Limit theorems for random walks

We consider a random walk $S_{\tau }$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau }$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{\alpha }$. We also prove asymptotic formula for the transition function of $S_{\tau }$ similar to the Pólya’s asymptotic formula for $B^{\alpha }$.     

Front velocity and directed polymers in random medium

We consider a stochastic model of N$N$ evolving particles studied by Brunet and Derrida. This model can be seen as a directed polymer in random medium with N$N$ sites in the transverse direction. Cook and Derrida use heuristic arguments to obtain a formula for the ground state energy of the polymer. We formalize their argument and show that there is an additional term in the formula in the critical case. We also consider a generalization of the model, and show that in the noncritical case the behavior is basically the same, whereas in the critical case a new correction appears.     

Nonquasilinear evolution of particle velocity in incoherent waves with random amplitudes

The one-dimensional motion of N particles in the field of many incoherent waves is revisited numerically. When the wave complex amplitudes are independent, with a Gaussian distribution, the quasilinear approximation is found to always overestimate transport and to become accurate in the limit of infinite resonance overlap.     

Limit theorems for the convex hull of random points in higher dimensions

We give a central limit theorem for the number of vertices of the convex hull of independent and identically distributed random vectors, being sampled from a certain class of spherically symmetric distributions in that includes the normal family. Furthermore, we prove that, among these distributions, the variance of exhibits the same order of magnitude as the expectation as The main tools are Poisson approximation of the point process of vertices of the convex hull and (sub/super)-martingales.

     

Limit theorem for a random walk in a reversible random environment

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 4, pp. 627–644, October–December, 1989.     

Zero White Noise Limit through Dirichlet forms,with application to diffusions in a random medium

Summary We study the Zero White Noise Limit for diffusions in a continuous multidimensional medium: given a continuous function on ⁿ ,W, we consider diffusions whose drift term is the gradient ofW and whose diffusion coefficient is constant equal to . We describe the asymptotics of the exit time from a domain and of the law of the process when tends to zero. By applying these results to a random self-similar mediumW we prove limit theorems for a diffusion in a random medium. Our theorems agree with results usually proved through the large deviation principle, although, in our setup, this last tool is not available. We extend to the multidimensional case properties of diffusions in a random medium already known in one dimension.     

Limit theorems for random walks with dynamical random transitions

Let be an ergodic and conservative non-singular transformation of (, ,m) (thedynamic environment), let _w be a random probability on a locally compact second countable groupG, and define

Limit theorems for random processes with random time substitution

In this paper we prove a theorem on sufficient conditions for the convergence in the Skorokhod space D[0, 1] of a sequence of random processes with random time substitution. We obtain almost sure versions of this theorem.     

Stability for a random evolution equation with Gaussian perturbation

In this paper we consider a random evolution equation which is perturbed by Gaussian type noise, and show how to construct its coupled solutions. On the basis of the coupling results, we discuss the asymptotic flatness and the stability in total variation norm for the solutions of the equation. In addition, we also prove the Feller continuity, and the existence and uniqueness of invariant measures of the solutions.     

Local Limit Theorems for random walks in a 1D random environment

It is proved that the setwise limit of a bounded sequence of signed topological measures is a signed topological measure; here the signed measures and proper signed topological measures which are the components of the decomposition of the members of the sequence setwise converge to the corresponding components of the decomposition of the limit signed topological measure. The results thus obtained give a negative answer to the question concerning the possibility to represent a regular Borel measure (nonzero) as a setwise limit of a sequence of proper topological measures.     

Asymptotic direction for random walks in random environments

In this paper we study the existence of an asymptotic direction for random walks in random i.i.d. environments (RWRE). We prove that if the set of directions where the walk is transient contains a non-empty open set, the walk admits an asymptotic direction. The main tool to obtain this result is the construction of a renewal structure with cones. We also prove that RWRE admits at most two opposite asymptotic directions.     

Limit theorems for a supercritical branching process with immigration in a random environment

Let (Z_n) be a supercritical branching process with immigration in a random environment. Firstly, we prove that under a simple log moment condition on the offspring and immigration distributions, the naturally normalized population size W_n converges almost surely to a finite random variable W. Secondly, we show criterions for the non-degeneracy and for the existence of moments of the limit random variable W. Finally, we establish a central limit theorem, a large deviation principle and a moderate deviation principle about log Z_n.     

Well-posedness for a transport equation with nonlocal velocity

We study a one-dimensional transport equation with nonlocal velocity which was recently considered in the work of Córdoba, Córdoba and Fontelos [A. Córdoba, D. Córdoba, M.A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (3) (2005) 1377-1389]. We show that in the subcritical and critical cases the problem is globally well-posed with arbitrary initial data in Hmax{3/2−γ,0}. While in the supercritical case, the problem is locally well-posed with initial data in H3/2−γ, and is globally well-posed under a smallness assumption. Some polynomial-in-time decay estimates are also discussed. These results improve some previous results in [A. Córdoba, D. Córdoba, M.A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (3) (2005) 1377-1389].     

Limit theorems for random permanents with exchangeable structure

Permanents of random matrices extend the concept of U-statistics with product kernels. In this paper, we study limiting behavior of permanents of random matrices with independent columns of exchangeable components. Our main results provide a general framework which unifies already existing asymptotic theory for projection matrices as well as matrices of all-iid entries. The method of the proofs is based on a Hoeffding-type orthogonal decomposition of a random permanent function. The decomposition allows us to relate asymptotic behavior of permanents to that of elementary symmetric polynomials based on triangular arrays of rowwise independent rv's.     

On granting credit in a random environment

The paper extends the Bierman-Hausman credit granting model by incorporating economic factors, which may vary randomly over time. Sufficient conditions are found for the optimality of simple credit policies. The monotonicity results are based on nonstandard orderings induced by the probability of collection. Moreover, monotonicity of the credit policy with respect to the prior information is studied.     

Superprocesses with spatial interactions in a random medium

We construct a class of interactive measure-valued diffusions driven by a historical super-Brownian motion and an independent white noise by solving a certain stochastic equation. In doing so, we show that the approach of Perkins (2002) [3] can be used to study the problem examined by Dawson et al. (2001) [1]. This unifies and extends both Dawson et al. (2001) [1] and Perkins (2002) [3] and establishes a new class of measure-valued diffusions. The existence and pathwise uniqueness of the solutions are proved, and the solutions are shown to satisfy the natural martingale problem.     

Limit theory for random walks in degenerate time-dependent random environments

We study continuous-time (variable speed) random walks in random environments on ${\mathbb{Z}}^d$, $d\ge 2$, where, at time t, the walk at x jumps across edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume stationary and ergodic with respect to space–time shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The proofs invoke Moser iteration to prove sublinearity of the corrector in pointwise sense; a key additional input is a conversion of certain weighted energy norms to ordinary ones. Our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the Helffer–Sjöstrand representation of gradient models with certain non-strictly convex potentials.