Limiting distributions and estimation in generalized Polya random walk scheme

Asymptotic behavior of distributions generated by Polya random walks is investigated. Unbiased estimators of these distributions are constructed for closed first-arrival plans.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 20–36, 1986.     

Statistical estimation in the scheme of continuous Markov random walks

A constructive method of obtaining unbiased estimates of the unknown parameters and characteristics of a continuous finite Markow chain (CMC) ν(t), t≥0, with states 1, 2, ..., k and constant transition intensitiesλ _i,j<∞, i≠j, i, j=1, 2,..., k; $$\lambda _{i,i} = 0,\quad q_i = \sum\limits_{j = 1}^k {\lambda _{i,j} } ,\quad i = 1, 2,..., k,$$ is considered in the present paper for a wide class of stopping rules.     

Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment

Summary We consider a model of random walk on ℤ^ν, ν≥2, in a dynamical random environment described by a field ξ={ξ _t (x): (t,x)∈ℤ^ν+1}. The random walk transition probabilities are taken as P(X _{t +1}= y|X _t = x,ξ _t =η) =P ₀( y−x)+ c(y−x;η(x)). We assume that the variables {ξ _t (x):(t,x) ∈ℤ^ν+1} are i.i.d., that both P ₀(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P ₀, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X _t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X _t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L ^p estimates for a class of functionals of the field. Received: 4 January 1996/In revised form: 26 May 1997     

Local asymptotics of a Markov modulated random walk with heavy-tailed increments

In this paper, we obtain sufficient and necessary conditions for local asymptotics for the maximum of a Markov modulated random walk with long-tailed increments and negative drifts, where the local asymptotics means asymptotic behaviour of P(· ∈ (x, x + z]) for each z > 0, as x→∞. Our results extend and improve the existing ones in the literature.     

Reinforced random walk

Summary Leta _i,i1, be a sequence of nonnegative numbers. Difine a nearest neighbor random motion =X ₀,X ₁, ... on the integers as follows. Initially the weight of each interval (i, i+1), i an integer, equals 1. If at timen an interval (i, i+1) has been crossed exactlyk times by the motion, its weight is . Given (X ₀,X ₁, ...,X _n)=(i₀, i₁, ..., i_n), the probability thatX _n+1 isi _n–1 ori _n+1 is proportional to the weights at timen of the intervals (i _n–1,i _n) and (i _n,ii_n+1). We prove that either visits all integers infinitely often a.s. or visits a finite number of integers, eventually oscillating between two adjacent integers, a.s., and that X _n /n=0 a.s. For much more general reinforcement schemes we proveP ( visits all integers infinitely often)+P ( has finite range)=1.Supported by a National Science Foundation Grant     

Vertex-reinforced random walk

Summary This paper considers a class of non-Markovian discrete-time random processes on a finite state space {1,...,d}. The transition probabilities at each time are influenced by the number of times each state has been visited and by a fixed a priori likelihood matrix,R, which is real, symmetric and nonnegative. LetS _i (n) keep track of the number of visits to statei up to timen, and form the fractional occupation vector,V(n), where . It is shown thatV(n) converges to to a set of critical points for the quadratic formH with matrixR, and that under nondegeneracy conditions onR, there is a finite set of points such that with probability one,V(n)p for somep in the set. There may be more than onep in this set for whichP(V(n)p)>0. On the other handP(V(n)p)=0 wheneverp fails in a strong enough sense to be maximum forH.This research was supported by an NSF graduate fellowship and by an NSF postdoctoral fellowship     

A phase transition in the random transposition random walk

Our work is motivated by Bourque and Pevzner's (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on n elements. Consider this walk in continuous time starting at the identity and let D _t be the minimum number of transpositions needed to go back to the identity from the location at time t. D _t undergoes a phase transition: the distance D _{cn /2}∼u(c)n, where u is an explicit function satisfying u(c)=c/2 for c≤1 and u(c)<c/2 for c>1. In addition, we describe the fluctuations of D _{cn /2} about its mean in each of the three regimes (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Renyi random graph model.     

Simple random walk on the line in random environment

Summary We obtain strong limiting bounds for the maximal excursion and for the maximum reached by a random walk in a random environment. Our results derive from a simple proof of Pólya's theorem for the recurrence of the random walk on the line. As applications, we obtain bounds for the number of visits of the random walk at the origin.     

Interpolating between random walk and rotor walk

We introduce a family of stochastic processes on the integers, depending on a parameter and interpolating between the deterministic rotor walk () and the simple random walk (). This p‐rotor walk is not a Markov chain but it has a local Markov property: for each the sequence of successive exits from is a Markov chain. The main result of this paper identifies the scaling limit of the p‐rotor walk with two‐sided i.i.d. initial rotors. The limiting process takes the form , where is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation (1) for all . Here is a standard Brownian motion and are constants depending on the marginals of the initial rotors on and respectively. Chaumont and Doney have shown that Equation 1 has a pathwise unique solution , and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, and . This last result, together with the main result of this paper, implies that the p‐rotor walk is recurrent for any two‐sided i.i.d. initial rotors and any .     

Random walk in random groups

((no abstract)) . Submitted: January 2002, Final version: December 2002.     

A note on random walk in random scenery

We consider a random walk in random scenery {Xn=η(S₀)+?+η(Sn),n∈N}, where a centered walk {Sn,n∈N} is independent of the scenery {η(x),x∈Zd}, consisting of symmetric i.i.d. with tail distribution P(η(x)>t)∼exp(−cαtα), with 1?α<d/2. We study the probability, when averaged over both randomness, that {Xn>ny} for y>0, and n large. In this note, we show that the large deviation estimate is of order exp(−ca(ny)), with a=α/(α+1).     

Annealed deviations of random walk in random scenery

Let (Zn)_n∈N be a d-dimensional random walk in random scenery, i.e., with (Sk)_k∈N0 a random walk in Zd and (Y(z))_z∈Zd an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of for various choices of sequences n(bn) in [1,∞). Depending on n(bn) and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497-527] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004].     

Ballisticity conditions for random walk in random environment

Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions d ?? 2. In 2002, Sznitman introduced for each ${\gamma\in (0, 1)}$ the ballisticity conditions (T) _?? and (T??), the latter being defined as the fulfillment of (T) _?? for all ${\gamma\in (0, 1)}$ . He proved that (T??) implies ballisticity and that for each ${\gamma\in (0.5, 1)}$ , (T) _?? is equivalent to (T??). It is conjectured that this equivalence holds for all ${\gamma\in (0, 1)}$ . Here we prove that for ${\gamma\in (\gamma_d, 1)}$ , where ?? _d is a dimension dependent constant taking values in the interval (0.366, 0.388), (T) _?? is equivalent to (T??). This is achieved by a detour along the effective criterion, the fulfillment of which we establish by a combination of techniques developed by Sznitman giving a control on the occurrence of atypical quenched exit distributions through boxes.     

Adaptive stochastic numerical scheme in parallel random walk models for transport problems in shallow water

This paper deals with the simulation of transport of pollutants in shallow water using random walk models and develops several computation techniques to speed up the numerical integration of the stochastic differential equations (SDEs). This is achieved by using both random time stepping and parallel processing.We start by considering a basic stochastic Euler scheme for integration of the diffusion and drift terms of the SDEs, with a strong order 1 in the strong sense. The errors due to this scheme depend on the location of the pollutant; it is dominated by the diffusion term near boundaries, and by the deterministic drift further away from the boundaries. Using a pair of integration schemes, one of strong order 1.5 near the boundary and one of strong order 2.0 elsewhere, we can estimate the error and approximate an optimal step size for a given error tolerance. The resulting algorithm is developed such that it allows for complete flexibility of the step size, while guaranteeing the correct Brownian behaviour.Modelling pollutants by non-interacting particles enables the use of parallel processing in the simulation. We take advantage of this by implementing the algorithm using the MPI library. The inherent asynchronic nature of the particle simulation, in addition to the parallel processing, makes it difficult to get a coherent picture of the results at any given points. However, by inserting internal synchronisation points in the temporal discretisation, the code allows pollution snapshots and particle counts to be made at times specified by the user.