Non-homogeneous Random Walks Research Articles

Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are more general than existing results in the literature, which consider only the case of sums of independent (typically, identically distributed) random variables. We do not assume the Markov property. Existing results that we generalize include a circle of ideas related to the Marcinkiewicz-Zygmund strong law of large numbers, as well as more recent work of Kesten and Maller. Our proofs are robust and use martingale methods. We demonstrate the benefit of the generality of our results by applications to some non-classical models, including random walks with heavy-tailed increments on two-dimensional strips, which include, for instance, certain generalized risk processes.

Rate of escape and central limit theorem for the supercritical Lamperti problem

The study of discrete-time stochastic processes on the half-line with mean drift at x given by μ 1 ( x ) → 0 as x → ∞ is known as Lamperti’s problem. We give sharp almost-sure bounds for processes of this type in the case where μ 1 ( x ) is of order x − β for some β ∈ ( 0 , 1 ) . The bounds are of order t 1 / ( 1 + β ) , so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of ( 2 + 2 β + ε ) -moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where x β μ 1 ( x ) has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β = 0 . We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.

Nonhomogeneous random walks systems on ℤ

We consider a random walks system on ℤ in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on ℤ, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to 1 but not fast enough.

Nonhomogeneous random walks systems on ℤ

We consider a random walks system on ℤ in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on ℤ, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to 1 but not fast enough.

Up-and-Down Designs for Selecting the Dose with Maximum Success Probability

Assume that the probability of success is unimodal as a function of dose, such as may be the case when too much of a drug is toxic and too little is ineffective. We characterize a class of up-and-down designs, that is, treatment allocation methodologies, for identifying the dose that maximizes the patients' success probability. These designs are constructed to use accruing information to limit the number of patients that are exposed to doses with high probabilities of failure. This treatment allocation procedure is motivated by Kiefer–Wolfowitz's stochastic approximation procedure. However, we take the response to be binary and the possible treatment space to be a lattice. The procedure is shown to allocate treatments to pairs of subjects in a way that causes the treatment distribution to center around the treatment with maximum success probability. The procedure defines a nonhomogeneous random walk, so well-known theory is used to explicitly characterize the treatment distribution. As an estimator of the best dose, the mode of the empirical treatment distribution is shown to converge faster than does the last dose allocated, which is used as an estimator of the optimal dose in stochastic approximation procedures.

Scale-free aggregation in sensor networks

Sensor networks are distributed data collection systems, frequently used for monitoring environments in which “nearby” data have a high degree of correlation. This induces opportunities for data aggregation, that are crucial given the severe energy constraints of the sensors. Thus, it is very desirable to take advantage of data correlations in order to avoid transmitting redundancy. In our model, we formalize a notion of correlation, that can vary according to a parameter k. Then we relate the expected collision time of “nearby” walks on the grid to the optimum cost of scale-free aggregation. We also propose a very simple randomized algorithm for routing information on a grid of sensors that satisfies the appropriate collision time condition. Thus, we prove that this simple scheme is a constant factor approximation (in expectation) to the optimum aggregation tree simultaneously for all correlation parameters k. The key contribution in our randomized analysis is to bound the average expected collision time of non-homogeneous random walks on the grid, i.e. the next hop probability depends on the current position.

New possibilities for the mathematical modeling of turbulent transport processes in plasma

A new mathematical model is proposed for the probability distributions of the characteristics of the processes observed in turbulent plasmas. The model is based on formal theoretical considerations related to probabilistic limit theorems for a nonhomogeneous random walk and has the form of a finite mixture of Gaussian distributions. The reliability of the model is confirmed by the results of a statistical analysis of the experimental data on density fluctuations in high-temperature plasmas of the L-2M, LHD, and TJ-II stellarators and the local fluctuating flux in the TAU-1 linear device and in the edge plasma of the L-2M stellarator with the use of the estimation-maximization algorithm. It is shown that low-frequency structural turbulence in a magnetized plasma is related to non-Brownian transport, which is determined by the characteristic temporal and spatial scales of the ensembles of stochastic plasma structures. Mechanisms that could be responsible for the random nature of time samples of the local turbulent flux in TAU-1 are indicated. A new physical concept of the intermittence of plasma turbulent pulsations is developed on the basis of the statistical separation of mixtures in terms of the model proposed. The intermittence of plasma pulsations is shown to be associated with the generation of plasma structures (solitons and vortices) and their nonlinear interaction, as well as with their damping and drift.

Central Limit Theorem for a Class of Nonhomogeneous Random Walks

A spatially nonhomogeneous random walk ηt on the grid ℤν=ℤm X ℤn is considered. Let ηt 0 be a random walk homogeneous in time and space, and let ηt be obtained from it by changing transition probabilities on the set A=Ā X ℤn, |Ā| m ≥ 2 or the drift is distinct from zero, then the central limit theorem holds for ηt.

Invariance principle for nonhomogeneous random walks on the grid ℤ1

A nonhomogeneous random walk on the grid ℤ1 with transition probabilities that differ from those of a certain homogeneous random walk only at a finite number of points is considered. Trajectories of such a walk are proved to converge to trajectories of a certain generalized diffusion process on the line. This result is a generalization of the well-known invariance principle for the sums of independent random variables and Brownian motion.

A Local Limit Theorem for Nonhomogeneous Random Walk on a Lattice

In this paper, we study the walk of a particle on the $\nu$-dimensional lattice ${\bf Z}^\nu ,\nu = 1,2,3$, of which the one-step transition probabilities ${\bf Pr} (y \to x)$ differ from those of the homogeneous symmetric walk only in a finite neighborhood of the point $x = 0$. For such a walk, the main term (having the order $O(1/t^{\nu/2} )$) of the asymptotics of the probability ${\bf Pr} (x_t = x|x_0 = y)$ as $t \to \infty $ is studied, $x,y \in {\bf Z}^\nu ,x_t $ being the position of the particle at time t. It turns out that, for $\nu = 2,3$, this main term of the asymptotics differs from the corresponding term of the asymptotics for the homogeneous walk (which has a usual Gaussian form) by a quantity of the order $O(t^{t - \nu/2} (|y| + 1)^{ - (\nu - 1)/2} )$. Thus the correction to the Gaussian term is comparable with it only in a finite neighborhood of the origin. In the case $\nu = 1$, this correction has the form \[ \frac{{{\text{const}}}}{{\sqrt t }}\left( {{\text{sign}}\,\exp \left\{ { - \frac{{{\text{const}}}} {t}(|x| + |y|)^2 } \right\} + o\left( {\frac{1}{{|x|}}} \right)} \right), \] i.e., remains of the same order as the Gaussian term on distances $|y| \sim \sqrt t $. The proof of these results is obtained by a detailed study of the structure of the resolvent $(\mathcal{T} - zE)^{ - 1} $ of the stochastic operator $\mathcal{T}$ of our model for z lying in a small neighborhood of the point $z = 1$ (the right boundary of the continuous spectrum of $\mathcal{T}$).

On the frequency count for a random walk with absorbing boundaries: a carcinogenesis example. I

A nonhomogeneous random walk on non-negative integers with transition probabilities p0i= delta 0i, pNi= delta Ni, Pi,i+1= lambda i, pi,i-1= mu i, and pi,i= rho i, lambda i+ mu i+ rho i=1, is studied. In particular, when the transition probabilities are independent of position, a general expression for the joint probability generating function (JPGF) of the frequency count of the stages 1,2,...,N-1 is derived. The appropriate marginal forms of this JPGF yield the PGF of the frequency count at any pair of stages, and at any particular single stage. Some moment formulae associated with the frequency count are derived. A random walk conditional on absorption at a specified boundary is also considered. The random walk model proposed is eminently suitable for the example of carcinogenesis.

A random walk for the solution sought: Remarks on the difference scheme approach to nonlinear semigroups and evolution operators

In [2], Crandall and Evans show existence of mild solution to an abstract Cauchy Problem: u′(t)+Au(t)∋f(t), 0≤t≤T, u(0)=x0, where A is an accretive operator in a general Banach space X and f e L1(0,T;X). Their method involves proving convergence in the L∞-norm of a sequence of step function approximations αn(σ, τ) to the solution of a first order partial differential equation. We consider a more general Cauchy Problem and show a.e. existence of mild solution by proving convergence of the step functions αn(σ, τ) in the L1-norm. Fundamental to the proof is a nonhomogeneous random walk in the plane.

Laplace operator and random walk on one-dimensional nonhomogeneous lattice

A classical result of probability theory states that under suitable space and time renormalization, a random walk converges to Brownian motion. We prove an analogous result in the case of nonhomogeneous random walk on onedimensional lattice. Under suitable conditions on the nonhomogeneous medium, we prove convergence to Brownian motion and explicitly compute the diffusion coefficient. The proofs are based on the study of the spectrum of random matrices of increasing dimension.