On the non-recurrent random walk in a random environment

2018 ◽  
Vol 28 (3) ◽  
pp. 139-156 ◽  
Author(s):  
Valeriy I. Afanasyev
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Environment ◽  
Recurrent Random Walk ◽  
High Level ◽  
Transient Random Walk

Abstract For weakly transient random walk in a random environment that tend at −∞ the limit theorem for the time of hitting a high level is proved.

scholarly journals Recurrence and Transience of Critical Branching Processes in Random Environment with Immigration and an Application to Excited Random Walks

2014 ◽  
Vol 46 (03) ◽  
pp. 687-703 ◽  
Author(s):  
Elisabeth Bauernschubert
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Environment ◽  
Branching Processes ◽  
Random Environments ◽  
Recurrent Random Walk ◽  
The Right

We establish recurrence and transience criteria for critical branching processes in random environments with immigration. These results are then applied to the recurrence and transience of a recurrent random walk in a random environment on ℤ disturbed by cookies inducing a drift to the right of strength 1.

scholarly journals Empirical processes for recurrent and transient random walks in random scenery

2020 ◽  
Vol 24 ◽  
pp. 127-137
Author(s):  
Nadine Guillotin-Plantard ◽  
Françoise Pène ◽  
Martin Wendler
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Random Walks ◽  
Stable Distribution ◽  
Random Variables ◽  
Empirical Processes ◽  
Independent Random Variables ◽  
Random Scenery ◽  
Recurrent Random Walk ◽  
Transient Random Walk

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s,t))s,t∈[0,1] with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(\mathds{1}_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where (ξx, x ∈ ℤd) is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n ∈ ℕ is a random walk evolving in ℤd, independent of the ξ’s. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787–2799], the case where (Sn)n ∈ ℕ is a recurrent random walk in ℤ such that (n−1/αSn)n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (Sn)n ∈ ℕ is either: (a) a transient random walk in ℤd, (b) a recurrent random walk in ℤd such that (n−1/dSn)n≥1 converges in distribution to a stable distribution of index d ∈{1, 2}.

On the local time of a recurrent random walk on ℤ²

2021 ◽  
Vol 105 (0) ◽  
pp. 69-78
Author(s):  
V. Bohun ◽  
A. Marynych
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Process ◽  
Local Time ◽  
Functional Limit Theorem ◽  
Functional Limit ◽  
Content Type ◽  
Finite Set ◽  
Number Of Visits ◽  
Recurrent Random Walk

We prove a functional limit theorem for the number of visits by a planar random walk on Z 2 \mathbb {Z}^2 with zero mean and finite second moment to the points of a fixed finite set P ⊂ Z 2 P\subset \mathbb {Z}^2 . The proof is based on the analysis of an accompanying random process with immigration at renewal epochs in case when the inter-arrival distribution has a slowly varying tail.

Sign in / Sign up

Export Citation Format

Share Document