195 research outputs found

    Branching structure for the transient random walk on a strip in a random environment

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    An intrinsic branching structure within the transient random walk on a strip in a random environment is revealed. As applications, which enables us to express the hitting time explicitly, and specifies the density of the absolutely continuous invariant measure for the "environments viewed from the particle".Comment: 16 page

    A note on the passage time of finite state Markov chains

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    Consider a Markov chain with finite state {0,1,...,d}\{0, 1, ..., d\}. We give the generation functions (or Laplace transforms) of absorbing (passage) time in the following two situations : (1) the absorbing time of state dd when the chain starts from any state ii and absorbing at state dd; (2) the passage time of any state ii when the chain starts from the stationary distribution supposed the chain is time reversible and ergodic. Example shows that it is more convenient compared with the existing methods, especially we can calculate the expectation of the absorbing time directly

    Scaling limit of the local time of the (1,L)βˆ’(1,L)-random walk

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    It is well known (Donsker's Invariance Principle) that the random walk converges to Brownian motion by scaling. In this paper, we will prove that the scaled local time of the (1,L)βˆ’(1,L)-random walk converges to that of the Brownian motion. The results was proved by Rogers (1984) in the case L=1L=1. Our proof is based on the intrinsic multiple branching structure within the (1,L)βˆ’(1,L)-random walk revealed by Hong and Wang (2013)

    Branching structure for an (L-1) random walk in random environment and its applications

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    By decomposing the random walk path, we construct a multitype branching process with immigration in random environment for corresponding random walk with bounded jumps in random environment. Then we give two applications of the branching structure. Firstly, we specify the explicit invariant density by a method different with the one used in Br\'emont [3] and reprove the law of large numbers of the random walk by a method known as the environment viewed from particles". Secondly, the branching structure enables us to prove a stable limit law, generalizing the result of Kesten-Kozlov-Spitzer [11] for the nearest random walk in random environment. As a byproduct, we also prove that the total population of a multitype branching process in random environment with immigration before the first regeneration belongs to the domain of attraction of some \kappa -stable law.Comment: 31 page

    Branching structure for the transient (1;R)-random walk in random environment and its applications

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    An intrinsic multitype branching structure within the transient (1;R)-RWRE is revealed. The branching structure enables us to specify the density of the absolutely continuous invariant measure for the environments seen from the particle and reprove the LLN with an drift explicitly in terms of the environment, comparing with the results in Br\'emont (2002).Comment: 25 page

    Tail asymptotic of the stationary distribution for the state dependent (1,R)-reflecting random walk: near critical

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    In this paper, we consider the (1,R)(1,R) state-dependent reflecting random walk (RW) on the half line, allowing the size of jumps to the right at maximal RR and to the left only 1. We provide an explicit criterion for positive recurrence and the explicit expression of the stationary distribution based on the intrinsic branching structure within the walk. As an application, we obtain the tail asymptotic of the stationary distribution in the "near critical" situation

    Limit theorems for the minimal position of a branching random walk in random environment

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    We consider a branching system of random walk in random environment (in location) in N\mathbb{N}. We will give the exact limit value of Mnn\frac{M_{n}}{n}, where MnM_{n} denotes the minimal position of branching random walk at time nn. A key step in the proof is to transfer our branching random walks in random environment (in location) to branching random walks in random environment (in time), by use of Bramson's "branching processes within a branching process"

    Scaling limit of the local time of the Sinai's random walk

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    We prove that the local times of a sequence of Sinai's random walks convergence to those of Brox's diffusion by proper scaling, which is accord with the result of Seignourel (2000). Our proof is based on the convergence of the branching processes in random environment by Kurtz (1979)

    Limit theorems for supercritical MBPRE with linear fractional offspring distributions

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    We investigate the limit behavior of supercritical multitype branching processes in random environments with linear fractional offspring distributions and show that there exists a phase transition in the behavior of local probabilites of the process affected by strongly and intermediately supercritical regimes. Some conditional limit theorems can also be obtained from the representation of generating functions.Comment: 25 page

    Light-tailed behavior of stationary distribution for state-dependent random walks on a strip

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    In this paper, we consider the state-dependent reflecting random walk on a half-strip. We provide explicit criteria for (positive) recurrence, and an explicit expression for the stationary distribution. As a consequence, the light-tailed behavior of the stationary distribution is proved under appropriate conditions. The key idea of the method employed here is the decomposition of the trajectory of the random walk and the main tool is the intrinsic branching structure buried in the random walk on a strip, which is different from the matrix-analytic method
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