Law of Large Numbers for Random Walk with Unbounded Jumps and Birth and Death Process with Bounded Jumps in Random Environment

Hua-Ming Wang ()
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Hua-Ming Wang: Anhui Normal University

Journal of Theoretical Probability, 2018, vol. 31, issue 2, 619-642

Abstract: Abstract We study a random walk with unbounded jumps in random environment. The environment is stationary and ergodic, uniformly elliptic and decays polynomially with speed $$Dj^{-(3+\varepsilon _0)}$$ D j - ( 3 + ε 0 ) for some $$D>0$$ D > 0 and small $$\varepsilon _0>0.$$ ε 0 > 0 . We prove a law of large numbers under the condition that the annealed mean of the hitting time of the lattice of the positive half line is finite. As the second part, we consider a birth and death process with bounded jumps in stationary and ergodic environment whose skeleton process is a random walk with unbounded jumps in random environment. Under a uniform ellipticity condition, we prove a law of large numbers and give the explicit formula of its velocity.

Keywords: Random walk; Random environment; Unbounded jumps; Birth and death process; Skeleton process; 60K37; 60J80 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-016-0731-3

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