For a class of stationary Markov-dependent sequences $(A_n,B_n)\in\mathbb{R}^2,$ we consider the random linear recursion $S_n=A_n+B_nS_{n-1},$ $n\in\mathbb{Z},$ and show that the distribution tail of its stationary solution has a power law decay.Comment: Published at http://dx.doi.org/10.1214/105051606000000844 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We obtain non-Gaussian limit laws for one-dimensional random walk in a random environment in the case that the environment is a function of a stationary Markov process. This is an extension of the work of Kesten, M. Kozlov and Spitzer [13] for random walks in i.i.d. environments. The basic assumption is that the underlying Markov chain is irreducible and either with finite state space or with transition kernel dominated above and below by a probability measure.MSC2000: primary 60K37, 60F05; secondary 60J05, 60J80.

We use observed transmission line outage data to make a Markovian influence graph that describes the probabilities of transitions between generations of cascading line outages. Each generation of a cascade consists of a single line outage or multiple line outages. The new influence graph defines a Markov chain and generalizes previous influence graphs by including multiple line outages as Markov chain states. The generalized influence graph can reproduce the distribution of cascade size in the utility data. In particular, it can estimate the probabilities of small, medium and large cascades. The influence graph has the key advantage of allowing the effect of mitigations to be analyzed and readily tested, which is not available from the observed data. We exploit the asymptotic properties of the Markov chain to find the lines most involved in large cascades and show how upgrades to these critical lines can reduce the probability of large cascades.

We consider a multitype branching process with immigration in a random environment introduced by Key in [Ann. Probab. 15 (1987) 344-353]. It was shown by Key that, under the assumptions made in [Ann. Probab. 15 (1987) 344-353], the branching process is subcritical in the sense that it converges to a proper limit law. We complement this result by a strong law of large numbers and a central limit theorem for the partial sums of the process. In addition, we study the asymptotic behavior of oscillations of the branching process, that is, of the random segments between successive times when the extinction occurs and the process starts again with the next wave of the immigration.

We consider transient random walks on a strip in a random environment. The model was introduced by Bolthausen and Goldsheid [Comm. Math. Phys. 214 (2000) 429--447]. We derive a strong law of large numbers for the random walks in a general ergodic setup and obtain an annealed central limit theorem in the case of uniformly mixing environments. In addition, we prove that the law of the "environment viewed from the position of the walker" converges to a limiting distribution if the environment is an i.i.d. sequence.Comment: Published in at http://dx.doi.org/10.1214/08-AOP393 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We introduce random walks in a sparse random environment on Z and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. More specifically, a random walk in a sparse random environment can be characterized as a "locally strong" perturbation of a simple random walk by a random potential induced by "rare impurities," which are randomly distributed over the integer lattice. Interestingly, in the critical (recurrent) regime, our model generalizes Sinai's scaling of (log n) 2 for the location of the random walk after n steps to (log n) α , where α > 0 is a parameter determined by the distribution of the distance between two successive impurities. Similar scaling factors have appeared in the literature in different contexts and have been discussed in [29] and [31]. MSC2010: primary 60K37; secondary 60F05.

In this paper, we formulate and analyze a Markov process modeling the motion of DNA nanomechanical walking devices.We consider a molecular biped restricted to a well-defined one-dimensional track and study its asymptotic behavior.Our analysis allows for the biped legs to be of different molecular composition, and thus to contribute differently to the dynamics. Our main result is a functional central limit theorem for the biped with an explicit formula for the effective diffusivity coefficient in terms of the parameters of the model. A law of large numbers, a recurrence/transience characterization and large deviations estimates are also obtained.Our approach is applicable to a variety of other biological motors such as myosin and motor proteins on polymer filaments.

a b s t r a c tWe consider the equation R n = Q n + M n R n−1 , with random non-i.i.d. coefficients (Q n , M n ) n∈Z ∈ R 2 , and show that the distribution tails of the stationary solution to this equation are regularly varying at infinity.