scholarly journals Random Walks with Stationary Increments and Renewal Theory.

1980 ◽  
Vol 143 (3) ◽  
pp. 373
Author(s):  
J. F. C. Kingman ◽  
H. C. P. Berbee
Keyword(s):  
Random Walks ◽  
Renewal Theory ◽  
Stationary Increments

scholarly journals Frog models on trees through renewal theory

2018 ◽  
Vol 55 (3) ◽  
pp. 887-899 ◽  
Author(s):  
Sandro Gallo ◽  
Pablo M. Rodriguez
Keyword(s):  
Random Walks ◽  
Critical Parameter ◽  
Renewal Theory ◽  
Percolation Model ◽  
Critical Parameters

Abstract We study a class of growing systems of random walks on regular trees, known as frog models with geometric lifetime in the literature. With the help of results from renewal theory, we derive new bounds for their critical parameters. Our approach also improves the existing bounds for the critical parameter of a percolation model on trees known as cone percolation.

Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks

2001 ◽  
Vol 33 (3) ◽  
pp. 652-673 ◽  
Author(s):  
Cheng-Der Fuh ◽  
Tze Leung Lai
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Renewal Theory ◽  
Passage Time ◽  
Boundary Crossing ◽  
High Threshold ◽  
Renewal Theorem ◽  
First Passage ◽  
Markov Random ◽  
General State Space

We prove a d-dimensional renewal theorem, with an estimate on the rate of convergence, for Markov random walks. This result is applied to a variety of boundary crossing problems for a Markov random walk (Xn,Sn), n ≥0, in which Xn takes values in a general state space and Sn takes values in ℝd. In particular, for the case d = 1, we use this result to derive an asymptotic formula for the variance of the first passage time when Sn exceeds a high threshold b, generalizing Smith's classical formula in the case of i.i.d. positive increments for Sn. For d > 1, we apply this result to derive an asymptotic expansion of the distribution of (XT,ST), where T = inf { n : Sn,1 > b } and Sn,1 denotes the first component of Sn.

Special Topic: A Comprehensive Renewal Theory for General Random Walks

2021 ◽  
pp. 305-328
Author(s):  
Rabi Bhattacharya ◽  
Edward C. Waymire
Keyword(s):  
Random Walks ◽  
Renewal Theory ◽  
Special Topic
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