Two explicit Skorokhod embeddings for simple symmetric random walk

Motivated by problems in behavioural finance, we provide two explicit constructions of a randomized stopping time which embeds a given centred distribution $\mu$ on integers into a simple symmetric random walk in a uniformly integrable manner. Our first construction has a simple Markovian structure: at each step, we stop if an independent coin with a state-dependent bias returns tails. Our second construction is a discrete analogue of the celebrated Azéma–Yor solution and requires independent coin tosses only when excursions away from maximum breach predefined levels. Further, this construction maximizes the distribution of the stopped running maximum among all uniformly integrable embeddings of $\mu$.     

On the structure of stable random walks

We show that the Cauchy random walk on the line, and the Gaussian random walk on the plane are similar as infinite measure preserving transformations.     

An elementary approach to Brownian local time based on simple,symmetric random walks

Summary In this paper we define Brownian local time as the almost sure limit of the local times of a nested sequence of simple, symmetric random walks. The limit is jointly continuous in <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(t,x)$. The rate of convergence is $n^{\frac14} (\log n)^{\frac34}$ that is close to the best possible. The tools we apply are almost exclusively from elementary probability theory.     

On the range of reversible random walks onZ 2 in a random environment

In this paper, a weak law of large numbers is obtained for the range of two dimensional reversible random walk in a random environment.Partly supported by NSF of China.     

An intersection property of the simple random walks inZ d

Let {S _d (n)} _n0 be the simple random walk inZ ^d , and ^(d)(a,b)={S _d (n)Z ^d :anb}. Supposef(n) is an integer-valued function and increases to infinity asn tends to infinity, andE _n ^(d) ={^(d)(0,n)^(d)(n+f(n),)}. In this paper, a necessary and sufficient condition to ensureP(E _n ^d) ,i.o.)=0, or 1 is derived ford=3, 4. This problem was first studied by P. Erdös and S.J. Taylor.This work is partly supported by the National Natural Sciences Foundation of China.     

Individual behaviors of oriented walks

Given an infinite sequence t=(_k)_k of −1 and +1, we consider the oriented walk defined by S_n(t)=∑_k=1ⁿ₁₂…_k. The set of t's whose behaviors satisfy S_n(t)bn^τ is considered ( and 0<τ1 being fixed) and its Hausdorff dimension is calculated. A two-dimensional model is also studied. A three-dimensional model is described, but the problem remains open.     

Diffusivity of a random walk on random walks

We consider a random walk with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor with respect to the case of the classical simple random walk without constraint. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 267–283, 2015     

Random walks on random simple graphs

This paper looks at random regular simple graphs and considers nearest neighbor random walks on such graphs. This paper considers walks where the degree d of each vertex is around (log n)^a where a is a constant which is at least 2 and where n is the number of vertices. By extending techniques of Dou, this paper shows that for most such graphs, the position of the random walk becomes close to uniformly distributed after slightly more than log n/log d steps. This paper also gets similar results for the random graph G(n, p), where p = d/(n − 1). © 1996 John Wiley & Sons, Inc.     

A random environment for linearly edge-reinforced random walks on infinite graphs

We consider linearly edge-reinforced random walk on an arbitrary locally finite connected graph. It is shown that the process has the same distribution as a mixture of reversible Markov chains, determined by time-independent strictly positive weights on the edges. Furthermore, we prove bounds for the random weights, uniform, among others, in the size of the graph.      

Expected cover times of random walks on symmetric graphs

We give very simple proofs for an (N–1)H _N–1 lower bound and anN ² upper bound for the expected cover time of symmetric graphs.     

Self-intersections of random walks on lattices

Let {X _n ^d }_n≥0be a uniform symmetric random walk on Z^d, and Π^(d) (a,b)={X _n ^d ∈ Z^d : a ≤ n ≤ b}. Suppose f(n) is an integer-valued function on n and increases to infinity as n↑∞, and let

Laws of iterated logarithm for transient random walks in random environments

We consider laws of iterated logarithm for one-dimensional transient random walks in random environments. A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environments, including independent identically distributed environments and uniformly ergodic environments.     

Random walks in random environments on metric groups

Random walks in random environments on countable metric groups with bounded jumps of the walking particle are considered. The transition probabilities of such a random walk from a pointx εG (whereG is the group in question) are described by a vectorp(x) ε ℝ^|W| (whereW ⊏G is fixed and |W|<∞). The set {p(x),x εG} is assumed to consist of independent identically distributed random vectors. A sufficient condition for this random walk to be transient is found. As an example, the groups ℤ ^d , free groups, and the free product of finitely many cyclic groups of second order are considered. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 129–135, January, 2000.     

Branching random walks with random environments in time

We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the convergence of the free energy n-llog Zn（t）, large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn（t）/E[Zn（t）ξ].     

On the support of the simple branching random walk

Connectivity of the support of the simple branching random walk is established in certain asymmetric cases, extending a previous result of Grill.     

The type problem for random walks on trees

We consider the question of whether the simple random walk (SRW) on an infinite tree is transient or recurrent. For random-trees (all vertices of distancen from the root of the tree have degreed _n , where {d _n } are independent random variables), we prove that the SRW is a.s. transient if lim inf _n n E(log(d _n-1))>1 and a.s. recurrent if lim sup _n n E(log(d _n-1))<1. For random trees in which the degrees of the vertices are independently 2 or 3, with distribution depending on the distance from the root, a partial classification of type is obtained.Research supported in part by NSF DMS 8710027.     

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.      

Deterministic random walks on finite graphs

The rotor‐router model, also known as the Propp machine, is a deterministic process analogous to a random walk on a graph. Instead of distributing tokens to randomly chosen neighbors, the Propp machine deterministically serves the neighbors in a fixed order by associating to each vertex a “rotor‐router” pointing to one of its neighbors. This paper investigates the discrepancy at a single vertex between the number of tokens in the rotor‐router model and the expected number of tokens in a random walk, for finite graphs in general. We show that the discrepancy is bounded by O (mn) at any time for any initial configuration if the corresponding random walk is lazy and reversible, where n and m denote the numbers of nodes and edges, respectively. For a lower bound, we show examples of graphs and initial configurations for which the discrepancy at a single vertex is Ω(m) at any time (> 0). For some special graphs, namely hypercube skeletons and Johnson graphs, we give a polylogarithmic upper bound, in terms of the number of nodes, for the discrepancy. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 46,739–761, 2015     

Series of tail distributions of finitely inhomogeneous random walks

Let Sn=X₁+?+Xn be a random walk, where the steps Xn are independent random variables having a finite number of possible distributions, and consider general series of the form

(∗)