Non–linear renewal theory for lattice random walks

1982 ◽  
Vol 1 (3) ◽  
pp. 193-205 ◽  
Author(s):  
Steve Lalley
Keyword(s):  
Random Walks ◽  
Renewal Theory ◽  
Non Linear ◽  
Lattice Random Walks

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.

Limit theorems for first-passage times in linear and non-linear renewal theory

1984 ◽  
Vol 16 (04) ◽  
pp. 766-803 ◽  
Author(s):  
S. P. Lalley
Keyword(s):  
Limit Theorems ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Renewal Theory ◽  
Local Limit ◽  
Linear Boundary ◽  
First Passage ◽  
Non Linear ◽  
First Time ◽  
Passage Times

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

scholarly journals Multiphase partitions of lattice random walks

2019 ◽  
Vol 126 (5) ◽  
pp. 50002
Author(s):  
Massimiliano Giona ◽  
Davide Cocco
Keyword(s):  
Random Walks ◽  
Lattice Random Walks

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.

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