimensional Simple Random Walk: Green’s Function and Harmonic Measure

We study the Doob's $h$-transform of the two-dimensional simple random walk with respect to its potential kernel, which can be thought of as the two-dimensional simple random walk conditioned on never hitting the origin. We derive an explicit formula for the Green's function of this random walk, and also prove a quantitative result on the speed of convergence of the (conditional) entrance measure to the harmonic measure (for the conditioned walk) on a finite set.

On the range of a two-dimensional conditioned simple random walk

We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient and we show that it is recurrent in the sense that each infinite set is visited infinitely often, almost surely. We prove that, for a large set, the proportion of its sites visited by the conditioned walk is approximately a Uniform$[0,1]$ random variable. Also, given a set $G\subset\mathbb{R}^2$ that does not surround the origin, we prove that a.s. there is an infinite number of $k$'s such that $kG\cap \mathbb{Z}^2$ is unvisited. These results suggest that the range of the conditioned walk has fractal behavior.

Connectivity properties of Branching Interlacements

We consider connectivity properties of the Branching Interlacements model in $\mathbb{Z}^d,~d\ge5$, recently introduced by Angel, R\'ath and Zhu in 2016. Using stochastic dimension techniques we show that every two vertices visited by the branching interlacements are connected via at most $\lceil d/4\rceil$ conditioned critical branching random walks from the underlying Poisson process, and that this upper bound is sharp. In particular every such two branching random walks intersect if and only if $5\le d\le 8$. The stochastic dimension of branching random walk result is of independent interest. We additionally obtain heat kernel bounds for branching random walks conditioned on survival.

On the maximum of conditioned random walks and tightness for pinning models

We consider real random walks with finite variance. We prove an optimal integrability result for the diffusively rescaled maximum, when the walk or its bridge is conditioned to stay positive, or to avoid zero. As an application, we prove tightness under diffusive rescaling for general pinning and wetting models based on random walks.

A one-dimensional version of the random interlacements

We base ourselves on the construction of the two-dimensional random interlacements (Comets et al., 2016) to define the one-dimensional version of the process. For this, we consider simple random walks conditioned on never hitting the origin. We compare this process to the conditional random walk on the ring graph. Our results are the convergence of the vacant set on the ring graph to the vacant set of one-dimensional random interlacements, a central limit theorem for the interlacements’ local time and the convergence in law of the local times of the conditional walk on the ring graph to the interlacements’ local times.

Conditioned random walks and interaction-driven condensation

We consider a discrete-time continuous-space random walk under the constraints that the number of returns to the origin (local time) and the total area under the walk are fixed. We first compute the joint probability of an excursion having area a and returning to the origin for the first time after time τ. We then show how condensation occurs when the total area constraint is increased: an excursion containing a finite fraction of the area emerges. Finally we show how the phenomena generalises previously studied cases of condensation induced by several constraints and how it is related to interaction-driven condensation which allows us to explain the phenomenon in the framework of large deviation theory.

Universality of local times of killed and reflected random walks

In this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential distribution. The proof is based on known results for conditioned random walks, which allow to determine the asymptotic behaviour of moments of local times. Using this information we also show that the field of local times of a reflected random walk converges in the sense of finite dimensional distributions. This is in the spirit of the seminal result by Knight [10] who has shown that for the symmetric simple random walk local times converge weakly towards a squared Bessel process. Our result can be seen as an extension of the second Ray-Knight theorem to all asymptotically stable random walks.

Conditioned random walks from Kac-Moody root systems

Random paths are time continuous interpolations of random walks. By using Littelmann path model, we associate to each irreducible highest weight module of a Kac Moody algebra g a random path W. Under suitable hypotheses, we make explicit the probability of the event E: W never exits the Weyl chamber of g. We then give the law of the random walk defined by W conditioned by the event E and proves this law can be recovered by applying to W the generalized Pitmann transform introduced by Biane, Bougerol and O'Connell. This generalizes the main results of [10] and [16] to Kac Moody root systems and arbitrary highest weight modules. Moreover, we use here a completely new approach by exploiting the symmetry of our construction under the action of the Weyl group of g rather than renewal theory and Doob's theorem on Martin kernels.

Graph-based clustering and ranking for diversified image search

In this paper, we consider the problem of clustering and re-ranking web image search results so as to improve diversity at high ranks. We propose a novel ranking framework, namely cluster-constrained conditional Markov random walk (CCCMRW), which has two key steps: first, cluster images into topics, and then perform Markov random walk in an image graph conditioned on constraints of image cluster information. In order to cluster the retrieval results of web images, a novel graph clustering model is proposed in this paper. We explore the surrounding text to mine the correlations between words and images and therefore the correlations are used to improve clustering results. Two kinds of correlations, namely word to image and word to word correlations, are mainly considered. As a standard text process technique, tf-idf method cannot measure the correlation of word to image directly. Therefore, we propose to combine tf-idf method with a novel feature of word, namely visibility, to infer the word-to-image correlation. By latent Dirichlet allocation model, we define a topic relevance function to compute the weights of word-to-word correlations. Taking word to image correlations as heterogeneous links and word-to-word correlations as homogeneous links, graph clustering algorithms, such as complex graph clustering and spectral co-clustering, are respectively used to cluster images into topics in this paper. In order to perform CCCMRW, a two-layer image graph is constructed with image cluster nodes as upper layer added to a base image graph. Conditioned on the image cluster information from upper layer, Markov random walk is constrained to incline to walk across different image clusters, so as to give high rank scores to images of different topics and therefore gain the diversity. Encouraging clustering and re-ranking outputs on Google image search results are reported in this paper.

Continuity of asymptotic characteristics for random walks on hyperbolic groups

We describe a new approach to proving the continuity of asymptotic entropy as a function of a transition measure under a finite first moment condition. It is based on using conditional random walks and amounts to checking uniformity in the strip criterion for the identification of the Poisson boundary. It is applicable to word hyperbolic groups and in several other situations when the Poisson boundary can be identified with an appropriate geometric boundary.