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Summary: The wireless sensor network (WSN) has been assimilated into modern society and is utilized in many crucial application domains, including animal monitoring, border surveillance, asset monitoring, and so forth. These technologies aid in protecting the place of the event's occurrence from the adversary. Maintaining privacy concerning the source location is challenging due to the sensor nodes' limitations and efficient routing strategies. Hence, this research introduces a novel source location privacy preservation using the awareness‐based Gannet with random‐Dijkstra's algorithm (AGO‐RD). The network is initialized by splitting the hotspot and non‐hotspot region optimally using the proposed awareness‐based Gannet (AGO) algorithm. Here, the multi‐objective fitness function is utilized to initialize the network based on factors like throughput, energy consumption, latency, and entropy. Then, the information is forwarded to the phantom node in the non‐hotspot region to preserve the source location's privacy, which is far from the sink node. The proposed random‐Dijkstra algorithm is utilized to route the information from the phantom node to the sink with more security. Analysis of the proposed AGO‐RD‐based source location privacy preservation technique in terms of delay, throughput, network lifetime, and energy consumption accomplished the values of 6.52 ms, 95.68%, 7109.9 rounds, and 0.000125 μJ. [ABSTRACT FROM AUTHOR]

Influence maximization in online social networks is used to select a set of influential seed nodes to maximize the influence spread under a given diffusion model. However, most existing proposals have huge computational costs and only consider the dyadic influence relationship between two nodes, ignoring the higher-order influence relationships among multiple nodes. It limits the applicability and accuracy of existing influence diffusion models in real complex online social networks. To this end, in this paper, we present a novel information diffusion model by introducing hypergraph theory to determine the most influential nodes by jointly considering adjacent influence and higher-order influence relationships to improve diffusion efficiency. We mathematically formulate the influence maximization problem under higher-order influence relationships in online social networks. We further propose a hypergraph sampling greedy algorithm (HSGA) to effectively select the most influential seed nodes. In the HSGA, a random walk-based influence diffusion method and a Monte Carlo-based influence approximation method are devised to achieve fast approximation and calculation of node influences. We conduct simulation experiments on six real datasets for performance evaluations. Simulation results demonstrate the effectiveness and efficiency of the HSGA, and the HSGA has a lower computational cost and higher seed selection accuracy than comparison mechanisms. [ABSTRACT FROM AUTHOR]

Growth in agricultural total factor productivity (TFP), which explains most of the long‐term growth in U.S. agricultural output, may be slowing. The Economic Research Service (ERS) of the USDA is confident that current levels of below‐average growth will eventually regain the long‐term trend line. Others disagree, arguing instead that due to declining public expenditures on agricultural research, TFP growth experienced a downward and seemingly permanent structural shift about 30 years ago. In this paper, I argue that neither perspective is accurate since agricultural TFP is best modeled as a random walk with drift and thus not governed by a deterministic trend line. When I use a first difference model to accommodate the unit root, I do not find a structural break in the rate of drift. However, I acknowledge that this finding may not be general because I show that my test for a structural break has low power. To add theoretical relevance, I develop a simple model of stochastic innovation and farm technology adoption, and then use simulation results from my model to explain why a random walk for agricultural TFP is a theoretically sound proposition. [ABSTRACT FROM AUTHOR]

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan [Probab. Theory Related Fields 93 (1992), pp. 377–392]. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero–one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk. [ABSTRACT FROM AUTHOR]

We 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. [ABSTRACT FROM AUTHOR]

In this paper, we show that traditional comparisons of mean squared prediction error (MSPE) between two competing forecasts may be highly controversial. This is so because when some specific conditions of efficiency are not met, the forecast displaying the lowest MSPE will also display the lowest correlation with the target variable. Given that violations of efficiency are usual in the forecasting literature, this opposite behavior in terms of accuracy and correlation with the target variable may be a fairly common empirical finding that we label here as "the MSPE paradox." We characterize "paradox zones" in terms of differences in correlation with the target variable and conduct some simple simulations to show that these zones may be non‐empty sets. Finally, we illustrate the relevance of the paradox with a few empirical applications. [ABSTRACT FROM AUTHOR]

In a previous work, we defined and studied random walk labelings of graphs. These are graph labelings that are obtainable by performing a random walk on the graph, such that each vertex is labeled upon its first visit. In this work, we calculate the number of random walk labelings of several natural graph families: The wheel, fan, barbell, lollipop, tadpole, friendship, and snake graphs. Additionally, we prove several combinatorial identities that emerged during the calculations. [ABSTRACT FROM AUTHOR]

The study of drug–target interaction plays an important role in the process of drug development. The subject of DTI forecasting has advanced significantly in the last several years, yielding numerous significant research findings and methodologies. Heterogeneous data sources provide richer information and comprehensive perspectives for drug–target interaction prediction, so many existing methods rely on heterogeneous networks, and graph embedding technology becomes an important technology to extract information from heterogeneous networks. These approaches, however, are less concerned with potential noisy information in heterogeneous networks and more focused on the extent of information extraction in those networks. Based on this, a potential DTI predictive network model called FBRWPC is proposed in this paper. It uses a fine-grained similarity selection program to first integrate similarity on similar networks and then a bidirectional random walk graph embedding learning method with restart to obtain an updated drug target interaction matrix. Through the use of similarity selection and fine-grained selection similarity integration, the framework can effectively filter out the noise present in heterogeneous networks and enhance the model's prediction performance. The experimental findings demonstrate that, even after being split up into four distinct types of data sets, FBRWPC can still retain great prediction performance, a sign of the model's resilience and good generalization. [ABSTRACT FROM AUTHOR]

We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate μ between open and closed, following a Glauber dynamics for the random cluster model with parameters p, q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n 2 / μ . In our proof we construct a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable. [ABSTRACT FROM AUTHOR]

A complex network is an abstract modeling of complex systems in the real world, which plays an important role in analyzing the function of complex systems. Community detection is an important tool for analyzing network structure. In this paper, we propose a new community detection algorithm (RWBS) based on different seed nodes which aims to understand the community structure of the network, which provides a new idea for the allocation of resources in the network. RWBS provides a new centrality metric ( M C ) to calculate node importance, which calculates the ranking of nodes as seed nodes. Furthermore, two algorithms are proposed for determining seed nodes on networks with and without ground truth, respectively. We set the number of steps for the random walk to six according to the six degrees of separation theory to reduce the running time of the algorithm. Since some traditional community detection algorithms may detect smaller communities, e.g., two nodes become one community, this may make the resource allocation unreasonable. Therefore, modularity (Q) is chosen as the optimization function to combine communities, which can improve the quality of detected communities. Final experimental results on real-world and synthetic networks show that the RWBS algorithm can effectively detect communities. [ABSTRACT FROM AUTHOR]

A celebrated result in probability theory is that a simple symmetric random walk on the d-dimensional lattice \mathbb {Z}^d is recurrent for d=1,2 and transient for d\geq 3. In this note, we derive a closed-form expression, in terms of the Lauricella function F_C, for the return probability for all d\geq 3. Previously, a closed-form formula had only been available for d=3. [ABSTRACT FROM AUTHOR]

The effectiveness of agricultural spraying processes depends considerably on the ability of the atomized droplets to reach the target site in the desired amount. In this work, two mathematical models to study the trajectories and deposition of atomized droplets are implemented and compared. On the one hand, a computational fluid dynamics (CFD) coupled with discrete phase model (DPM) is implemented to calculate the trajectories of atomized droplets and determine distances at which the droplets are deposited. The continuous phase (atmospheric air) is modelled by continuity, momentum, and energy equations. On the other hand, a Lagrangian random‐walk (LRW) model based on force and energy balances to predict the pulverization process of a nozzle is formulated and implemented in Python. Both models take into account the effects of drag, gravity, buoyancy, and evaporation on individual droplets, as well as the impact of atmospheric stability and dispersion. By tracking a large number of trajectories, meaningful estimates of dispersal statistics can be obtained. The LRW model accurately replicated the trajectories, deposition distances, and final diameters of atomized droplets for three atmospheric stability cases, compared with CFD simulation results. The results of both models agreed that 100 μm droplets were most susceptible to wind‐induced spray drift, depositing at the furthest distances from the nozzle. In addition, 50 μm droplets exhibited a significant tendency to evaporate entirely before reaching the ground. The LRW model is found to be a cost‐effective alternative for estimating spray drift compared to the computationally intensive CFD approach. [ABSTRACT FROM AUTHOR]

In this paper, we provide the mathematical foundation for an explicit and universal feature of cover time for a large class of random work processes, which was previously observed by Chupeau et al (2015 Nat. Phys. 11 844–7). Specifically, we rigorously establish that the fluctuations of the cover time, normalized by the mean first passage time, follow a Gumbel distribution, for finite-range, symmetric, irreducible random walks on a torus of dimension three or higher. The result contributes to a better understanding of cover-time behavior in random search processes, especially on the efficiency of exhaustive searches. Our approach builds upon the work of Belius (2013 Probab. Theory Relat. Fields 157 635–89) on cover times for simple random walks, leveraging a strong coupling between the random walk and random interlacements. [ABSTRACT FROM AUTHOR]

The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two point function as a sum over random walks. An analogous expansion for Fermions on non-Euclidean geometries is still lacking. Casiday et al (2022 Linear Multilinear Algebr. 72 325–65) proposed a classical 'Dirac walk' diffusing on vertices and edges of an oriented graph with a square root of the graph Laplacian. In contrast to the simple random walk, each step of the walk is given a sign depending on the orientation of the edge it goes through. In a toy model, we propose here to study the Green functions, spectrum and the spectral dimension of such 'Dirac walks' on the Bethe lattice, a d -regular tree. The recursive structure of the graph makes the problem exactly solvable. Notably, we find that the spectrum develops a gap and that the spectral dimension of the Dirac walk matches that of the simple random walk ( d s = 1 for d = 2 and d s = 3 for d ⩾ 3 ). [ABSTRACT FROM AUTHOR]

By the dimension reduction idea, overshoot for random walks, coupling and martingale arguments, we obtain a simpler and easily computable expression for the first-order correction constant between discrete harmonic measures for random walks with rotationally invariant step distribution in R d (d ≥ 2) and the corresponding continuous counterparts. This confirms and extends a conjecture in Jiang and Kennedy (J Theor Probab 30(4):1424–1444, 2017), and simplifies the related expression of Wang et al. (Bernoulli 25(3):2279–2300, 2019). Furthermore, we propose a universality conjecture on high-order corrections for error estimation between generalized discrete harmonic measures and their continuous counterparts, which generalizes the universality conjecture of the first-order correction in Kennedy (J Stat Phys 164(1):174–189, 2016); and we prove this conjecture heuristically for the rotationally invariant case, and also provide several examples of second-order error corrections to check the conjecture by a numerical simulation argument. [ABSTRACT FROM AUTHOR]

A random walk with counterbalanced steps is a process of partial sums Š(n) = X1 + • • • + Xn whose steps Xn are given recursively as follows. For each n ≥ 2, with a fixed probability p, Xn is a new independent sample from some fixed law μ, and with complementary probability 1 - p, Xn = -Xυ(n) counterbalances a previous step, with υ(n) uniform random pick from {1,...,n -1} We determine the asymptotic behavior of Š(n) in terms of p and the first two moments of μ. Our approach relies on a coupling with a reinforcement algorithm due to H. A. Simon, and on properties of random recursive trees and Eulerian numbers, which may be of independent interest. The method can be adapted to the situation where the step distribution μ belongs to the domain of attraction of a stable law. [ABSTRACT FROM AUTHOR]

The article offers a method for estimating the volatility covariance matrix of vectors of financial time series data using a change point approach. The proposed method supersedes general varying-coefficient parametric models, such as GARCH, whose coefficients may vary with time, by a change point model. In this study, an adaptive pointwise selection of homogeneous segments with a given right-end point by a local change point analysis is introduced. Sufficient conditions are obtained under which the maximum likelihood process is adaptive against the covariance estimate to yield an optimal rate of convergence with respect to the change size. This rate is preserved while allowing the jump size to diminish. Under these circumstances, argmax results of a two-sided negative Brownian motion or a two-sided negative drift random walk under vanishing and non-vanishing jump size regimes, respectively, provide inference for the change point parameter. Theoretical results are supported by the Monte–Carlo simulation study. A bivariate data on daily log returns of two US stock market indices as well as tri-variate data on daily log returns of three banks are analyzed by constructing confidence interval estimates for multiple change points that have been identified previously for each of the two data sets. [ABSTRACT FROM AUTHOR]

The Grey Wolf Optimization Algorithm (GWO) is a member of the swarm intelligence algorithm family, which possesses the highlights of easy realization, simple parameter settings and wide applicability. However, in some large-scale application problems, the grey wolf optimization algorithm easily gets trapped in local optima, exhibits poor global exploration ability and suffers from premature convergence. Since grey wolf's update is guided only by the best three wolves, it leads to low population multiplicity and poor global exploration capacity. In response to the above issues, we design a multi-strategy collaborative grey wolf optimization algorithm (NOGWO). Firstly, we use a random walk strategy to extend the exploration scope and enhance the algorithm's global exploration capacity. Secondly, we add an opposition-based learning model influenced by refraction principle to generate an opposite solution for each population, thereby improving population multiplicity and preventing the algorithm from being attracted to local optima. Finally, to balance local exploration and global exploration and elevate the convergence effect, we introduce a novel convergent factor. We conduct experimental testing on NOGWO by using 30 CEC2017 test functions. The experimental outcomes indicate that compared with GWO and some swarm intelligence algorithms, NOGWO has better global exploration capacity and convergence accuracy. In addition, we also apply NOGWO to three engineering problems and an unmanned aerial vehicle path planning problem. The outcomes of the experiment suggest that NOGWO performs well in solving these practical problems. [ABSTRACT FROM AUTHOR]

Abstract The study of drug–target interaction plays an important role in the process of drug development. The subject of DTI forecasting has advanced significantly in the last several years, yielding numerous significant research findings and methodologies. Heterogeneous data sources provide richer information and comprehensive perspectives for drug–target interaction prediction, so many existing methods rely on heterogeneous networks, and graph embedding technology becomes an important technology to extract information from heterogeneous networks. These approaches, however, are less concerned with potential noisy information in heterogeneous networks and more focused on the extent of information extraction in those networks. Based on this, a potential DTI predictive network model called FBRWPC is proposed in this paper. It uses a fine-grained similarity selection program to first integrate similarity on similar networks and then a bidirectional random walk graph embedding learning method with restart to obtain an updated drug target interaction matrix. Through the use of similarity selection and fine-grained selection similarity integration, the framework can effectively filter out the noise present in heterogeneous networks and enhance the model's prediction performance. The experimental findings demonstrate that, even after being split up into four distinct types of data sets, FBRWPC can still retain great prediction performance, a sign of the model's resilience and good generalization.

The discovery of novel therapeutic targets, defined as proteins which drugs can interact with to induce therapeutic benefits, typically represent the first and most important step of drug discovery. One solution for target discovery is target repositioning, a strategy which relies on the repurposing of known targets for new diseases, leading to new treatments, less side effects and potential drug synergies. Biological networks have emerged as powerful tools for integrating heterogeneous data and facilitating the prediction of biological or therapeutic properties. Consequently, they are widely employed to predict new therapeutic targets by characterizing potential candidates, often based on their interactions within a Protein-Protein Interaction (PPI) network, and their proximity to genes associated with the disease. However, over-reliance on PPI networks and the assumption that potential targets are necessarily near known genes can introduce biases that may limit the effectiveness of these methods. This study addresses these limitations in two ways. First, by exploiting a multi-layer network which incorporates additional information such as gene regulation, metabolite interactions, metabolic pathways, and several disease signatures such as Differentially Expressed Genes, mutated genes, Copy Number Alteration, and structural variants. Second, by extracting relevant features from the network using several approaches including proximity to disease-associated genes, but also unbiased approaches such as propagation-based methods, topological metrics, and module detection algorithms. Using prostate cancer as a case study, the best features were identified and utilized to train machine learning algorithms to predict 5 novel promising therapeutic targets for prostate cancer: IGF2R, C5AR, RAB7, SETD2 and NPBWR1.

In recent years, with the strong development of information technology, detecting communities in large real networks is a very important issue which is of interest to many scientists. Community detection in large real networks with millions of nodes is often difficult. To solve this problem, many online community search algorithms have been proposed with many different approaches. One of the approaches is to coordinate the vertices of the graph and build a reasonable distance between those vertices. It has been observed that vertices in the same community have approximately the same probability of reaching other vertices through a random walk. Based on this principle, the authors propose a way to coordinate vertices and build distances between vertices in the graph that reduces computational complexity compared to existing techniques. This approach involves representing peaks as vectors and using the Kmeans++ algorithm for community detection, whose effectiveness is evaluated through experimental results presented.

Recommender systems provide important functions in areas such as dealing with data overload, providing personalized consulting services, and assisting clients in investment decisions. However, the cold start problem in recommender systems has always been in urgent need of solution and optimization. Based on this, this paper classifies the traditional methods and cutting-edge methods to solve the cold start problem, and expounds the research progress and excellent methods in recent years. Firstly, three traditional solutions to the cold start problem are summarized: recommendation based on content filtering, recommendation based on collaborative filtering, and hybrid recommendation. Secondly, the current cutting-edge recommendation algorithms to solve the cold start problem are summarized, and they are classified into the data-driven strategy and the method-driven strategy. The method-driven strategy is divided into algorithms based on meta-learning, algorithms based on context information and session str-ategy, algorithms based on random walk, algorithms based on heterogeneous graph information and attribute graph, and algorithms based on adversarial mechanism. According to the type of cold start problem, the algorithms are divided into two categories: new users and new items. Then, according to the particularity of the recommendation field, the cold start problem of the recommendation in the multimedia information field and the online e-commerce platform field is expounded. Finally, the possible research directions to solve the cold start problem in the future are summarized.

We consider fractional integral operators (I − T) d , d ∈ (− 1 , 1) acting on functions g : Z ν → R , ν ≥ 1 , where T is the transition operator of a random walk on Z ν . We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ (s ; d) , s ∈ Z ν of (I − T) d . The asymptotic behavior of τ (s ; d) as | s | → ∞ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Z ν solving the difference equation (I − T) d X = ε with white noise on the right-hand side is discussed and their scaling limits. Several examples, including fractional lattice Laplace and heat operators, are studied in detail. [ABSTRACT FROM AUTHOR]

We investigate three aspects of weak* convergence of the n-step distributions of random walks on finite volume homogeneous spaces $ G/\Gamma $ G / Γ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesàro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesàro convergence towards Haar measure for uniquely ergodic random walks. [ABSTRACT FROM AUTHOR]

We discuss the qualitatively new properties of random walks on groups that arise in the situation when the entropy of the step distribution is infinite. [ABSTRACT FROM AUTHOR]

This paper deals with a reliability system hit by three types of shocks ranked as harmless, critical, or extreme, depending on their magnitudes, being below H 1 , between H 1 and H 2 , and above H 2 , respectively. The system's failure is caused by a single extreme shock or by a total of N critical shocks. In addition, the system fails under occurrences of M pairs of shocks with lags less than some δ (δ -shocks) in any order. Thus, the system fails when one of the three named cumulative damages occurs first. Thus, it fails due to the competition of the three associated shock processes. We obtain a closed-form joint distribution of the time-to-failure, shock count upon failure, δ -shock count, and cumulative damage to the system on failure, to name a few. In particular, the reliability function directly follows from the marginal distribution of the failure time. In a modified system, we restrict δ -shocks to those with small lags between consecutive harmful shocks. We treat the system as a generalized random walk process and use an embellished variant of discrete operational calculus developed in our earlier work. We demonstrate analytical tractability of our formulas which are also validated, through Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

Internal DLA (IDLA) is an internal aggregation model in which particles perform random walks from the origin, in turn, and stop upon reaching an unoccupied site. Levine and Peres showed that, when particles start instead from fixed multiple-point distributions, the modified IDLA processes have deterministic scaling limits related to a certain obstacle problem. In this paper, we investigate the convergence rate of this "extended source" IDLA in the plane to its scaling limit. We show that, if δ is the lattice size, fluctuations of the IDLA occupied set are at most of order δ 3 / 5 from its scaling limit, with probability at least 1 - e - 1 / δ 2 / 5 . [ABSTRACT FROM AUTHOR]

Neighborhood rough sets, as an effective tool for processing numerical data, is widely used in many fields, such as data mining, machine learning and decision-making system. However, most of the existing neighborhood rough set-based attribute reduction algorithms have low efficiency. To address the limitation, this paper has proposed an efficient positive region search algorithm based on multiple hash buckets and multiple granularity mechanisms. This algorithm achieves a more accurate neighborhood extent by superimposing the effects of multiple hash buckets, and accelerates positive region searching through the idea of multiple granularity. In addition, on the foundation the positive region search algorithm, we improved the existing algorithm and proposed an attribute reduction algorithm based on multi-hash bucket and multi-granularity. To further remove the redundant attributes, the two algorithms mentioned above are applied into a novel attribute reduction approach based on random walk. Experiments conducted on UCI datasets show that our attribute reduction algorithm has high efficiency. Moreover, attribute reduction approach we proposed can further compress the reduced attribute set, and the results maintain similar or even better classification accuracy. [ABSTRACT FROM AUTHOR]

We study a counterbalanced random walk S ˇ n = X ˇ 1 + ⋯ + X ˇ n , which is a discrete time non-Markovian process and X ˇ n are given recursively as follows. For n ≥ 2, X ˇ n is a new independent sample from some fixed law μ ≠ 0 with a fixed probability p, and X ˇ n = − X ˇ v (n) with probability 1 − p, where v(n) is a uniform random variable on {1, ⋯, n − 1}. We apply martingale method to obtain a strong invariance principle for Šn. [ABSTRACT FROM AUTHOR]

We consider an asymptotically stable multidimensional random walk S (n) = (S 1 (n) , ... , S d (n)) . For every vector x = (x 1 ... , x d) with x 1 ≥ 0 , let τ x : = min { n > 0 : x 1 + S 1 (n) ≤ 0 } be the first time the random walk x + S (n) leaves the upper half space. We obtain the asymptotics of p n (x , y) : = P (x + S (n) ∈ y + Δ , τ x > n) as n tends to infinity, where Δ is a fixed cube. From that, we obtain the local asymptotics for the Green function G (x , y) : = ∑ n p n (x , y) , as | y | and/or | x | tend to infinity. [ABSTRACT FROM AUTHOR]

For any fixed real a > 0 and x ∈ R d , d ≥ 1 , we consider the real-valued random process (S n) n ≥ 0 defined by S 0 = a , S n = a + ln | g n ⋯ g 1 x | , n ≥ 1 , where the g k , k ≥ 1 , are i.i.d. nonnegative random matrices. By using the strategy initiated by Denisov and Wachtel to control fluctuations in cones of d-dimensional random walks, we obtain an asymptotic estimate and bounds on the probability that the process (S n) n ≥ 0 remains nonnegative up to time n and simultaneously belongs to some compact set [ b , b + ℓ ] ⊂ R ∗ + at time n. [ABSTRACT FROM AUTHOR]

In this paper, we consider (upward skip-free) discrete-time and discrete-space Markov additive chains (MACs) and develop the theory for the so-called W ~ and Z ~ scale matrices, which are shown to play a vital role in the determination of a number of exit problems and related fluctuation identities. The theory developed in this fully discrete set-up follows similar lines of reasoning as the analogous theory for Markov additive processes in continuous time and is exploited to obtain the probabilistic construction of the scale matrices, identify the form of the generating function and produce a simple recursion relation for W ~ , as well as its connection with the so-called occupation mass formula. In addition to the standard one- and two-sided exit problems (upwards and downwards), we also derive distributional characteristics for a number of quantities related to the one- and two-sided 'reflected' processes. [ABSTRACT FROM AUTHOR]

We study the simple random walk on trees and give estimates on the mixing and relaxation times. Relying on a seminal result by Basu, Hermon and Peres characterizing cutoff on trees, we give geometric criteria that are easy to verify and allow to determine whether the cutoff phenomenon occurs. We provide a general characterization of families of trees with cutoff, and show how our criteria can be used to prove the absence of cutoff for several classes of trees, including spherically symmetric trees, Galton–Watson trees of a fixed height, and sequences of random trees converging to the Brownian continuum random tree. [ABSTRACT FROM AUTHOR]

We consider an SIR (susceptible $\to$ infective $\to$ recovered) epidemic in a closed population of size n , in which infection spreads via mixing events, comprising individuals chosen uniformly at random from the population, which occur at the points of a Poisson process. This contrasts sharply with most epidemic models, in which infection is spread purely by pairwise interaction. A sequence of epidemic processes, indexed by n , and an approximating branching process are constructed on a common probability space via embedded random walks. We show that under suitable conditions the process of infectives in the epidemic process converges almost surely to the branching process. This leads to a threshold theorem for the epidemic process, where a major outbreak is defined as one that infects at least $\log n$ individuals. We show further that there exists $\delta \gt 0$ , depending on the model parameters, such that the probability that a major outbreak has size at least $\delta n$ tends to one as $n \to \infty$. [ABSTRACT FROM AUTHOR]